Spectrally resolved optical study of transient spin
dynamics in semiconductors
byZhigang Chen
B.S., Nanjing University, 1998
M.S., University of Colorado, 2005

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulﬁllmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2006

This thesis entitled:
Spectrally resolved optical study of transient spin dynamics in semiconductors
written by Zhigang Chen
has been approved for the Department of Physics

Prof. Steven T. Cundiﬀ

Prof. Daniel S. DessauDate

The ﬁnal copy of this thesis has been examined by the signatories, and we ﬁndthat both the content and the form meet acceptable presentation standards ofscholarly work in the above mentioned discipline.

iii
Chen, Zhigang (Ph.D., Physics)
Spectrally resolved optical study of transient spin dynamics in semiconductors
Thesis directed by Prof. Steven T. Cundiﬀ

Using the spin of the electron to carry information, instead of or in addition to its charge, could provide advances in the capabilities of microelectronics.
Successful implementation of spin-based electronics requires preservation of theelectron spin coherence. Long spin coherence times have been observed in lightly
n-doped semiconductors, with a maximum at a “magic” electron density. We systematically study the spin dynamics of the electron in a GaAs quantum well, where
the electron density in the well can be varied through optical excitation. We showthat spin coherence is lost due to the interplay between localization by disorderand dynamical scattering. The disorder potential is characterized by measuring
the electron Landé g factor dependence on density. Our results show that thelongest spin coherence is obtained for weakly localized spins, which may dictate
a compromise in the design of devices between increasing the spin coherence timeand improving transport properties.
We also explore the intimate connection between electron spin and opticalexcitation that initiates and controls the spin states. We study the interplay of spin
dynamics between excitons, negatively charged excitons and the two-dimensional
electron gas in a lightly n-doped semiconductor quantum well. The spin of theelectron gas can be polarized through interband transitions, and the electron spin
can persist long after the recombination of optically excited carriers. We ﬁndthat the excitation of spin polarization of the resident electrons depends on therecombination times and spin relaxation times of the optically excited carriers andthe energy chosen for the light pulses.

Dedication

To my wife, Yingqi.

v

Acknowledgements

First, I would like to thank my advisor, Professor Steve Cundiﬀ. I am
grateful for his constant support and encouragement throughout my graduate
career. His enthusiasm for research has been a great inspiration and his brilliant
scientiﬁc insight has always helped me and kept me motivated. I especially thankhim for giving me opportunities to try many exciting projects, and providing good
advices whenever needed. I would also like to thank Professor Dan Dessau for his
advices and encouragement, and other committee members, Professor Charles
Rogers, Professor Victor Gurarie and Professor Rafael Piestun, for their supportand helpful comments.
Finally, my thanks to my colleagues, family and friends for their support.
I would like to thank my fellow Cundiﬀ group members, past and present, fortheir friendship and aid. Sam Cater, who contributed so much to my graduate
research, gave me countless ideas and advices. My wife, Yingqi, sacriﬁced so muchto give me support throughout my career. Many thanks my friends, Wenjian Cai,
Wengang Zheng and Ying-ju Wang for their great friendship.

vi

Contents

Chapter
1 Introduction1

1.1

A perspective on spintronics . . . . . . . . . . . . . . . . . . . . .

1

1.2

General background . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.2.1

Spin coherence . . . . . . . . . . . . . . . . . . . . . . . .

3

1.2.2

Spin relaxation mechanisms . . . . . . . . . . . . . . . . .

6

1.2.3

Electron Landé g factor . . . . . . . . . . . . . . . . . . .

8Outline of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

1.3

2 Experimental Techniques

12

2.1

Pump-probe spectroscopy . . . . . . . . . . . . . . . . . . . . . .

13

2.2

Generation of spin polarized carriers . . . . . . . . . . . . . . . .

14

2.3

Transient Faraday and Kerr rotation . . . . . . . . . . . . . . . .

18

2.3.1

Faraday eﬀect . . . . . . . . . . . . . . . . . . . . . . . . .

18

2.3.2

Experimental details . . . . . . . . . . . . . . . . . . . . .

19

2.3.3

Two-color measurements . . . . . . . . . . . . . . . . . . .

23

Preliminary measurements on bulk GaAs . . . . . . . . . . . . . .

24

2.4.1

Undoped GaAs . . . . . . . . . . . . . . . . . . . . . . . .

25

2.4.2

n-doped GaAs . . . . . . . . . . . . . . . . . . . . . . . . .

26

2.4

vii
3 Spectral study of transient Faraday and Kerr eﬀects

29

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

3.2

Sample characteristics and experiments . . . . . . . . . . . . . . .

31

3.3

Linear spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

3.3.1

Reﬂection and transmission . . . . . . . . . . . . . . . . .

33

3.3.2

Characteristic matrix method . . . . . . . . . . . . . . . .

34

3.3.3

Linear spectra: calculation . . . . . . . . . . . . . . . . . .

37

Faraday and Kerr spectra . . . . . . . . . . . . . . . . . . . . . .

38

3.4

4 Interaction induced spin polarization in a 2DEG
4.1

44

Exciton and trion spin dynamics . . . . . . . . . . . . . . . . . . .

46

4.1.1

Exciton spin dynamics . . . . . . . . . . . . . . . . . . . .

46

4.1.2

Trion spin dynamics . . . . . . . . . . . . . . . . . . . . .

47

4.2

Sample characteristics and experiments . . . . . . . . . . . . . . .

48

4.3

Spin dynamics of excitons, trions and 2DEG: experiments . . . . .

50

4.4

Spin dynamics of excitons, trions and 2DEG: analysis . . . . . . .

58

4.4.1

Trion spin dynamics . . . . . . . . . . . . . . . . . . . . .

58

4.4.2

Exciton spin dynamics . . . . . . . . . . . . . . . . . . . .

62

4.4.3

Spin coherence loss via exciton spin relaxation . . . . . . .

66

4.5

Spin dynamics in highly doped sample . . . . . . . . . . . . . . .

67

4.6

Electron spin coherence vs. 2DEG density . . . . . . . . . . . . .

70

5 Eﬀects of disorder on electron spin dynamics in semiconductor quantum
wells

74

5.1

Electron Landé g factor. . . . . . . . . . . . . . . . . . . . . . .

75

5.2

Sample characteristics and experiments . . . . . . . . . . . . . . .

77

5.2.1

Mixed type I/type II quantum wells . . . . . . . . . . . . .

77

5.2.2

Experimental setup . . . . . . . . . . . . . . . . . . . . . .

79

viii

5.3

5.2.3

Photoluminescence . . . . . . . . . . . . . . . . . . . . . .

80

5.2.4

Photoluminescence excitation and absorption spectra . . .

83

5.2.5

Electron density estimation . . . . . . . . . . . . . . . . .

86

Electron spin dynamics . . . . . . . . . . . . . . . . . . . . . . . .

88

5.3.1

Data analysis . . . . . . . . . . . . . . . . . . . . . . . . .

88

5.3.2

Spectral study . . . . . . . . . . . . . . . . . . . . . . . . .

89

5.3.3

Electron localization . . . . . . . . . . . . . . . . . . . . .

91

5.3.4

Electron g factor nonlinearity . . . . . . . . . . . . . . . .

93

5.3.5

Spin dephasing time vs. electron density . . . . . . . . . .

94

5.3.6

Electron g factor vs. temperature and excitation intensity

96

6 Conclusion

101

Bibliography

103

Appendix
A Birefringence of cryostat windows

110

A.1 Birefringence detected by Kerr rotation measurements . . . . . . . 110
A.2 Calibration of birefringence . . . . . . . . . . . . . . . . . . . . . 112
A.3 Birefringence compensation . . . . . . . . . . . . . . . . . . . . . 116

ix

FiguresFigure1.1

Scheme of a spin ﬁeld-eﬀect transistor . . . . . . . . . . . . . . . .

2

1.2

Electron spin Larmor precession . . . . . . . . . . . . . . . . . . .

5

2.1

Pump-probe experiment . . . . . . . . . . . . . . . . . . . . . . .

13

2.2

Semiconductor band structure and selection rules . . . . . . . . .

15

2.3

Transient Faraday rotation setup . . . . . . . . . . . . . . . . . .

19

2.4

Two-color transient Kerr rotation setup . . . . . . . . . . . . . . .

22

2.5

Transient Kerr rotation data under magnetic ﬁeld . . . . . . . . .

22

2.6

Temperature dependence of T2∗ and g ∗ in undoped bulk GaAs . .

25

2.7

Doping density dependence of electron g factor . . . . . . . . . . .

27

2.8

Optical excitation density dependence of T2∗ . . . . . . . . . . . .

27

3.1

Schematic of photo-induced Faraday eﬀect near a resonance . . .

32

3.2

Faraday and Kerr spectra resulting from an energy shift. . . . .

32

3.3

Sample layer structure and absorption spectrum . . . . . . . . . .

34

3.4

Reﬂectivity spectra across the sample . . . . . . . . . . . . . . . .

35

3.5

Measured and calculated reﬂectivity spectra . . . . . . . . . . . .

39

3.6

Measured and calculated Faraday and Kerr spectra . . . . . . . .

41

4.1

PL and PLE spectra for the lightly doped CdTe quantum well . .

49

x
4.2

Transient Kerr rotation signals under resonant exciton and trion
excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

4.3

Spectral and intensity study of transient Kerr rotation . . . . . .

54

4.4

Temperature dependence of electron spin dephasing time . . . . .

55

4.5

Hole spin ﬂip time on temperature, excitation energy and intensity

57

4.6

Spin interplay between trions and 2DEG . . . . . . . . . . . . . .

60

4.7

Spin interplay between excitons, trions and 2DEG . . . . . . . . .

65

4.8

PL and PLE spectra for the highly doped CdTe quantum well . .

68

4.9

Transient Kerr rotation data for the highly doped CdTe sample .

69

4.10 PL and PLE spectra for samples with diﬀerent doping density . .

70

4.11 Magnetic ﬁeld dependence of spin dephasing rate . . . . . . . . .

71

5.1

Band diagram for the mixed type I/type II quantum well . . . . .

78

5.2

PL spectra for various HeNe intensities . . . . . . . . . . . . . . .

81

5.3

PLE spectra for various HeNe intensities . . . . . . . . . . . . . .

83

5.4

Absorption spectra for various HeNe intensities . . . . . . . . . .

85

5.5

Time resolved PL from pulsed HeNe excitation . . . . . . . . . . .

87

5.6

Typical Kerr rotation transients and nonlinear ﬁts . . . . . . . . .

89

5.7

Amplitude spectra of Kerr rotation . . . . . . . . . . . . . . . . .

90

5.8

Electron Localization in quantum wells . . . . . . . . . . . . . . .

92

5.9

Electron density dependence of g factor . . . . . . . . . . . . . . .

93

5.10 Density and ﬁeld dependence of electron spin dephasing . . . . . .

95

5.11 Temperature dependence of g factor . . . . . . . . . . . . . . . . .

97

5.12 g factor dependence on intensity and degree of spin polarization .

98

A.1 Kerr rotation amplitude vs. λ/4 waveplate angle . . . . . . . . . . 111
A.2 Setup for birefringence calibration . . . . . . . . . . . . . . . . . . 112
A.3 Power vs. analyzer angle α . . . . . . . . . . . . . . . . . . . . . . 114

xi
A.4 Birefringence for the front and back windows . . . . . . . . . . . . 115

Chapter 1
Introduction

1.1

A perspective on spintronics
Spintronics, or spin-based electronics, involves the study of active control

and manipulation of spin degrees of freedom in solid-state systems [1, 2, 3]. Using
the spin of the electron to carry information, instead of or in addition to its
charge, could provide advances in the capabilities of microelectronics [4]. In a
broad sense, there are already commercial spintronic devices. One example is theread heads for magnetic hard discs that use the giant magnetoresistance eﬀect.
Those devices normally consist of magnetic and non-magnetic metals rather thansemiconductors, and are more appropriately described as magnetoelectronics [4, 5].
Traditional electronics, on the other hand, is dominated by semiconductors for
obvious reasons. Combining the advantages of semiconductors with the conceptof spins and thereby introducing spin into semiconductor devices is therefore alogical step and opens a new pathway for spintronics.
The current goal of spintronics is to understand the interaction between
the particle spin and its solid-state (especially semiconductor) environments andto make useful devices using the acquired knowledge. Fundamental studies in
spintronics include the generation of spin polarization, spin transport, spin dynamics and relaxation, and spin detection. We illustrate the above aspects byshowing a prototypical spintronic device, a spin ﬁeld-eﬀect transistor [6], which is

2Figure 1.1: Scheme of a spin ﬁeld-eﬀect transistor. The current is controlled by
the gate voltage, being large if the electron spin at the drain is parallel to its
initial direction (top), and small if antiparallel (bottom).

depicted in Figure 1.1. The spin ﬁeld-eﬀect transistor consists of a source, a drain,a narrow channel, and a gate for controlling the current. The source and drain
are ferromagnets acting as the injector and detector of electron spins. Specifically, the source injects electrons with spins parallel to the electron transport
direction, and the drain detects electron spins near the drain after the electronstransport through the channel. An eﬀective magnetic ﬁeld (in the direction of Ωin Figure 1.1), arising from the spin-orbit coupling due to the conﬁnement of thechannel and the applied electrostatic potential of the gate [7], will cause electron
spins to precess as the electrons travel through the channel. The magnitude ofthis eﬀective magnetic ﬁeld is tunable by the gate voltage. By modifying the gate
voltage, we can have either parallel or antiparallel electron spins at the drain, andtherefore eﬀectively control the current. Key issues of the device include how longthe electron can preserve its spin coherence as it propagate through the channel
and how fast the electron spin will precess under the magnetic ﬁeld.

3
A quantum-coherent approach to electronics oﬀers the advantage of ultrafast device operation resulting from the interference between two coherently occupied quantum states. This requires the quantum coherence being able to evolve
coherently long enough to be manipulated. In semiconductors, the optical coherence (the electric dipoles between the electron-hole pairs after optical pulse
excitation) time is usually tens of picoseconds or less. The electron spin states,on the other hand, show a more robust resistance to the environmental sourcesof decoherence. Long spin coherence times exceeding hundreds of nanoseconds
has been observed in lightly n-doped semiconductors [8, 9]. Spin coherence isalso preserved for transport over distances exceeding 100 micrometers [10]. Spintherefore oﬀers opportunities to store and manipulate phase coherence over lengthand time scales much larger than is typically possible in charge-based devices, andshows promise for constructing quantum bits in quantum information processing[11]. Time-resolved optical experiments have been proven to be a good candidatefor generating, manipulating and detecting the spin coherence in semiconductor
systems [12, 13]. Optical pulses are used to create a superposition of the basis
spin states deﬁned by an applied magnetic ﬁeld, and follow the phase, amplitude
and location of the resulting electronic spin precession (coherence). The work presented in this thesis will focus on the coherent optical generation and manipulationof electron spins.
1.2

General background

1.2.1

Spin coherence

The term “coherence” refers to a well-deﬁned phase, or phase memory ina system. In a quantum system, the evolution of the wavefunction follows its
Hamiltonian in a predictable way, which we call the quantum-coherent evolution

4of the state. Consider an electron spin in a basis deﬁned by a transverse appliedmagnetic ﬁeld (Voigt geometry), i.e. the electron is in a pure spin state initiallywhose direction is perpendicular to the magnetic ﬁeld (Figure 1.2). The electron
Hamiltonian under the magnetic ﬁeld is
H = gµB S · B/h̄ = ωL Sz ,(1.1)

with ωL ≡ gµB B/h̄ the electron Larmor precession frequency. Here g is theelectron Landé g factor, and µB is the Bohr magneton. The eigenstates in thesystem are |Sz ; +i and |Sz ; −i, simply denoted as |+i and |−i, which are split in
the magnetic ﬁeld by the Zeeman energy ∆E = gµB B = h̄ωL . The initial electron
spin wavefunction is in the +x direction,11|Ψ(0)i = |Sx ; +i = √ |+i + √ |−i.
2
2

(1.2)

It is a coherent superposition of the spin-up (|+i) and spin-down (|−i) eigenstates, deﬁned by the applied magnetic ﬁeld. The electron spin will evolve withtime under the spin Hamiltonian, resulting in oscillation between the two spin
eigenstates. Speciﬁcally, the electron spin wavefunction |Ψ(t)i at a later time t
follows|Ψ(t)i = exp(

iωL t
iωL t
−iHt
1
1
)|Ψ(0)i = √ exp(−)|+i + √ exp(
)|−i.
h̄
2
2
2
2

(1.3)

If the projection of electron spin in the x direction is measured, the measured
quantity (averaged over an ensemble) is the expectation value:hSx (t)i = hΨ(t)|Sx |Ψ(t)i =h̄cos(ωL t).
2

(1.4)

This is obtained by expressing the operator Sx in the z-basis,Sx =h̄[(|+ih−|) + (|−ih+|)] .
2

(1.5)

5

Figure 1.2: An initial spin vector in the +x direction precesses under an appliedmagnetic ﬁeld.

Classically, the oscillation in Equation 1.4 corresponds to Larmor precession of
the spin vector at a frequency ωL in a plane normal to the applied ﬁeld, with the
projection of the spin vector along its initial direction oscillating as a cosine.
An electron can lose its spin coherence through spin scattering oﬀ anothertype of spin, spin-orbit scattering oﬀ a lattice vibration or impurity, or particle
annihilation through optical recombination with a hole. Spin decoherence canbe represented by adding a phenomenological damping term in the Hamiltonianshown in Equation 1.1,
H = gµB S · B/h̄ − ih̄Γ.

(1.6)

Equation 1.3 and 1.4 can then be rewritten as|Ψ(t)i = exp(iωL texp(−Γt)
iωL t
−iHt
√
exp(−
)|Ψ(0)i =)|+i + exp()|−i ,
h̄
2
2
2
(1.7)




h̄
exp(−2Γt) cos(ωL t).
2

(1.8)

and
hSx (t)i = hΨ(t)|Sx |Ψ(t)i =The characteristic time 1/2Γ is usually termed the transverse spin relaxation time,or spin coherence time, T2 . For an ensemble of electron spins, inhomogeneous
factors such as variations in the electron g factor or in the local magnetic ﬁeld cancause diﬀerent spins to precess at slightly diﬀerent frequencies. This will produce

6
a spreading in the relative phase of spins within the electron distribution, evenwhen all spins are evolving coherently. This additional decay of spin polarization
due to inhomogeneities is termed spin dephasing, T2∗ . T2∗ gives the lower boundon T2 ,
1
1
1
≈
+
,
∗
T2
T2 Tinh(1.9)

unless inhomogeneous eﬀects can be eliminated.
1.2.2

Spin relaxation mechanisms

In this portion of the chapter, some of the most relevant spin relaxation
mechanisms for conduction electrons in semiconductors are discussed. A detailed
review of spin relaxation mechanisms in semiconductors and metals can be foundin Reference [2].
1.2.2.1

Elliott-Yafet mechanism

In the Elliott-Yafet mechanism, the conduction electron spins relax throughordinary momentum scattering (spin-independent processes). In semiconductors,
the electron wavefunction experiences spin-orbit coupling induced by the latticeions, due to the spin-orbit Hamiltonian
Hso =

h̄
∇V (r) × p̂ · σ̂,
4m2 c2(1.10)

where V (r) is the crystal potential, p̂ is the momentum operator and σ̂ are thePauli spin matrices. Due to the spin-orbit coupling, the energy eigenstates are
no longer the spin eigenstates (Pauli spin-up | ↑i and spin-down | ↓i states), i.e.
the spin magnetic quantum number ms is no longer a good quantum number.
Instead, the energy eigenstates are a mixture of | ↑i and | ↓i states, and are labeled as pseudospin eigenstates. The spin quantization axis for those pseudospin
states varies with momentum k. One of the consequences of this is that the spin

7
quantum state of the electrons is not preserved for an elastic scattering event(even for spin-independent scattering). The spin decoherence can then be understood as an eﬀective k-dependent rotation of the pseudospin resulting from themomentum scattering. In the Elliott-Yafet mechanism, the spin relaxation timeτs is proportional to momentum scattering time τp , meaning the more orbital
scattering present in the material the more rapid the spin decoherence.
1.2.2.2

D’yakonov-Perel’ mechanism

Another eﬃcient spin relaxation mechanism for conduction electrons is theD’yakonov-Perel’ mechanism, resulting from the spin-orbit coupling in systems
lacking inversion symmetry. The zinc-blende semiconductors, such as GaAs and
CdTe, are crystals without inversion symmetry. The inversion symmetry is also
broken by the presence of asymmetric conﬁning potentials in heterostructures, orby applying an electric ﬁeld. In this category of crystals, even with the absenceof an applied magnetic ﬁeld, the pseudospin states are not degenerate. This
spin splitting can be described by introducing an intrinsic k-dependent eﬀective
magnetic ﬁeld B(k), and Ω(k) = (e/m)B(k) is the resulting Larmor precession
vector. The spin-orbit coupling for conduction electrons now has the form
Hso =

h̄
∇V (r) × p̂ · σ̂ = h̄Ω · σ̂/2.
4m2 c2(1.11)The presence of this k-dependent internal magnetic ﬁeld implies that spins ofindividual electrons will precess at diﬀerent rate according to their momenta, andthe ensemble electron spins will dephase inhomogeneously in momentum space.
Now consider the case when the momentum scattering time τp is shorter
than the spin precession time due to internal magnetic ﬁelds (τp Ω(k) ≤ 1). Spinsof individual electrons will precess about ﬂuctuating magnetic ﬁelds, whose magnitude and direction change randomly. The faster the momentum scattering, the8more random the electron spin precession and the slower the electron spin dephasing. This is the case for D’yakonov-Perel’ mechanism, which is in the motional
narrowing regime. The spin relaxation time τs is therefore inversely proportional
to the momentum scattering time τp . The most important diﬀerence between the
Elliott-Yafet and the D’yakonov-Perel’ mechanisms is their opposite dependence
on momentum scattering time τp .
1.2.2.3

Hyperfine interaction mechanism

The hyperﬁne interaction, which is the magnetic interaction between themagnetic moments of electrons and nuclei, provides an important mechanism forensemble spin dephasing or single spin decoherence of localized electrons. The
spatial variation in the nuclear spins, experienced by localized electrons via thehyperﬁne interaction, leads to electron spin dephasing. In the limit of small orbitaland spin correlation between separated electron states and nuclear spin states,
the spins of the ensemble electron dephase inhomogeneously. In a system where astrong orbital correlation (electron hopping) or spin correlation (electron-electron
exchange interaction) between neighboring electron states is present, the inhomogeneous spin dephasing is motionally narrowed, as in the case of D’yakonov-Perel’mechanism. The spin dephasing due to the hyperﬁne interaction is shown to beimportant in low-dimensional semiconductors but rather weak in bulk semiconductors.
1.2.3

Electron Landé g factor

The electron Landé g factor is one of the basic parameters that describe
the magnitude of the Zeeman splitting of electronic states in magnetic ﬁelds. The

9Landé g factor for an atomic electron is given bygJ =3 S(S + 1) − L(L + 1)+,
2
2J(J + 1)(1.12)

with an energy splitting under a magnetic ﬁeld B∆E = gJ µB J · B.

(1.13)

Here, J is the total electronic angular momentum, L is the orbital angular momentum, and S is the spin angular momentum. In the special case of free electrons(L = 0), g ≈ 2.0.
An eﬀective Landé g factor (g ∗ ) is commonly used in solid state physics,
with its deﬁnition shown in the Zeeman splitting ∆E = g ∗ µB B. g ∗ of conduction
band electrons in semiconductors diﬀers from the free electron value, 2.0, due tothe spin-orbit interaction. Speciﬁcally, the g ∗ values at the conduction-band edge
are smaller than the free-electron value, due to the spin-orbit interaction between
the s-like (the lowest conduction band) and p-like (both the valence bands andthe next higher conduction bands) states. The k · p perturbation theory predicts
electronic properties such as the eﬀective mass m∗ and the Landé g factor g ∗ inthe vicinity of a semiconductor band extremum. A ﬁve-band k · p model [14]was developed to calculate the g ∗ , where the spin-orbit interaction of the valencebands and higher conduction bands were taken into account:"EPg ∗ ≈ g0 1 −
3



′

E
1
1
− P
−
E E + ∆0
3


1
1
−
c
c
E(Γ7 ) − E E(Γ8 ) − E

!

′

#

+C .
(1.14)

Here E is the electron energy measured from the valence band maximum, EP ,′EP , E(Γc7 ), E(Γc8 ) and ∆0 are energy parameters of the bands. As shown inEquation 1.14, the g ∗ dependence on the energies of fundamental gaps in semiconductors is established.

10
1.3

Outline of thesis
This thesis is focused on optical studies of spin dynamics in semiconductormaterials, especially that in quantum well systems. Chapter 2 gives an explanation of the optical techniques used to measure the spin dynamics of the carriers.
Preliminary results on the spin dynamics in bulk GaAs samples are also presented.
In Chapter 3 we present a spectral study of the spin dynamics in semiconductor
quantum wells, where spin dependent nonlinearities of the materials are discussed.
Chapter 4 explores the intimate connection between electron spin and optical excitation that initiates and controls the spin states. This involves studies of spin
interplay between resident electrons and interband transitions. In Chapter 5 wesystematically study the spin dynamics of the electron in a GaAs quantum well.
The focus is on extending spin coherence time through optimum doping. We study
the eﬀects of disorder and dynamical scattering on electron spin decoherence. A
conclusion is presented in Chapter 6. An appendix is included that discusses calibration and compensation of the birefringence of the cryostat windows, which
is important in polarization-sensitive measurements such as the transient Faraday/Kerr rotation experiments.

Publications[1] Z. Chen, S. G. Carter, R. Bratschitsch, P. Dawson and S. T. Cundiﬀ, “Eﬀects
of Disorder on Electron Spin Dynamics in a Semiconductor Quantum Well,” Nature Physics (in press).

[2] Z. Chen, R. Bratschitsch, S. G. Carter, S. T. Cundiﬀ, D. R. Yakovlev, G.
Karczewski, T. Wojtowicz, and J. Kossut, “Electron Spin Polarization throughInteractions between Excitons, Trions and the Two-Dimensional Electron Gas,”

11
Phys. Rev. B. (in press).

[3] S. G. Carter, Z. Chen and S. T. Cundiﬀ, “Optical Measurement and Control of
Spin Diﬀusion in n-Doped GaAs Quantum Wells,” Phys. Rev. Lett. 97, 136602(2006).

[4] R. Bratschitsch, Z. Chen, S. T. Cundiﬀ, E. A. Zhukov, D. R. Yakovlev, and M.
Bayer, G. Karczewski, T. Wojtowicz, and J. Kossut, “Electron Spin Coherence in
n-Doped CdTe/CdMgTe Quantum Wells,” Appl. Phys. Lett. 89, 221113 (2006).

[5] R. Bratschitsch, Z. Chen, S. T. Cundiﬀ, D. R. Yakovlev, G. Karczewski, T. Wojtowicz and J. Kossut, “Electron Spin Dephasing in n-Doped CdTe/(Cd, Mg)Te
Quantum Wells,” Phys. Status Solidi B 243, 2290-2292 (2006).

[6] Z. Chen, R. Bratschitsch and S. T. Cundiﬀ, “Interference Eﬀects in Transient
Kerr Spectra of a Semiconductor Multilayer Structure,” Opt. Lett. 30, 2320-2322(2005).

[7] R. Bratschitsch, Z. Chen and S. T. Cundiﬀ, “Ultrafast Spin Phenomena in
Highly Excited n-Doped GaAs,” Phys. Status Solidi C 0, 1506-1508 (2003).

Chapter 2
Experimental Techniques

For the study of carrier dynamics in semiconductors, ultrafast pump-probe
spectroscopy techniques are widely used [15]. A pump pulse excites carriers insemiconductors, and a delayed probe pulse monitors the time evolution of optically
excited carriers. The work presented in this thesis is focused on the study of spin
dynamics in semiconductor materials. For such study, optical spin orientation
is implemented into the pump-probe experiment, to enable optical generationand detection of carrier spins in semiconductors. The transient Faraday/Kerrrotation technique, which are essential for most of the experiments presented
in this thesis, uses Faraday/Kerr eﬀects induced by spin magnetization in themedium to extract dynamical information of carrier spins. With the addition ofan applied magnetic ﬁeld, the time evolution of the spin coherence, deﬁned by
the applied ﬁeld, can be followed by optical pulses (refer to Section 1.2.1). We
start this chapter by presenting general aspects of the pump-probe experiments.
The selection rules in semiconductors are then discussed to illustrate carrier spinpolarization through optical orientation. The transient Faraday/Kerr rotation
technique is introduced, with the experimental setup discussed. We conclude
the chapter by presenting preliminary results measured on bulk GaAs samples,
which illustrate information regarding spin dynamics that can be obtained fromFaraday/Kerr rotation measurements.

13

Figure 2.1: Schematic for a pump-probe measurement. The temporal dynamics ismonitored by varying the relative time delay between the pump and probe pulses.

2.1

Pump-probe spectroscopy
Pump-probe experiments using ultrafast lasers enable direct measurementsof carrier dynamics, with a temporal resolution limited only by the duration of
the laser pulse. Carrier dynamics initiated by a pump pulse can be monitoredby varying the relative time delay of a probe pulse (Figure 2.1). This is done inpractice by using a translation stage that varies the optical path length of one
beam relative to the other. A pump-probe experiment measures the diﬀerential
transmission by the probe pulse, with and without the pump excitation. Witha pump excitation, the photoexcited electrons ﬁll states in the conduction band,
which would otherwise be available for excitation of valence electrons by probe
photons, therefore induce a change in the transmitted probe intensity. The direct
application of pump-probe experiments is to track the carrier population decay
as a function of time, giving information of the optically created carrier lifetime.
To monitor spin dynamics of the carrier, circularly polarized pump and probe
pulses are used. Pump pulse preferentially excite carrier spins in one of the spin
subband, and a co-circular probe pulse measure the spin dynamics in the same spin
subband. In Section 2.3, we will introduce the transient Faraday/Kerr rotation
technique, which is a special version of the pump-probe experiment. The principal

14advantage of Faraday/Kerr rotation over traditional pump-probe is the sensitivityof the measurement. Before proceeding to the Faraday/Kerr rotation technique,we ﬁrst discuss the selection rules through which spin polarized carriers can begenerated with light.
2.2

Generation of spin polarized carriers
The carrier spins in semiconductors can be oriented using optical techniques

in which circularly polarized photons transfer their angular momenta to electrons(optical orientation). Speciﬁcally, a photon of right or left circularly polarized
light has a projection of its angular momentum in the light direction equal to+1 or -1, respectively. As a consequence of angular momentum conservation, thephotoexcited electron and hole are spin-polarized following the absorption of thephoton, and the angular momentum is distributed between the electron and holeaccording to the selection rules.
We illustrate basic principles of optical orientation by the example of GaAs,
which is representative of a large class of zinc-blende semiconductors. A schematicband structure of GaAs is depicted in Figure 2.2(a). GaAs is a direct bandbap
material, i.e. both the minimum of the conduction band and maximum of thevalence band lie at k = 0 in momentum space. The bandgap is Eg = 1.52 eV
at low temperature, and energy of the spin split-oﬀ band is lower than the heavy
and light hole bands by ∆so = 0.34 eV (we ignore optical orientation due to thesplit-oﬀ band, because of its much higher interband energy).
For the lowest-lying bands in the semiconductors of interest here, the character of the conduction band is “s-like” while that of the valence bands is “p-like”.
This reﬂects the relative contributions to the density of states in each band fromatomic s and p orbitals of the atoms that form the solid. An electron state in the

15

Figure 2.2: (a) Schematic band structure of GaAs near the interband gap. Eg isthe bandgap and ∆so is the energy splitting of split-oﬀ band; cb, conduction band;hh (lh), heavy (light) hole valence band; so, split-oﬀ valence band. (b) Selection
rules for interband transitions between the heavy and light hole valence bands
and conduction band for σ + , σ − and π polarized light. The numbers denote therelative transition intensities.

conduction band is described by the Bloch wavefunctionc
ψ~km= um (~r) exp(i~k · ~r).

(2.1)

Here m represents the electron spin direction (m = ±1/2), ~k is the crystal momentum and um is the Bloch amplitude, i.e. an invariant under symmetry transformations of the crystal. For the conduction band states, the Bloch amplitudes
can be approximately separated into orbital and spin parts [16]:u1/2 (~r) = S ↑, u−1/2 (~r) = S ↓,(2.2)

with S denoting the coordinate part, and ↑ (↓) the spin part of the s-like Blochamplitude. Unlike the “s-like” conduction band, the valence band is “p-like”(L=1), so that the spin-orbit coupling is sizable. This means that the Bloch16
amplitudes for the valence band electrons should be expressed as eigenstates of
the total angular momentum, J = L + S, and its z-component, Jz . We denote theBloch amplitudes according to the total angular momentum J and its projection
onto the positive z axis mj as |J, mj i. Speciﬁcally, the eigenstates for the heavy
hole valence band are written as |3/2, ±3/2i, |3/2, ±1/2i for light holes, and|1/2, ±1/2i for split-oﬀ holes. The spin-orbit coupling removes the degeneracy
between J = 3/2 and J = 1/2, so that the split-oﬀ hole band is ∆so below theheavy and light hole valence band.
In analogy to atoms, interband optical excitation excites electrons fromthe valence to conduction band in semiconductors following selection rules. The
transition probability for an electron excited from the valence to conduction band
~ is given by the Fermi’s golden rule
under an incident electromagnetic ﬁeld E
Rvb→cb =22π~
hcb| − e~r · E|vbi
δ(Eg − h̄ω),
h̄

(2.3)

with Eg the energy of the bandgap. The interband selection rules are then represented by the dipole moments, hcb|D̂|vbi, with D̂ ≡ −e~r, corresponding totransitions between the conduction band and diﬀerent sub-bands of the valenceband. The matrix elements of the dipole moment between the heavy hole valenceband and the conduction band follows [16]qhS ↑ D̂ 3/2, 3/2i = − 1/2(λ̂ + iµ̂), hS ↓ D̂ 3/2, 3/2i = 0,hS ↑ D̂ 3/2, −3/2i = 0,hS ↓ D̂ 3/2, −3/2i =q

1/2(λ̂ − iµ̂), (2.4)

and the matrix elements of the dipole moment between the light hole valence bandand the conduction band follows
hS ↑ D̂ 3/2, 1/2i =qhS ↑ D̂ 3/2, −1/2i =2/3ν̂,

q

q

hS ↓ D̂ 3/2, 1/2i = − 1/6(λ̂ + iµ̂),

1/6(λ̂ − iµ̂), hS ↓ D̂ 3/2, −1/2i =q

2/3ν̂.

(2.5)

17Here λ̂ and µ̂ are unit vectors in a plane perpendicular to the momentum, and ν̂
is a unit vector along the momentum. The same constant has been removed fromthe dipole moments in Equation 2.4 and 2.5 for simplicity. The electric ﬁeld will
couple to diﬀerent interband transitions depending on the dot-product between
2~ · E|vbi~
the ﬁeld vector and diﬀerent dipoles (transition probability R ∝ hcb|D).

A σ ± circularly polarized photon, with the electric ﬁeld inq

1/2(λ̂ ± iµ̂) direc-

tion, will couple to both the heavy and light hole transitions, while a π polarized
photon (with ﬁeld in ν̂ direction) can also couple to the light hole transition. The
complete selection rules for interband transitions between the heavy and light holevalence bands and the conduction band are depicted in Figure 2.2(b). The relativetransition intensities for the diﬀerent transitions are also denoted in the ﬁgure.
The spin polarization of the photoexcited conduction band electrons depends on the photon energy. For a photon energy tuned to the interband transition below the split-oﬀ energy, only the heavy and light hole subbands contribute.
We deﬁne the electron spin polarization as
P = (n+ − n− )/(n+ + n− ),(2.6)

with n+ and n− the densities of electrons polarized parallel (ms = 1/2) andantiparallel (ms = −1/2) to the light propagation direction. For a zinc-blende
structure (such as that of GaAs and CdTe), the spin polarization is equal to−50% (+50%) at the moment of a σ + (σ − ) photoexcitation. This is due to thediﬀerent transition probabilities between the heavy hole and light hole subbands,
as depicted in Figure 2.2(b). Figure 2.2(b) also suggests a 100% spin polarizationin the conduction band in semiconductors where the degeneracy of the heavy andlight hole states is lifted. This is achieved in strained bulk zinc-blende crystals orin quantum well heterostructures.

18
2.3

Transient Faraday and Kerr rotation

2.3.1

Faraday effect

The Faraday eﬀect is the rotation of linearly polarized light as it passes
through a material with a circular birefringence induced by an applied magnetic
ﬁeld. The medium equation (electric displacement) for materials exhibiting the
Faraday eﬀect is [17]~ = ǫE
~ + iǫ0 γ B~ × E,~
D(2.7)

where γ is the magnetogyration coeﬃcient that determines the rotatory power.
~ = B ẑ, the invariant
With a magnetic ﬁeld in the light propagating direction B~ = E0 (x̂ ± iŷ) for σ ± circular polarization. The
states to the above equation are E
corresponding index of refraction can be written as
′

n± =

q

n ± γB(2.8)

for σ + and σ − helicities, respectively. The induced circular birefringence corresponds to the diﬀerent indices of refraction for σ + and σ − light. Because linearlypolarized light can be viewed as a superposition of σ + and σ − light, the circularbirefringence will add a phase diﬀerence between the two components while thelight travels through the medium. This phase diﬀerence corresponds to an angleof rotation of the linear polarization, hence the name Faraday rotation, with therotation angle′′

π(n+ − n− )L
πγBL
≈
,
θF =
λ0
nλ0

(2.9)

with L the medium thickness and λ0 the free space wavelength of the light.
The spin dynamics in semiconductors is measured by implementing Faraday
eﬀect into pump-probe experiments. The Faraday eﬀect is now induced by carrier
spins, which can be generalized as a pump-excited spin magnetization in thelight propagation direction (the origin of Faraday rotation resulting from optical

19

Figure 2.3: Schematic setup for transient Faraday rotation measurements. PBS,
polarizing beam splitter; λ/2, half-wave plate; λ/4, quarter-wave plate. An optional quarter-wave plate is inserted for the Faraday ellipticity measurement.

orientation will be discussed in Chapter 3). By monitoring the rotation angle ofthe probe linear polarization as a function of the time delay, the dynamics of spinpolarization in the beam direction is directly measured. A schematic of transient
Faraday rotation setup is shown in Figure 2.3.
2.3.2

Experimental details

The pump and probe pulses are provided by a mode-locked Ti:Sapphirelaser system (Coherent Mira 900), pumped by a 5 W CW Diode-Pumped Laser
at 532 nm (Coherent Verdi V5). It provides optical pulses in the near-infrared
wavelength range, with a repetition rate of 76 MHz and a duration less than 200
fs (due to spatial chirp caused by optical components in the setup, the pulse duration is about 300 fs before it arrives at the samples, based on autocorrelation
measurements). The normal spectral linewidth of the laser output is about several
nm, with the center wavelength tuned near the interband transition of the semiconductor samples. The laser output is split into pump and probe pulses, with20
the probe pulse time-delayed by a 60 cm translation stage (Newport IMS600PP),
up to a delay of 4 ns. A retroreﬂecting mirror is mounted on the stage to minimize
beam misalignment due to stage movement. The pump beam is chopped either
by an optical chopper (Stanford Research SR540) that goes up to 3.7 kHz, or an
acousto optical modulator (Isomet 1250C), so that a lock-in ampliﬁer can be used
for phase-sensitive detection.
The pump and probe beams are focused to a ∼ 100 µm and ∼ 50 µm spot
on the sample, respectively. This is to ensure that the probe beam will only
measure a part of the area within that excited by the pump beam and henceprobe uniform excitation. The sample is mounted on a sample mount within amagneto-optical cryostat (Janis split-coil superconducting magnet cryostat, model
12-CNDT) capable of temperatures from 2 − 300 K and magnetic ﬁelds from 0-6.5
Tesla.
As shown in Figure 2.3, a circularly polarized pump pulse excites spin polarized carries in the semiconductor samples, with the spin polarization oriented
in the pump beam direction. They start to Larmor-precess in an in-plane appliedmagnetic ﬁeld (Voigt geometry, with the magnetic ﬁeld perpendicular to the beam
direction). A linear probe pulse at a later time is used to detect the spin throughthe Faraday eﬀect. The circular birefringence created by the spin magnetization
will produce a rotation of the linear polarization of the transmitted probe, which
is detected by a polarization bridge. By varying the time delay between pumpand probe pulse, one can monitor the spin precession of the carriers in the time
domain. A balance detection scheme is employed in order to sensitively detect
small rotation signals. At negative time delays (when the pump has no eﬀect),the probe linear polarization is rotated by a half-wave plate to be at 45◦ withrespect to a polarizing beam splitter (a Glan Laser prism polarizer in our experiments). This ensures equal intensities for the orthogonal polarizations that exit

21
the beam splitter, which are directed to two photodiodes. When balanced, thediﬀerence current between the two photodiodes is zero. At positive delays after
pump excitation, however, the Faraday rotation of the probe is detected as a slight
imbalance of the diﬀerence current between the two photodiodes. The nonzero
diﬀerence current is ampliﬁed by an op-amp (with several ampliﬁcation settingsfrom 0.1 MΩ to 10 MΩ), and passed on to a lock-in ampliﬁer (Stanford Research
SR830). The main advantage of the balance detection scheme is that ﬂuctuationsof the laser intensity are manifested simultaneously in both arms of the bridge
and are thus canceled. This is, however, not the case for diﬀerential transmission
pump-probe measurements.
Similar to Faraday rotation, a diﬀerence in absorption coeﬃcients of themedium for the two circularly polarized components will induce ellipticity in thetransmitted light, which is called Faraday ellipticity. We will study the Faraday
ellipticity spectrum in Chapter 3. Experimentally, the Faraday ellipticity is measured by inserting an additional quarter-wave plate before the polarization bridge,
as shown in Figure 2.3. The purpose of the quarter-wave plate is to convert theleft and right circular polarizations into orthogonal linear polarizations, so thatthe two photodiodes essentially measure the diﬀerence intensity of the left andright circular polarization components in a linear probe beam. The diﬀerence
current from the balanced detector is then proportional to the Faraday ellipticity.
For the study of thick, opaque samples, a transient Kerr rotation technique
is used, which is similar to a transient Faraday rotation measurement, except that
it is the polarization of the reﬂected probe beam that is measured. A transient
Kerr rotation setup is shown in Figure 2.4. A typical transient Kerr rotation signal
is plotted in Figure 2.5, which is measured in a lightly n-doped CdTe quantum
well sample. The data for delay times over 100 ps can be well described by an

22

Figure 2.4: Schematic setup for transient Kerr rotation measurements. Bothpump and probe pulses are spectrally ﬁltered through pulse shaping technique,to enable independent tuning of their photon energies. The spectral linewidths ofthe pulses are also narrowed for spectrally resolved measurements.

Transient Kerr rotation (arb. units)

n-doped CdTe quantum wellL

= 2

43.3 GHz (at

g

B=2T)

| *|=1.55

T = 1100 ps
*
2

0

200

400

600

800

1000

Time (ps)Figure 2.5: Kerr rotation data from a lightly n-doped CdTe quantum well. Pump
and probe photon energies are tuned near the exciton resonance of the quantum
well. The long-delay signal can be well ﬁtted by an exponential oscillation decay,
giving information of the electron spin dephasing time (T2∗ ), Larmor precession
frequency (ωL ) and the eﬀective g factor (g ∗ ).

23
exponential oscillation decay in the form ofθK = A exp(−t/T2∗ ) cos(ωL t),(2.10)

with T2∗ the spin dephasing time and ωL = g ∗ µB B/h̄ the Larmor precession frequency. From a nonlinear ﬁt of the data with Equation 2.10, we can obtain thespin dephasing time, Larmor precession frequency under the magnetic ﬁeld, andthe eﬀective Landé g factor. A Larmor precession frequency of ωL = 2π 43.3 GHz
is measured at B = 2 T, corresponding to |g ∗ | = 1.55. A T2∗ over 1 ns is retrieved
from the data. As this time constant exceeds lifetimes of optically created carriers (∼ 100 ps), it is the spin dephasing time of the resident electrons. A fast
exponential decay at short time delay is related to exciton spin dephasing, which
will be detailed in Chapter 4.
2.3.3

Two-color measurements

In a typical transient Faraday/Kerr rotation measurement, both the pumpand probe photon energies (degenerate) are tuned near the vicinity of some absorption resonances for resonant excitation and detection sensitivity. A two-color
measurement, where the pump and probe photon energies can be independently
tuned, is preferred when multiple resonances in the samples are present. A twocolor measurement is useful for selective excitation and detection through diﬀerent
resonances (carrier species). It is especially helpful for spectral studies of the Faraday/Kerr rotation, where the pump photon energy is varied with the probe energy
ﬁxed near a resonance, and vice versa, so that diﬀerent eﬀects from excitation anddetection can be well separated. We perform a two-color transient Faraday/Kerrrotation measurement by using a pulse shaping technique [18]. Both pump andprobe pulses are spectrally ﬁltered through a 4-f pulse shaper with a movableslit, to enable independent tuning of their photon energies, which is shown in24
Figure 2.4.
Figure 2.4 shows the basic pulse shaping apparatus, which consists of a pairof diﬀraction gratings and lenses and a movable slit. The lenses are 2-inch biconvex lenses with f = 15 cm. The diﬀraction gratings are of 1200 grooves/mm
with a blaze angle at 750 nm. The width of the slit is adjustable with width
ranging from 0 to 4 mm (Melles Griot SLT 201). The frequency components
contained within the incident laser pulse are angularly dispersed by the ﬁrst diffraction grating, and are focused at the back focal plane of the ﬁrst lens, where
diﬀerent frequency components are spatially separated. The second lens and grating recombine all frequency components into a collimated beam. The adjustableslit, placed in the back focal plane of the ﬁrst lens, narrows the pulse spectrum
depending on its width. The center wavelength is controlled by translating theslit in the focal plane. The 4-f pulse shaper used has a spectral resolution of∼ 0.1 nm. The spectral linewidths of the pulses are normally adjusted to be ∼0.2
nm for spectrally resolved measurements. By using separate pulse shapers forpump and probe beams, their photon energies can be independently tuned, hence
a two-color measurement.
2.4

Preliminary measurements on bulk GaAs
A preliminary study is performed by measuring spin dynamics in bulk GaAs.

The importance of GaAs for spintronics and quantum computing applications has
been recently underlined by the observation of rather long spin relaxation times(on the order of 100 ns) in n-doped samples [19]. We study nominally undoped
and n-doped bulk GaAs (doping densities of 2.4 − 5.3 × 1016 , 2 − 3 × 1017 , and2 − 3.3 × 1018 cm−3 ) by performing transient Faraday rotation experiments in the
Voigt geometry, under a magnetic ﬁeld of 4 Tesla. The GaAs samples are thinned
down to 30 µm and mounted on a sapphire disk, for transmission measurements.

25

-0.42

Faraday rotation (a. u.)g factor

-0.40

-0.44

-0.46

-0.48

T=4.5K
0

200

400

600

800

Time (ps)

0

50

100

T (K)

150

200

Spin dephasing time (ps)

450
(a)

-0.38

(b)

400

350

300

250

200
0

50

100

150

200

T (K)

Figure 2.6: (a) g factor and (b) spin dephasing time (T2∗ ) as a function of temperature in undoped bulk GaAs. Solid line in (a) is a linear ﬁt for data above 40 K.
B = 4 T. Inset: Transient Faraday rotation signal at T = 4.5 K.

The pump and probe pulses are degenerate in photon energy and are tuned near
the bandgap of GaAs.
2.4.1

Undoped GaAs

We start with the nominally undoped GaAs sample. The transient Faraday
rotation signal at T = 4.5 K and B = 4 T is shown in the inset of Figure 2.6.
A spin dephasing time of T2∗ ∼ 200 ps and g factor of g ∗ = −0.452 are retrieved
from a nonlinear ﬁt using Equation 2.10. We measure the g-factor as a functionof temperature, which is shown in Figure 2.6(a). For temperature above 40 K,
a rather good linear dependence on temperature is observed, with a linear dependence of g ∗ (T ) = −0.46 + 5 × 10−4 T. This is in agreement with the linear
temperature dependence reported in Reference [20]. Below 40 K, the g factor
substantially deviates from the linear dependence. The donor binding energy for
GaAs is ∼ 4 meV, or 40 K [21, 22]. Below 40 K, electrons are largely localized by

26
binding with the donors or impurities, and above 40 K, are largely free. We believe
the deviation of g factor at low temperature is related to electron localization at
donor sites. A theory including eﬀects from donors is however unavailable. In Section 1.2.3, a g ∗ dependence on the energies of fundamental gaps in semiconductors
is established. The g ∗ dependence on temperature can therefore be represented
by the temperature variation of the energy gaps, which are well known with highaccuracies [23]. A calculation of g ∗ dependence on temperature was carried out[23] by using the ﬁve-band k · p model discussed in Section 1.2.3 [14]. The theoretical prediction by k · p theory is in the opposite direction than the experimental
results, manifesting the need for appreciable, temperature dependent correctionsof this band model.
The temperature dependence of the spin dephasing time is plotted in Figure 2.6(b). The spin dephasing time T2∗ at ﬁrst increases with temperature, andthen starts to decrease at temperature over 80 K. The decrease of T2∗ with temperature above 80 K has been found to be consistent with the D’yakonov-Perel’mechanism [24], which is the dominant spin dephasing mechanism for delocalized electrons. At low temperature, the increase of T2∗ with temperature is rather
counter-intuitive, because momentum scattering rate is usually increased with increasing temperature. It can be qualitatively explained with a motional narrowing
picture, in which the correlation time decreases with increasing temperature much
faster than the dispersion of local Larmor frequencies.
2.4.2

n-doped GaAs

We study spin dynamics in the four bulk GaAs samples with diﬀerent doping
concentrations [25]. A comprehensive study of the electron spin decoherence timeover a large range of doping levels in bulk n-GaAs has already been presented
elsewhere [8], with spin coherence time reaching its maximum near the metal-

27

data
-0.1

calculationg factor

-0.2

-0.3

-0.4

T=4.5K
16

17

10

18

10

10
-3

Doping density (cm)

Figure 2.7: Measured g factors of the four GaAs samples are plotted as a functionof their doping density. The solid line is the calculated g ∗ as described in the text.

16

undoped GaAs

650

(a)

180

Spin dephasing time (ps)n-doped GaAs (2.4 - 5.3x10

-3

cm

)

(b)

170
600
160
550
150

T = 4.5 K
140

B = 4 T

0

5x10

15

1x10

16

2x10

16

2x10

Optical Excitation Density

16

3x10

(cm )
-3

500
16

0

1x10

16

16

2x10

16

3x10

Optical Excitation Density

16

4x10

(cm )
-3

Figure 2.8: Electron spin dephasing time T2∗ in (a) undoped GaAs and (b) ndoped GaAs (2.4 − 5.3 × 1016 cm−3 ) as a function of optical excitation density.
T = 4.5 K and B = 4 T.

28
insulator transition. We focus our study on the doping concentration dependenceof the electron Landé g factor. In Figure 2.7, g factors for the four GaAs samples
measured at 4.5 K are plotted with their doping concentration. The g factor
increases monotonically with increasing doping density. In Section 1.2.3, g ∗ of the
conduction electron is shown to depend on its energy. This energy dependence
can be further simpliﬁed as a linear dependence g ∗ (E) = −0.44 + 6.3 × E (eV),within a small energy range near the conduction band edge [26]. The variationof g factor with carrier concentration can then be understood as its dependence
on the Fermi energy, with the electron energy at the Fermi edge increased withincreasing doping density due to ﬁlling of conduction band from dopant electrons.
The g factor measured through Faraday rotation should be that of electrons at the
Fermi energy. With the assumption of a linear energy dependence of g ∗ , a FermiDirac distribution with T = 0 K, and a parabolic conduction band near the bandedge, we are able to calculate the g factor dependence on electron density, whichis shown as a solid line in Figure 2.7. At low densities, the calculated g ∗ is higher
than the experimental values due to the set value (g0 = −0.44) at conduction
band edge. The discrepancy at high electron concentration is probably becauseof the assumption of linear energy dependence, which is not valid for energies far
away from the conduction band edge [27].
Figures 2.8(a) and (b) show the electron spin dephasing time T2∗ in n-doped
GaAs (2.4−5.3×1016 cm−3 ) and nominally undoped GaAs as a function of opticalexcitation density. While T2∗ increases almost linearly with increasing opticalexcitation density for the undoped sample, it shows an opposite behavior for then-doped sample. By increasing the optical excitation density, we also increase thecarrier temperature of the electrons. The results in Figure 2.8 indicate dramatic
diﬀerence in spin dephasing mechanisms under diﬀerent doping concentration.

Chapter 3
Spectral study of transient Faraday and Kerr effects

For the study of carrier spins in semiconductors, the magneto-optical Faraday and Kerr eﬀects, which are measured in transmission and reﬂection geometry,
respectively, are extensively used [28, 29]. The transient versions of these techniques are especially powerful for extracting spin dynamics in semiconductors[13, 30]. Spectral studies of the transient Faraday and Kerr eﬀects reveal the underlying spin-dependent optical nonlinearities [31, 32], improve understanding of
the techniques themselves, and, as shown in Chapter 4, distinguish the spin relaxation from diﬀerent species. In this chapter, we experimentally study the spectraof the transient Faraday and Kerr eﬀects near exciton resonances of an undoped
GaAs multiple quantum well. A multilayer calculation using a characteristic matrix method is performed to account quantitatively for the linear transmission(reﬂectivity) spectra and the Faraday (Kerr) spectra. We found that the Faraday
and Kerr spectra do not obey a Kramers–Kronig relation, which should be valid
for a simple slab of nonlinear material. This anomalous behavior is found to be
due to interference with multiple reﬂections within the sample structure on theKerr spectra [33].

30
3.1

Introduction
When a linearly polarized light is incident on a sample, a diﬀerence in theindex of refraction for the two circularly polarized components causes a rotationof the polarization axis of the transmitted light, which is referred to as Faraday
rotation. Likewise, a diﬀerence in the absorption coeﬃcients induces ellipticity inthe transmitted light, which is called Faraday ellipticity. A diﬀerence in opticalcoeﬃcients can be created by an external magnetic ﬁeld or, as in the case of thephoto-induced Faraday eﬀect, by optical orientation due to photoexcitation. In atransient Faraday measurement, a circularly polarized pump pulse preferentially
excites carriers into one of the two spin subbands and creates an imbalance ofspin population. This population imbalance will cause a change of the index ofrefraction and/or absorption coeﬃcient, which is detected by the Faraday eﬀect.
The resultant nonlinearities in the optical response can arise from:(1) saturation of the resonance owing to phase-space ﬁlling (arising from thePauli exclusion principle);(2) shift of the resonance energy due to band-gap renormalization and/or
Coulomb screening of the exciton binding energy;(3) enhancement of scattering due to excitation induced dephasing.
To distinguish the contribution of the above mechanisms to the Faraday
spectra, we start with the index of refraction and absorption spectra for the exciton
resonance in a quantum well. A good approximation to represent the exciton
resonance is to use the oscillator model [34]. The complex dielectric constant fora single resonance from the oscillator model follows
ǫ(ω) = ǫb +f
ω02 − ω 2 − iγω

,

(3.1)

31
where ǫb is the background dielectric constant, f is the oscillator strength, ω0 is
the resonance frequency and γ is the damping rate for the exciton. The complex
dielectric constant is equal to the square of the complex index of refraction ñ =n + iκ. The index of refraction n(ω) and the absorption coeﬃcient κ(ω) therefore
is equal to Re(ǫ(ω))1/2 and Im(ǫ(ω))1/2 , respectively. A dispersive refractive index
and a peak absorption are obtained around the resonance, as shown in Figure 3.1.
In Figure 3.1, schematic of the Faraday rotation and ellipticity near a resonance is shown for a shift of resonance energy and saturation. The Faraday spectra
are obtained based on the linear spectra for opposite helicities of circular polarization, with the nonlinearity described by a small change of resonant energy or adecrease of the oscillator strength for one helicity, and the linear spectra for theother helicity unchanged. Figure 3.2(a) shows the spectral behaviors of Faraday
rotation and ellipticity induced by an energy shift, calculated by using an energy
shift of 0.1 γ, with γ the full width half maximum linewidth of the resonance. The
rotation and ellipticity spectra are related by the Kramers–Kronig relation. Thisis expected because the rotation and ellipticity spectra are due to the diﬀerencesof the linear spectra, each of which obeys the Kramers–Kronig relation.
The Kerr eﬀect can be described as the Faraday eﬀect in reﬂection, which
is preferred for the study of thick, opaque samples. For a single interface, one
expects the Kerr spectrum to be related to the Faraday spectrum through the
Kramers–Kronig relation, as is the case for linear reﬂectivity and transmission
spectra. This is shown in Figure 3.2(b) for energy shift induced Faraday and Kerr
rotation spectra.
3.2

Sample characteristics and experiments
Experiments are carried out on a GaAs/Al0.3 Ga0.7 As multiple quantum well,which consists of 10 periods of a 10-nm well and a 10-nm barrier. The sample

32

Ro

tation (a.u.) Faraday (a.u.)

Figure 3.1: Schematic of the photo-induced Faraday rotation and ellipticity neara resonance due to shift of resonance energy and saturation.

rotation

ellipticity

Faraday

Kerr

(a)

(b)

-4

-2

0

Detuning (

2

4

)/

Figure 3.2: Faraday and Kerr eﬀect spectra resulting from an energy shift, calculated for a slab of nonlinear medium. Reﬂection from the front interface alone isconsidered for the Kerr eﬀect.

33
is grown by molecular beam epitaxy on a GaAs substrate. It consists of nominally a 0.5 µm GaAs buﬀer layer, a 0.3 µm Al0.4 Ga0.6 As etch stop layer, 120 nm
superlattice, 0.3 µm GaAs, 0.3 µm Al0.3 Ga0.7 As, the 10 periods quantum well,
0.2 µm Al0.3 Ga0.7 As and a 100 Å GaAs cap layer. The sample is mounted on asapphire plate (GaAs cap layer facing sapphire plate) with the opaque GaAs substrate removed by etching. The nominal sample structure after etching is shownin Figure 3.3(a).
The absorption spectrum of the sample is shown in Figure 3.3(b), where
the features at 1.539 eV and 1.545 eV correspond to the heavy-hole and lighthole exciton resonances in the quantum well. We measure spectrally the transient
Faraday and Kerr eﬀects under a magnetic ﬁeld of 4 T applied normal to thequantum well growth direction. The spectral width of the pump and probe pulses
is about 1 nm, and the laser energy is scanned in the vicinity of the heavyhole and light-hole exciton resonances of the quantum well. Average powers are
10 mW in the pump beam and 0.2 mW in the probe. The heavy-hole exciton
density due to pump excitation is estimated to be 1011 cm−2 per quantum well.
All experiments are performed at 4.5 K. Care is taken to compensate for straininduced birefringence in the cryostat windows (refer to Appendix A).
3.3

Linear spectra

3.3.1

Reflection and transmission

Linear transmission and reﬂectivity spectra of the sample are measured byusing a white light source and detected by a spectrometer. The reﬂectivity spectra are plotted in Figure 3.4. Both the transmission (not shown) and reﬂectivity
spectra show fringe patterns, with a fringe periodicity of ∼ 60 meV near the exciton resonances. The fringes are caused by interference between reﬂections from

34

Figure 3.3: (a) Layer structure of the GaAs quantum well used. (b) Absorption
spectrum near the heavy-hole (hh) and light-hole (lh) exciton resonances of thequantum well

the front and back surfaces of the sample. As the excitation spot moves acrossthe sample surface, we observe a shift of the fringes, which is illustrated in Figure 3.4. This indicates that the sample is slightly wedged (4% across the sample).
The heavy-hole exciton resonance may appear as an absorption dip lineshape, adispersive lineshape or a peak line shape, depending on where the exciton resonance sits on the fringes. The exciton resonances in the transmission spectra (notshown) are always a dip regardless of the sample thickness. The variation of sample thickness, which is not preferred in most cases, does provide us opportunities
to check the eﬀect of linear optical eﬀects (such as interference) on the Faraday
and Kerr eﬀect spectra, which in general depends on the nature of the opticalnonlinearities. We will discuss it in detail in Section 3.4.
3.3.2

Characteristic matrix method

A calculation of the linear spectra for the speciﬁc sample used is a necessary prerequisite to account for the experimental Faraday and Kerr spectra. The

35

heavy-holeacross sample

Reflectivity (a. u.)light-hole

1.48

1.50

1.52

1.54

1.56

1.58

1.60

Energy (eV)Figure 3.4: Measured reﬂectivity spectra across the sample surface. The heavyhole and light-hole exciton resonances are denoted in the ﬁgure. Spectra are oﬀset
vertically for clarity.

multilayer structure of the sample requires considerations of the multiple interfaces and interference eﬀects within the sample, as well as diﬀerent absorptions
associated with diﬀerent layers. We use a matrix method to treat the problem
by which the amplitudes and phases of the electromagnetic ﬁeld to the left of themultilayer are transformed to those to the right [35, 36]. The characteristic matrix
for each layer is obtained from the optical constants and layer thickness, and thecharacteristic matrix of the whole medium is the product of those for each layer.
The reﬂectivity and transmission are then easily calculated.
For a homogeneous dielectric ﬁlm situated between air and substrate, theincident electric ﬁeld is related to the transmitted electric ﬁeld byb E0t 




f
E0t



u0
1 
=



2u
0

u0



 1  f
E ,
M
 2t



−1 
1





u2

(3.2)

36
with the characteristic matrix


 m11

M≡






m12 

m21 m22=cos δ1iu1 sin δ1isin δ1 u1cos δ1.

(3.3)

Here the superscripts f and b refer to ﬁelds propagating forward and backward,
respectively, the subscript t refers to the tangential ﬁeld, and the subscripts 0, 1
and 2 refer to air, dielectric ﬁlm and substrate, respectively. The change of phase
after the ﬁeld propagating one-way in the dielectric layer is
δ1 =2π
n1 d1 cos θ1 ,
λ

(3.4)

where λ is the wavelength, n1 , d1 and θ1 are the index of refraction, thickness andangle of refraction of the dielectric layer, respectively. The tangential admittance
uj follows
uj =

s

ǫj
cos θj
µj

(3.5)

uj =

s

ǫj 1
µj cos θj(3.6)

for a TE wave, andfor a TM wave. From Equation 3.2 and 3.3 we obtain the reﬂection and transmission coeﬃcients of the ﬁlm
bE0t
u0 (m11 + m12 u2 ) − (m21 + m22 u2 ),f =
u0 (m11 + m12 u2 ) + (m21 + m22 u2 )
E0t

(3.7)

f
2u0
E2t
t= f =.
u0 (m11 + m12 u2 ) + (m21 + m22 u2 )
E0t

(3.8)

r=

The reﬂectivity and transmission in terms of r and t are
R = |r|2 ,

T =

u2 2
|t| .
u0

(3.9)

The above formula can be generalized to the case of a succession of stratiﬁed
media by using a characteristic matrix
M = M1 M2 . . . MN
for an N–layer structure.

(3.10)

37
3.3.3

Linear spectra: calculation

We calculate numerically the linear spectra of the sample by using the above
characteristic matrix method, based on the nominal sample structure and assuming Lorentzian lineshapes for both the heavy-hole and light-hole exciton resonances. There are some important aspects of this calculation that are worth
mentioning:(1) The sample is assumed homogeneous with interfaces parallel and opticallyﬂat.
(2) The absorption of the layers are included in the calculation by using complex indices of refraction (ñ) in Equation 3.4–3.6.
(3) Both the heavy-hole and light-hole exciton resonances are included in
the calculation for the GaAs quantum wells, with the complex index ofrefraction of the quantum well layers following that of an oscillator model(Equation 3.1).
(4) The interband transition of bulk GaAs (whose bandgap energy is ∼30meV
below that of the excitons in the quantum well) is also considered in thecalculation for thick GaAs layers. The changes of the index of refraction
and absorption associated with the transition is accounted for by using
the Elliott formula [37].
(5) The background indices of refraction used in the calculation for diﬀerent
layers are obtained from Reference [38], with nb = 3.656, 3.42, 3.36 and1.77 for GaAs, Al0.3 Ga0.7 As, Al0.4 Ga0.6 As and sapphire, respectively.
(6) Both the linear and Faraday (Kerr) spectra are experimentally measured
under normal incident light. We therefore set θ = 0 in Equation 3.4–3.6.

38(7) The reﬂectivity and transmission coeﬃcients obtained from Equation 3.7
and 3.8 not only give information of the amplitudes, but also the phases.
The Faraday and Kerr rotation are related to the phase changes of thetransmitted and reﬂected beams, respectively. We therefore calculate boththe amplitudes and phases of the reﬂectivity and transmission spectra.
The calculated transmission (not shown) and reﬂectivity spectra (solid lines
in Figure 3.5) are very similar to the measurements. The reﬂectivity spectrum
shown in Figure 3.5(a) has an absorption dip lineshape for the heavy-hole exciton resonance. This is because the sample thickness allows for constructive
interference at the heavy-hole exciton wavelength. As the thickness changes, theheavy-hole exciton resonance appears as a dispersive lineshape, as shown in Figure 3.5(b), which corresponds to a sample thickness roughly 1/8 of a wavelength
diﬀerent from that in Figure 3.5(a). We note that the multiple internal interfaces play a minor role in the spectra due to the similar indices of refraction of
the GaAs and Alx Ga1−x As layers. We extract the linear optical parameters suchas resonance energies, linewidths and oscillator strengths of the heavy-hole andlight-hole exciton transitions from comparison of the calculated and measured
spectra, which will be used for the analysis of the Faraday (Kerr) spectra in thenext section.
3.4

Faraday and Kerr spectra
We measure the Faraday (Kerr) rotation and ellipticity spectra by scanning

the incident pump and probe pulse wavelength (degenerate) in 0.5 nm steps. The
observed Faraday (Kerr) signal for each wavelength oscillates in time after excitation, due to the Larmor precession of carrier spins about the applied magnetic
ﬁeld. A Fourier transform of the transient reveals multiple precession frequen-

39
(a)

Reflectivity

0.35

0.30
hh

lh

0.35
0.30
0.25
(b)

0.20

1.525 1.530 1.535 1.540 1.545 1.550 1.555
Energy (eV)Figure 3.5: Measured (dots) and calculated (solid lines) linear reﬂectivity spectra
near heavy-hole (hh) and light-hole (lh) exciton resonances (indicated by dashedlines). Spectra in (a) and (b) correspond to diﬀerent sample thicknesses, which
are described in the text.

cies, which originate from diﬀerent Landé g factors for diﬀerent carrier types. By
selecting the single Fourier peak due to the electron spin precession in the quantum well, irrelevant contributions such as Faraday (Kerr) signals from the bulk
GaAs layer, coherence eﬀects from the cross-correlation between pump and probe
pulses, and DC signals due to scattered pump light into the detecting system are
all eliminated. The Faraday and Kerr spectra are then obtained by plotting theamplitude of the Fourier peak as a function of photon energy, as shown in Figure 3.6(a)–(d) (circles). The spectra are measured at the same sample thickness asthat in Figure 3.5(a). One sees similar spectral shapes for the Faraday and Kerr
rotation (ellipticity), in contrast to the prediction for a simple absorbing slab, i.e. a
Kramers–Kronig relation, which is shown in Figure 3.2(b). The Faraday and Kerr
spectra are also measured at a diﬀerent sample thickness [same thickness as that in
Figure 3.5(b)], which is plotted in Figure 3.6(e)–(h) as circles. While the Faraday
eﬀects show similar spectral proﬁles regardless of the sample thickness, there is a40clear change of Kerr spectra as the sample thickness is changed by a fraction ofa wavelength. Note that for this sample thickness, the Kerr and Faraday spectra
do approximately obey a Kramers–Kronig relation.
We calculate Faraday (Kerr) spectra based on the calculated transmission(reﬂectivity) spectra in the last section. Without any loss of generality, we assume
a σ + circular polarization of the incident pump pulse, which preferentially excites
spin − 12 electrons in the quantum well. The optical constants for σ + helicity arevaried as a result of the nonlinearity created by spin − 21 electrons, with the nonlinearity described by small changes of exciton resonance energies, linewidths andoscillator strengths. The change of the linear spectra for σ + helicity is then calculated following the method described in the last section. The optical constants
and linear spectra for σ − helicity are assumed to be unchanged after optical excitation, because of the absence of spin + 12 electrons. The Faraday (Kerr) ellipticity
spectra are calculated by subtracting the transmission (reﬂectivity) spectra for σ −helicity from those for σ + helicity. The Faraday (Kerr) rotation spectra are obtained as the product of transmission (reﬂectivity) spectra with the diﬀerence ofthe phase of transmission (reﬂectivity) spectra between σ + and σ − helicities. The
nonlinear parameters are varied to obtain good agreement between the calculated
and measured Faraday ellipticity spectra alone [Figure 3.6(b)]. We then use thesame nonlinear parameters to calculate the rest of the spectra, and the results areshown in Figure 3.6 as solid lines. The only diﬀerence in the calculation between
Figure 3.6(a)–(d) and (e)–(h) is the use of diﬀerent sample thickness. To represent
the experimental condition of ﬁnite bandwidths of both pump and probe pulses,
averaging of the exciton excitation with the pump spectrum, and averaging of the
Faraday (Kerr) signal with the probe spectrum are included in the calculation.
Good agreement is found between theory and experiment. We note that the theoretical lineshapes reproduce the lack of a Kramers–Kronig relation found in the41

K err (mrad)

Faraday (mrad)

Rotation

10

Faraday (mrad)

20

0
10
-10

0

-20

(a)

0

(b)

5

-5

(c)

-10

1.535

K err (mrad)

Ellipticity

1.540

1.545

0
1.535

10

(d)

1.540

1.545

20

0
10

-10

0

-20

(e)

-30

(f)

0
0
-5
-5

-10

1.535

(g)

1.540

1.545

Energy (eV)

(h)

1.535

1.540

1.545

Energy (eV)Figure 3.6: Measured (circles) and calculated (solid lines) Faraday and Kerr eﬀect
spectra. Spectra in (a)–(d) are obtained for a sample thickness the same as that in
Figure 3.5(a), while spectra in (e)–(h) are for a slightly diﬀerent sample thickness,same as that in Figure 3.5(b).

42experiment.
Because of the diﬀerent spectra for diﬀerent nonlinear contributions [39],as illustrated in Figure 3.1, one can distinguish among them by the shape of the
Faraday (Kerr) spectrum. We deduce from the comparison between calculations
and measurements a reduction of the heavy-hole exciton transition strength of
1.7% and an increase of its damping by 5.3% after excitation. We also obtain adecrease (red shift) of the heavy-hole exciton resonance energy by 0.11 meV. The
extraction of parameters is, however, not the goal of this work. Detailed extraction of the nonlinear parameters requires full knowledge of the sample structure,
which is not available with suﬃcient precision. Rather, we want to show the ambiguities in interpreting the Kerr eﬀect spectra, which have been ignored in mostexperiments that use Kerr eﬀect.
For the quantum well sample we studied, the spectral proﬁles of the exciton resonances in the reﬂectivity spectra are dominated by the interference eﬀect
from the total sample thickness, which gives a dip (peak) lineshape for constructive (destructive) interference and a dispersive lineshape otherwise. The reﬂectivity spectrum for a multilayer structure is in general diﬀerent from that for asingle interface, which is a result of multiple interfaces within the sample andinterference eﬀect from ﬁnite sample thickness. As a consequence, the Kerr effect spectra should also diﬀer from what is expected for a single interface. Kerr
spectra of semiconductor heterostructures depend not only on the nature of the
optical nonlinearities, but also on linear optical eﬀects such as the interference
from layer structures. Failure to take this into account may lead to ambiguities inthe extracted nonlinearities from the Kerr eﬀect and in distinguishing signals from
diﬀerent species. One therefore needs to have complete information of the sample
structures and reﬂectivity spectra for a correct interpretation of the observed Kerr
spectra.

43
The Faraday (Kerr) eﬀect is measured by using degenerate pump and probe
pulses with narrow linewidth. In Chapter 4, a two color experiment is performed
with the pump and probe pulses spectrally ﬁltered individually to enable independent tuning of their photon energies and spectral linewidths. By using nondegenerate pump and probe pulses, diﬀerent eﬀects from excitation and detection
can be separated, which is not possible with the current degenerate setup.

Chapter 4Interaction induced spin polarization in a 2DEG

In Section 2.4, the spin dynamics in bulk GaAs samples with diﬀerent doping densities have been studied. For future spintronic devices and quantum information processing, certain applications perform better in quantum conﬁnement
systems, such as quantum wells and quantum dots. In this chapter and the next
chapter, we will focus our study on the spin dynamics in quantum wells. The distinction of the spin dynamics in quantum wells from that of the bulk arises fromseveral aspects due to the well structure: the subband electronic structures, theenhancement of excitonic interactions, and the high mobility of the carriers. An
example is the slow-down of the valence band hole spin relaxation in the quantum
well compared with the bulk, due to the lifting of the degeneracy between theheavy-hole and light-hole valence bands in quantum wells.
Long spin relaxation times of the carriers are essential to applications in
spintronic devices and quantum information processing, and have been measured
in lightly n-doped semiconductor quantum wells [9]. These spin relaxation times
are measured by exciting and probing optical transitions. In those systems, theinterband optical properties are dominated by excitons and negatively charged
excitons (trions) at low temperatures, but the electron spin can persist long after
recombination of optically excited carriers. Coherent optical generation and manipulation of the electron spins require an understanding of how optical excitation

45
initiates and controls the electron spin polarization. The purpose of this work is toprovide a full physical picture of how light pulses polarize the spin of the electron
gas in weakly doped quantum wells.
We use a two-color transient Kerr rotation technique to study a lightly
n-doped CdTe quantum well, through which diﬀerent eﬀects from excitation anddetection can be well separated [40]. We show that the spin of the two-dimensional
electron gas (2DEG) can be polarized instantaneously after resonant excitationof spin polarized trions. Under resonant generation of excitons, the 2DEG canalso be spin-polarized through the formation of trions via exciton states. The
exciton spin polarization is transferred to the electron gas coherently, with the spin
coherence partially lost through exciton spin relaxation. To characterize the spinpolarization process of the electron gas, the interplay between the exciton, trionand 2DEG spins and their individual spin relaxation need to be taken into account.
The observed transient Kerr rotation signals are quantitatively explained with rateequations involving the spin populations of exciton, trion and the electron gas.
Trion spin relaxation also reveals information on the hole spin ﬂip process, giving
the excitation energy dependence of the hole spin relaxation [41].
We also examine the spin coherence of a 2DEG at diﬀerent density in a setof modulation-doped CdTe/Cd0.85 Mg0.15 Te quantum wells with diﬀerent doping
densities. The spin dephasing times T2∗ of the 2DEG are measured as a function of
the applied magnetic ﬁeld, and the dephasing times in the limit of zero magnetic
ﬁeld are obtained. The spin dephasing time as a function of the electron gas
density from 5 × 109 to 2.4 × 1011 cm−2 shows a non-monotonic behavior with amaximum at 8 × 1010 cm−2 [42, 43].

46
4.1

Exciton and trion spin dynamics

4.1.1

Exciton spin dynamics

Exciton spin dynamics have been addressed mostly in undoped quantum
wells. The exciton spin can relax via simultaneous spin ﬂips of the electron andhole in an exciton, i.e. exciton spin ﬂip as a whole [44, 45]. This mechanism has
been recognized as the relevant depolarizing mechanism of excitons under zero
applied magnetic ﬁelds. The hole in an exciton undergoes fast spin relaxation due
to strong mixing of the heavy- and light-hole valence band. A combination of theexchange interaction between the electron and the hole and the hole spin ﬂip itself
is responsible for the spin ﬂip of the exciton from +1 to -1 and vice versa.
The electron and hole in an exciton behave quite diﬀerently under an inplane external magnetic ﬁeld (Voigt geometry). In quantum wells, the heavy-holespins are constrained to lie normal to the quantum well plane by the eﬀects ofquantum conﬁnements, and are not observed to precess under moderate magnetic
ﬁelds due to the vanishing Zeeman splitting in the transverse ﬁelds [30]. On theother hand, the electron spin does undergo Larmor precession about the magnetic
ﬁeld axis. Assuming a long hole spin ﬂip time in the exciton, this results in an
oscillation between the bright and dark exciton populations [46], and a modiﬁed
spin precession frequency due to the electron-hole exchange interaction [47, 48].
The manifestation of exciton spin precession under the electron-hole exchangeinteraction relies on the stability of the hole-spin orientation within an exciton.
Besides the individual spin relaxation of the electron and hole in an exciton, theexciton spin ﬂip as a whole still plays a role in its spin dynamics under a magneticﬁeld.
In lightly n-doped quantum wells, the interband optical properties are still
dominated by excitonic transitions at low temperatures. Little work has been

47
done on the interactions between the spin polarization of excitons and that of
the electron gas in lightly n-doped quantum wells [49], despite the importance of
exciton transition on optical orientation of the electron gas.
4.1.2

Trion spin dynamics

A negatively charged exciton, a trion, was ﬁrst observed in a CdTe quantum
well [50]. It is formed by two electrons bound to one hole, analogous to a negatively
charged hydrogen ion (H− ) in vacuum. Trions play an important role in the optical
spectra of moderately doped quantum wells [51, 52]. II-VI semiconductor quantum
wells (such as the CdTe quantum wells used in this work) are an adequate choicefor the study of trions, because the Coulomb interaction in II-VI semiconductors
is typically twice that in III-V compounds, and because the trion binding energy
is expected to be an order of magnitude larger than that in bulk materials due tothe quantum conﬁnement.
The unique signature of a trion in spin dynamics is that it can only contribute to the spin dynamics by its hole spin, because a trion in its singlet state
consists of a hole and two electrons with opposite spins. However, upon opticalexcitation of spin-polarized trions, electrons of a certain spin polarization are removed from the 2DEG to form trions, polarizing the 2DEG with an opposite spin.
The spin orientation of the electron gas from resonant trion excitation has beenpreviously reported [53, 54], with the 2DEG instantaneously polarized during thetrion excitation. Trion recombination at a later time will release a spin polarized electron to the 2DEG, with the coherent spin polarization of the electron gas
modiﬁed depending on the trion recombination time, trion spin relaxation time
and the magnetic ﬁeld applied [55].

48
4.2

Sample characteristics and experiments
The samples are modulation doped CdTe/Cd0.85 Mg0.15 Te heterostructuregrown by molecular beam epitaxy on a (100) GaAs substrate with a 2 µm CdTe
buﬀer layer. The samples contain two CdTe quantum wells with widths of 120
and 70 Å, separated by a 50 Å Cd0.85 Mg0.15 Te barrier. The 120 Å well is one-side
doped by an iodine impurity layer 100 Å from the well. By choosing diﬀerent
thicknesses of the iodine layer, four samples are grown with 2DEG densities of
5 × 109 (nominally undoped), 8 × 1010 , 1.6 × 1011 and 2.4 × 1011 cm−2 in the 120Å well. The 70 Å quantum well remains undoped. All samples are grown on thesame wafer by the wedge doping technique to guarantee a consistent sample set[56]. We study the 120 Å quantum well only. First we will focus on the lightly
doped sample (ne = 8 × 1010 cm−2 ) for the study of spin interactions between
excitons, trions and the 2DEG. In Section 4.6, the electron spin coherence for all
four samples will be discussed.
The low temperature photoluminescence (PL) and photoluminescence excitation (PLE) spectra for the lightly doped quantum well (ne = 8 × 1010 cm−2 ) are
shown in Figure 4.1. The spectra exhibit exciton and trion lines with an energy
separation of 2.6 meV, which is the trion binding energy. There is a small Stokes
shift (∼0.5 meV) between the trion peaks in the PL and PLE spectra. The large
trion binding energy (as compared with III-V semiconductors) and the relatively
narrow linewidths of the resonances allow us to selectively generate excitons andtrions by means of optical excitation. The PLE spectrum shows that the peak
exciton absorption strength is about twice that of the trion. The PL spectrum,
however, is dominated by trion recombination. This indicates that the majority ofthe optically created carriers relax to the trion states. Under resonant excitationof excitons, the conversion of excitons to trions is much faster than the exciton

PL and PLE (arb. unit)

49

T

X

PL

1.604

PLE

1.606

1.608

1.610

1.612

Energy (eV)Figure 4.1: Low temperature PL and PLE spectra for the lightly n-doped CdTe
quantum well (ne = 8 × 1010 cm−2 ), showing the exciton (X) and trion (T) lines.

recombination.
We measure the transient Kerr rotation with an applied magnetic ﬁeld of
2 T applied normal to the quantum well growth direction (Voigt geometry) [30].
Pump and probe pulses are spectrally ﬁltered (resulting in ∼5 ps pulses) to enable
independent tuning of their photon energies. The spectral linewidths of the pulses
are adjusted to be ∼0.2 nm for spectrally resolved measurements. By performing
two-color measurements, not only can we selectively excite diﬀerent populations,but we are also able to compare experimental results under diﬀerent excitationenergies with a ﬁxed probe energy. The pump and probe photon energies arescanned near the vicinity of the exciton and trion resonances of the 120 Å quantum well. Average intensity in the pump is about 1 W/cm2 (except as speciﬁed
otherwise in the text) and 0.1 W/cm2 in the probe beam. The pump (probe)
beam is focused onto the sample with a spot diameter of ∼ 100 µm (50 µm). All
experiments are performed at 4 K.

50
4.3

Spin dynamics of excitons, trions and 2DEG: experiments
In this section, we spectrally study the spin dynamics in the lightly n-doped

CdTe quantum well (ne = 8 × 1010 cm−2 ). Transient Kerr rotation signals under transverse magnetic ﬁeld for resonant excitations of excitons and trions areshown in Figure 4.2(a). The probe photon energy is tuned at the exciton resonance (1.6093 eV) for both pump energies. The oscillation signals correspondto the Larmor precession of carrier spins in an external magnetic ﬁeld. A precession frequency of 43.3 GHz is measured, corresponding to a Landé g factor
of |g| = 1.55. The Kerr rotation signal under exciton generation exhibits two
exponential decays of the oscillation with very diﬀerent time constants. A nonoscillating decay component in addition to the double exponential is observed
under trion excitation. We perform a least-square ﬁt to the transient signals withthe form θK (t) = (A1 e−t/τ1 +A2 e−t/τ2 )cos(ωL t), where ωL is the Larmor precession
frequency, τ1 and τ2 (A1 and A2 ) are the long and short spin dephasing times (precession amplitudes), respectively. A third term of A3 e−t/τ3 is included in the ﬁt to
account for the non-oscillating decay under trion excitation. The time constants
retrieved from the ﬁt are also displayed in Figure 4.2(a). Both signals show along-lived oscillation component that lasts over 1 ns. As this time constant exceeds lifetimes of excitons and trions (measured to be < 100 ps by a streak camera[52]), we conclude that this is the spin Larmor precession of the 2DEG. The shorttime constants should be related to the exciton and trion spin dynamics, which
will be discussed later.
To study how the probe photon energy aﬀects the transient Kerr signal, we
scan the probe energy with the pump photon energy ﬁxed on either the excitonor trion resonance. The Kerr rotation amplitude of the 2DEG spin polarization(A1 ) is plotted in Figure 4.2(b) as a function of probe energy. The similarity

K err rotation (arb. unit)

51

3=1125 ps

2

1

0

=35 psPump trion=1140 ps

2

100

200

Time (ps)Pump exciton

300

400

Pump exciton
Pump trion

1

(b)

T
0

X
Spin dephasing time (ns)K err rotation (arb. unit)(a)=33 ps-1-2

Pump exciton
2.0
1.5
1.0
0.5

T
1.604

X

1.608

1.612

Energy (eV)

1.604

1.606

1.608

1.610

1.612

Probe energy (eV)Figure 4.2: (a) Transient Kerr rotation signals under resonant excitations of excitons and trions, with probe photon energy at the exciton resonance. The time
constants shown are obtained from nonlinear ﬁtting. (b) The Kerr rotation amplitude of the 2DEG spin polarization (A1 ) as a function of probe photon energy
under resonant excitations of excitons (X) and trions (T). Inset: probe energy
spectrum of the electron spin dephasing time (τ1 ), under resonant exciton generation. B = 2 T and T = 4 K.

of the probe spectra obtained with exciton and trion excitation conﬁrms spinpolarization of the electron gas for both excitation conditions. The spectra also
indicate a diﬀerence in response sensitivity between detection through exciton andtrion resonances (given that the carrier density generated at exciton resonance istwice that at trion resonance). The remaining experiments in this chapter will

52
be performed with the probe energy ﬁxed, so that a comparison of amplitudes
resulting from diﬀerent excitation energy is possible. The spectra in Figure 4.2(b)show a peak shape for the exciton resonance and a mixture of dispersive andpeak lineshape for the trion resonance. The diﬀerent lineshapes of the resonances
result from diﬀerent nonlinear responses of excitons and trions from spin-polarized
electrons [33]. This is supported by a recent work on CdTe-based quantum wells[57], where the spin-polarized electron gas is shown to eﬀectively screen excitons,
but not trions. In the inset of Figure 4.2(b), the electron spin dephasing time(τ1 ) is measured as a function of the probe photon energy, with the pump energy
on the exciton resonance (similar probe spectrum is observed for resonant trion
generation, not shown). The electron spin dephasing time obtained by probing
the trion resonance is almost two times longer than that by probing the exciton
resonance. The observed results may be related to the inhomogeneous distributionof the electrons, while the trion resonance is more sensitive for detecting the
localized electrons [58], and the exciton resonance for detecting weakly localized
electrons. The weakly localized electrons may experience diﬀerent spin relaxation
mechanism than the localized ones, resulting in a shorter spin dephasing time. A
similar behavior has been recently reported [40].
To further explore the short time behavior in the transient Kerr rotation
signals and the origins of the electron gas spin polarization through optical excitation, we study the Kerr rotation spectrum with varying pump photon energy.
Speciﬁcally, we selectively generate exciton and trion populations, while the probe
photon energy remains ﬁxed at the exciton resonance (1.6093 eV). The spectral
data are again analyzed by least-square ﬁt to a double exponential, and we plot
the Kerr rotation amplitude for the slow (A1 ) and fast decay components (A2 )as a function of pump energy, as shown in Figure 4.3(a). The amplitude of thefast decay component exhibits a sharp peak at the exciton energy, with a FWHM

53
linewidth (0.7 meV) agrees well with the exciton linewidth in the PLE spectrum(0.8 meV). We assign the fast oscillation decay to exciton spin precession and dephasing. The decay time (τ2 ) is a combination of the exciton spin dephasing time
and trion formation time. The relatively weak signal of A2 near the trion energy isprobably related to the ionization of trions into excitons, as the trion itself cannot
contribute to the oscillatory signal. The amplitude of the slow decay component,
which is proportional to the 2DEG spin polarization, is enhanced at both excitonand trion resonances. This enhancement indicates spin polarization of the electron gas through optical excitation of excitons. We propose that excitons polarize
the spin of the 2DEG through their conversion to trions, which preserves the spinorientation, and polarizes the 2DEG. The spin-ﬂip scattering of excitons by single
electrons, for instance, could also polarize the 2DEG. However, since the majorityof the optically excited excitons relax to the trion states (evidenced in the PL
spectrum), we believe the trion formation process dominates.
We obtain further information on the exciton and electron gas spin polarization by varying the pump excitation intensity. In Figure 4.3(b), the excitation
intensity dependence of the Kerr rotation amplitudes is shown for resonant excitation at excitons and trions. The fast decay amplitude (A2 ) for resonant excitation
at the exciton, related to the exciton spin polarization, increases monotonicallywith intensity, and can be well ﬁtted with a saturation function P1 I/(P2 +I), with
P1 , P2 constants and I the intensity. The electron gas spin polarization (A1 ), on
the other hand, actually decreases at high intensity for excitation at both excitonand trion resonances. The absorption strength at the trion energy is about one
half of that at the exciton energy. With this correction [by using diﬀerent scalesof excitation intensity for the exciton and trion in Figure 4.3(b)], we observe very
similar intensity dependence of the 2DEG spin polarization for resonant excitationof excitons and trions. By exciting trions directly, the electron spin polarization isKerr rotation (arb. unit)

54

3

X

(a)

A

1

2

A

1

0
1.606

0

Kerr rotation (arb. unit)

2

T

1.608

1.610

Pump energy (eV)

10

20

30

(b)

40
A

(exciton)A(trion)A(exciton)

1

3

1.612

1
2

2
x0.1

1

0

0

5

10

15

20
2

Excitation intensity (W/cm )Figure 4.3: (a) Kerr rotation amplitudes as a function of pump photon energy forthe slow (A1 ) and fast decay components (A2 ). (b) Excitation intensity dependence of the Kerr rotation amplitudes (A1 and A2 ) under resonant excitation at
excitons and trions, with A2 scaled by 0.1 for clarity. The amplitude A2 for resonant exciton generation is ﬁtted with a saturation function (dash line). B = 2 T
and T = 4 K. Probe energy is tuned at exciton resonance.

about twice that from exciton generation [as indicated by the peak A1 amplitudes
in Figure 4.3(b) for the exciton and trion]. We will show later that this is due to asimilar trion formation time and exciton spin dephasing time, and spin coherenceis partially lost before an exciton forms a trion.

Electron spin dephasing time (ns)55

Pump exciton
Pump trion

1.0

0.8

0.6

0.4
0

10

20

30

40

Temperature (K)Figure 4.4: Electron spin dephasing time (τ1 ) as a function of temperature, with
τ1 measured under resonant excitation of excitons and trions. The dash line(τ1 ∝ T −0.4 ) is a nonlinear ﬁt to the data under exciton generation. B = 2 T.

We measure the temperature dependence of the Kerr rotation signal bypumping at the exciton and trion resonances (with probe energy ﬁxed at theexciton resonance). We do not observe signiﬁcant temperature dependence ofthe exciton dephasing time (τ2 ) up to 40 K. The electron spin dephasing time(τ1 ), however, decreases rapidly with increasing temperature, which is shown in
Figure 4.4. The τ1 measured under resonant excitation of both excitons andtrions are plotted in Figure 4.4, and close values under both excitation conditions
are obtained. A least-square ﬁt of the data under exciton generation yields atemperature dependence τ1 ∝ T −0.4 . It was suggested that both the D’yakonovPerel’ and Elliott-Yafet spin relaxation mechanisms (discussed in detail in thelast chapter) are expected to give a decreased electron spin relaxation time withincreasing temperature in quantum wells [59]. Due to a lack of a comprehensive
theory on temperature dependent spin relaxation in quantum wells, we are unable
to distinguish the two mechanisms in our sample.

56
In Figure 4.2(a), a non-oscillating component at short time delay is observed
for trion excitation, with a decay time of τ3 = 33 ps. The trion only contributes tothe spin dynamics through its hole spin, because of the two opposite electron spinsin a trion. The non-oscillating component observed in the transient Kerr signal
is therefore assigned to the hole spin ﬂip in trions, with τ3 the hole spin ﬂip time[54]. We measure the hole spin ﬂip time as a function of the pump photon energy.
For this measurement, the probe photon energy is set near the trion resonanceat 1.6066 eV, which gives the most sensitivity for measuring the hole spin ﬂip.
The pump energy is scanned, with its excitation intensity varied to ensure roughly
equal optical carrier density (calibrated through the PLE spectrum). The resulting
spectrum for the hole spin ﬂip time is shown in Figure 4.5(a). The hole spin
ﬂip time is maximum around the trion resonance, and decreases rapidly withincreasing excitation energy. Note that the hole spin ﬂip time measured throughtransient Kerr rotation belongs to that in trions. For non-resonant excitation oftrions, the measured time sets the upper limit of the hole spin ﬂip time, and needs
to be taken with caution. Hole states are, in general, an admixture of various spinstates, and any energy or momentum relaxation process will lead to their spin
relaxation [44, 49]. This explanation supports our experimental results, wherea fast hole spin ﬂip is observed for hot hole generation. We note that the holespin ﬂip under resonant excitation of excitons is much faster than that under
excitation of trions [also illustrated in Figure 4.2(a) as a vanishing non-oscillation
decay for exciton generation]. We believe the enhancement of the hole spin ﬂip
rate in excitons is due to exchange interactions between the electron-hole pairs in
excitons, as detailed in Reference [45]. With the two electrons in a trion having
opposite spins, the exchange interaction vanishes for the hole, giving a longer holespin ﬂip time for trions.
We also measure the hole spin ﬂip time as a function of temperature and57

Figure 4.5: The hole spin ﬂip time (in trions) as a function of (a) excitation energy,(b) temperature and (c) excitation intensity. B = 2 T.

excitation intensity, as shown in Figure 4.5(b) and (c). For those measurements,
the pump photon energy is on resonance with trions, while the probe energy is
set near the trion resonance (1.6066 eV). The amplitude of the non-oscillatory
signal (A3 ), that originated from the hole spin ﬂip in trions, drops substantiallywith increasing temperature, and we are unable to retrieve information of holespin relaxation above 12 K. The hole spin ﬂip time does not change too much inthe measured temperature range. It, however, decreases strongly with increasing
excitation intensity, as shown in Figure 4.5(c). Increasing the optically injected
carrier density will increase the momentum scattering rate of the carriers. We cantherefore exclude the D’yakonov-Perel’ spin relaxation mechanism for hole spins,
which gives a spin relaxation rate that is inversely proportional to the momentum

58
scattering rate. We believe the hole spin relaxation follows a Elliott-Yafet spin
relaxation mechanism, which arises from momentum scattering during collisions
with phonons or impurities.
4.4

Spin dynamics of excitons, trions and 2DEG: analysis
In this section, we theoretically study the spin dynamics in a lightly n-doped

quantum well system. We derive rate equations involving the spin populations of
exciton, trion and the 2DEG. The individual spin relaxation of excitons, trions
and the 2DEG, as well as their spin interactions, is taken into account. The
results concentrate on the spin polarization of the electron gas through opticaltransitions (exciton and trion states). A ﬁnite transverse magnetic ﬁeld (Voigt
geometry) is included to represent the experimental condition. We are able toquantitatively account for the experimental results in the last section by choosing
the right parameters.
4.4.1

Trion spin dynamics

We start with the case of resonant trion excitation. We consider a systemof trions and 2DEG after optical generation of trions (ionization of trions into
excitons is not considered). In Figure 4.6(a), a schematic diagram reﬂects theinterplay between trions and 2DEG. Without any loss of generality, we assume a
σ + polarized pulse at t = 0 with a spin + 12 electron will generate a spin + 32 trion,
and leave behind a spin − 21 electron in the 2DEG:hνσ+ + e+ 1 =⇒ T+ 3 .
2

2

(4.1)

The trion spin will not precess under the magnetic ﬁeld, because of the two opposite spins of electrons inside a trion, and a negligible Zeeman splitting of theheavy-hole spins in the transverse direction. Instead, the trion spin will relax with59
its hole spin ﬂip rate and decay exponentially. We write down the rate equations
for the densities of spin + 23 trion (N+T ) and spin − 23 trion (N−T ):
dN+T
NT
NT
NT
= − T+ − +T + −T ,
dt
τr
2τs
2τs
T
T
T
dN−
N−
N+T
N−
=− T − T + T,
dt
τr
2τs
2τs

(4.2)

with τsT the trion spin relaxation time and τrT the trion recombination time. The
total trion density then evolves with trion recombination as
N T (t) = N+T (t) + N−T (t) = N+T (0) exp(−t/τrT ),(4.3)

while the spin polarization of trions (the density diﬀerence between the spin + 32
and − 23 trions) evolves as∆N T (t) = N+T (t) − N−T (t) = N+T (0) exp(−t/τ T ).

(4.4)

Here N+T (0) is the initial trion density after σ + pulse excitation. The decay rate
1/τ T = 1/τsT + 1/τrT combines the decay of ∆N T through trion spin relaxation(1/τsT ) and trion recombination (1/τrT ). The non-oscillating decay component
in Figure 4.2(a) corresponds to trion spin decay described in Equation 4.4, thus
τ T = 33 ps.
The spin polarization of the 2DEG under trion excitation is generated
through two diﬀerent sources, initial polarization upon trion formation (instantaneously during pulse duration) and possible spin polarization through trion
recombination (release of a spin polarized electron to the 2DEG after trion recombination). The former, as described in Equation 4.1, creates a spin − 12 electronin the 2DEG after σ + pulse. The spin polarization of the 2DEG generated upontrion formation at t = 0 will precess around the magnetic ﬁeld, with the projectionin the quantum well growth direction∆N1e (t) = −N+T (0) cos(ωL t) exp(−t/τse ).

(4.5)

60

Figure 4.6: (a) A schematic diagram displaying the interplay between the spin
populations of trions and 2DEG under resonant trion excitation. (b) Calculatedtransients of the spin polarization of the 2DEG, trions and the combination of thetwo.

Here τse is the electron spin dephasing time, and ωL is the electron Larmor precession frequency. As described above, the trion recombination can also contributeto the electron gas spin polarization by returning spin polarized electrons backto the 2DEG after recombination. This is a reverse process of that shown inEquation 4.1:
T+ 3 =⇒ hνσ+ + e+ 1 .
2

2

(4.6)

This spin polarization, however, could be in phase or out of phase with thatgenerated through trion formation, depending on whether the trion spins haverelaxed and on the phase of the electron spins precessing under the magnetic ﬁeld.

61
The 2DEG spin polarization generated by trion recombination during [t′ , t′ + dt′ ]isd∆N2e (t′ ) = (∆N T (t′ )/τrT )dt′ ,(4.7)

and the 2DEG spin polarization at t is∆N2e (t) =Z t
0(∆N T (t′ )/τrT ) cos(ωL (t − t′ )) exp(−(t − t′ )/τse )dt′ .

(4.8)

The total 2DEG spin polarization ∆N e (including contributions from both trion
formation and recombination) is∆N e (t) = ∆N1e + ∆N2e = B1 cos(ωL t + φ) exp(−t/τse ) + B2 exp(−t/τ T ),

(4.9)

with
v
u
u
τT
T
B1 = −N+ (0)t1 − (2 − T )

τr

τT
,
((ωL τ T )2 + 1)τrTωL (τ T )2,((ωL τ T )2 + 1)τrT − τ T
τTT.
B2 = −N+ (0)((ωL τ T )2 + 1)τrT

φ = arctan(4.10)The results in Equations 4.9 and 4.10 are simpliﬁed by assuming a much longer
electron spin dephasing time (τse ) than the trion time constants (τsT and τrT ), which
is suﬃcient for a τse over 1 ns in our sample. The relative contribution of the trion
recombination to the 2DEG spin is related to the term τ T /(((ωL τ T )2 +1)τrT ), which
will only be signiﬁcant when the trion spin orientation is partially maintained
during its lifetime (τsT > τrT ), and the trion time constants are much shorter
than the electron precession period (ωL τ T < 1), so that the electron spin is notout of phase with that of the trions. In our sample, the oscillation period is 23
ps for a magnetic ﬁeld of 2 T, which is comparable to the trion time constants(τsT = 33 ps). The trion recombination therefore plays a negligible role in thespin dynamics of the electron gas in our sample (∆N2e ≈ 0), and the 2DEG spin

62
is generated mainly from trion formation (∆N e ≈ ∆N1e ). The total 2DEG spinpolarization therefore follows that described in Equation 4.5. The spin dynamics of
2DEG, trions and the combination of the two (which corresponds to the transient
Kerr rotation signal measured) are calculated and plotted in Figure 4.6(b), with
τ T = 33 ps, τse = 1100 ps and ωL = 2π 43.3 GHz. The combined spin polarization
represents well the measured transient Kerr rotation signal for trion excitation
shown in Figure 4.2(a).
4.4.2

Exciton spin dynamics

The exciton spin precession and dephasing under a transverse magnetic ﬁeld
is illustrated in Figure 4.2(a) as the fast oscillating decay. In quantum wells, theheavy-hole spins are constrained to lie normal to the quantum well plane by effects of quantum conﬁnements, and are not observed to precess under moderatemagnetic ﬁelds [30]. The spin of the electron in an exciton does undergo Larmor
precession. With a long hole spin ﬂip time in the exciton, this results in an oscillation between the bright and dark exciton populations [46], and a modiﬁed spin
precession frequency due to electron-hole exchange interaction.1

To understandexciton spin relaxation, both the single-particle spin ﬂips of the electrons andholes in excitons, and the exciton spin ﬂip as a whole (simultaneous spin ﬂip ofelectrons and holes) need to be taken into account. By solving the rate equations
for the spin polarized electrons and holes in excitons (not shown), we obtain theXX) and spin − 12 (Ne−) electrons in excitons as
density of the spin 12 (Ne+









1
−t
+ 1
1
X
Ne+
(t) = N+X (0)e τ XT τrX
2

1
+ 1
−t
1
X
Ne−
(t) = N+X (0)e τ XT τrX
2

1



−t

1 − cos(ωL t)e−t

1 + cos(ωL t)e1
1
X + τXτess1
1
X + τX
τes
s

!

,

!

,

(4.11)

We do not observe any difference of oscillation frequencies between the exciton and electron
spin precession for magnetic fields ranging from 0.5 to 5.0 T. This supports the argument thatthe hole spin in the exciton relaxes very quickly in our sample.

63
X
X
and the density of the spin 23 (Nh+) and spin − 32 (Nh−) holes in excitons as



−t
1
X
Nh+
(t) = N+X (0)e
2

1
+ 1X
τ XT
τr



−t
1
X
Nh−
(t) = N+X (0)e
2




1
1
+
−t
τX
τsX


hs
,
1 + e




1
1
1
+ X
−t
+ 1X 
τX
τs

τ XT
τr
hs
.
1 − e(4.12)X
X
Here N+X (0) is the initial exciton density after σ + pulse excitation, τes
(τhs
) is

the electron (hole) spin ﬂip time in exciton, τsX is the exciton spin ﬂip time (asa whole), τ XT is the trion formation time, and τrX is the exciton recombination
time. Both the electron and hole in an exciton lose their populations throughexciton recombination and conversion of excitons into trions. They will lose theirspins through exciton spin ﬂip and their individual spin ﬂips. The electron in an
exciton will also precess in a magnetic ﬁeld, but the hole will only undergo spin
ﬂip. We consider the diﬀerence between the bright and dark excitons, because
the measured transient Kerr rotation signal of the exciton corresponds to the spin
population of the bright excitons only. The density of the spin 1 bright excitons
X
X
X
X
X
X
Nbr+
∝ Nh+
Ne−
and that of the spin -1 bright excitons Nbr−
∝ Nh−
Ne+
. The

spin polarization of the bright excitons is therefore
1
X
X
X
∆Nbr
(t) = N+X (0)(cos(ωL t)e−t/τ1 + e−t/τ2 ),2(4.13)

with
1
1
1
1
1
= X + X + XT + X ,
X
τ1
τes
τs
τ
τr
1
1
1
1
1
= X + X + XT + X .
Xτ2τhs τsττr(4.14)

We note that Equation 4.13 consists of an oscillating decay term and a nonoscillating decay term. From comparison with the experimental results [Figure 4.2(a)], we ﬁnd τ1X = 35 ps, and τ2X very small with no observable nonoscillation decay (τ2X < 10 ps, half period of the oscillation). This indicates a64very short hole spin ﬂip time in our sample. Both the exciton spin ﬂip as a wholeX(τsX ) and the electron spin ﬂip in excitons (τes) are included in our model (Equa-tions 4.14). However, they can not be separately determined based on the current
experiments. We therefore treat their combined eﬀects as an overall exciton spindephasing time τsX . Equations 4.14 may be rewritten as1
1
1
1= X + XT + X ,
X
τ1
τs
τ
τr
1
1
∼ X.
X
τ2
τhs

(4.15)

The calculated TKR from exciton spin dephasing is plotted in Figure 4.7(b),assuming τ1X = 35 ps, τ2X = 0 ps and ωL = 2π 43.3 GHz.
The spin interactions between excitons and the 2DEG are proposed to bethrough trion formation via exciton states. The spin interplay between excitons,
trions and electrons under resonant exciton excitation is displayed in Figure 4.7(a).
Any time a trion is formed from an exciton that consists of a − 21 electron, aspin polarization of − 12 is added to the 2DEG. Both bright and dark excitons
can form trions, and contribute to the 2DEG spin polarization. The hole spinorientation in the excitons does not matter in the spin polarization of the 2DEG,
because it is the spin coherence of the electron in the exciton that is passed to the
2DEG. By considering the electron spin polarization in excitons, as described by
Equation 4.11, the 2DEG spin polarization generated by trion formation during[t′ , t′ + dt′ ] is determined to be′Xd∆N e (t′ ) = −(N+X (0)/τ XT ) cos(ωL t′ )e−t /τ1 dt′ .

(4.16)

The cos(ωL t′ ) term in Equation 4.16 indicates a coherent spin transfer from exciton
to the 2DEG, i.e. the spin of the electron gas will precess in phase with the exciton
spin under the magnetic ﬁeld. The resulting 2DEG spin polarization at t under

65

Figure 4.7: (a) A schematic diagram displaying the interplay between the spin
populations of excitons, trions and 2DEG under resonant exciton excitation. (b)Calculated transients of the spin polarization of the excitons, 2DEG and the
combination of the two.

resonant exciton generation is thenXe

∆N e (t) = (N+X (0)τ1X /τ XT ) cos(ωL t)(e−t/τ1 − e−t/τs ).

(4.17)

Equation 4.17 is obtained by assuming long electron spin dephasing time (τse )and negligible electron spin polarization from trion recombination process. The
2DEG spin precession is shown to ﬁrst rise due to trion formation, and then decay
according to the electron spin dephasing. The calculated 2DEG spin dynamics, aswell as the overall signal including the exciton spin polarization (corresponds tothe experimental results) is plotted in Figure 4.7(b). The overall spin polarization

66
agrees well the measured TKR signal shown in Figure 4.2(a).
4.4.3

Spin coherence loss via exciton spin relaxation

The 2DEG is spin-polarized through trion formation for resonant excitationsof both excitons and trions. The degree of spin polarization, however, diﬀers forthe two excitation conditions. Under resonant trion excitation, a fast hole spin,or as in the case of our experiments, a high precession frequency of the electrongas, favors a complete transfer of the spin coherence to the 2DEG (discussed indetail in Section 4.4.1). An incomplete transfer of the spin coherence, however,occurs under resonant exciton generation, resulting from exciton spin dephasing.
The 2DEG spin polarization under resonant excitation of trion and exciton are
described in Equations 4.5 and (4.17), respectively, and the ratio between the two
at long delay time is
N+T (0)N+T (0) τ XT
τ XT
≈
),
(1
+
N+X (0) τ1X
N+X (0)τsX(4.18)

under the assumption that the exciton recombination time is much longer than
the trion formation time (evidenced from the small exciton recombination signal
in PL spectrum). The ratio in Equation 4.18 compares the 2DEG spin polarization at a long delay time for exciton and trion excitations, which corresponds toamplitude A1 retrieved from the experimental results. With an equal number ofcarriers generated initially (N+T (0) = N+X (0)), the 2DEG spin polarization undertrion excitation is (1 + τ XT /τsX ) times that under the exciton generation. In Figure 4.3(b), the peak 2DEG polarization for trion generation is about 2 times thatfor exciton generation, which indicates similar time constants for the trion formation time and exciton spin dephasing time (τ XT ≈ τsX ). In our sample, excitons
form trions at the same rate as they lose their spin coherence through exciton spin
relaxation. Therefore, only half of the spin coherence is passed to the 2DEG.

67
4.5

Spin dynamics in highly doped sample
In the previous sections, we study the lightly doped CdTe sample (ne =8 × 1010 cm−2 ), where the interband optical transitions are still dominated by
the excitonic behaviors. In this section, we study the spin dynamics in a highly
n-doped CdTe quantum well (ne = 2.4 × 1011 cm−2 ). The sample belongs to a setof samples grown on the same wafer, and its characteristics are described in Section 4.2. In Figure 4.8(a), the PL and PLE spectra of the sample is shown, withbroad, asymmetric peaks replacing sharp excitonic peaks shown in the spectra forthe lightly doped sample (Figure 4.1). The PLE intensity rises relatively quickly
on the low energy side and falls oﬀ slowly on the high energy side, to merge into thecontinuum absorption. The enhancement of absorption at 1.6081 eV is identiﬁed
as Fermi edge singularity, resulting from multiple scattering processes involving
electrons near the Fermi energy [60]. The PL spectrum is also broadened as aresult of large electron density existing in the quantum well, with the spectrum
peaked near the bottom of the bandgap. In Figure 4.8(b), the Kerr rotation amplitude is measured as a function of probe photon energy under a pump excitationenergy of 1.6081 eV. The amplitude is measured as the peak-to-dip amplitude of
the oscillation for a pump-probe delay near 30 ps. The spectrum in Figure 4.8(b)shows a dispersive lineshape of a single resonance, centered near the energy of thepeak in the PLE spectrum.
Transient Kerr rotation signal is measured by tuning the pump photon energy to 1.6075 eV and probe energy to 1.6062 eV, which is shown in Figure 4.9(a).
The Kerr rotation signal again exhibits two exponential decays of the oscillation with diﬀerent time constants, as measured with the lightly-doped sample.
We ﬁt transient data with a double exponential oscillation decay, (A1 e−t/τ1 +A2 e−t/τ2 ) cos(ωL t), with τ1 and τ2 (A1 and A2 ) the long and short spin dephasing

PL and PLE (arb. unit)(a)1.6081 eV

PLE

PL

Kerr rotation (arb. unit)

68

(b)
10

0

1.6081 eV

-10

-20

0

1.600

1.605

1.610

Energy (eV)

1.615

1.605

1.610

Probe energy (eV)1.615

Figure 4.8: (a) Low temperature PL and PLE spectra for the highly doped CdTe
quantum well (ne = 2.4 × 1011 cm−2 ). (b) The Kerr rotation amplitude as afunction of probe photon energy under a pump excitation energy of 1.6081 eV.
Arrows in both ﬁgures indicate the energy of the peak in the PLE spectrum.

times (precession amplitudes), respectively. The time constants retrieved fromthe ﬁt are also displayed in Figure 4.9(a). It is a little surprising to observe a double exponential decay of the Kerr signal, since the excitonic transitions alreadydisappear for this highly doped sample, and the optical properties of the sample
are dominated by that of the 2DEG. A spectral study of the Kerr rotation is performed for the pump photon energy, and the amplitude spectra of A1 and A2 areplotted in Figure 4.9(b). The probe photon energy is ﬁxed at 1.6062 eV for thevarious pump energies. Both the A1 and A2 spectra show a similar shape as thatof the PLE spectra. This indicates that both Kerr rotation components probably
originate from the 2DEG spin polarization. We propose that the observed Kerr
rotation signals correspond to the spin relaxation of the 2DEG, with its spin relaxing with diﬀerent rates at diﬀerent stages. The short decay time (τ2 = 71 ps)
is very close to the hole lifetime measured by time-resolved PL experiments in asimilar sample [52]. It is, however, well beyond the typical hole spin ﬂip time inKerr rotation (arb. unit)

Kerr rotation (arb. unit)

69

(a)
2

=71 ps1=453 ps

0

0

200

400

600

800

1000

Time (ps)
(b)

A

1

A

2

0

1.606

1.608

1.610

1.612

Photon Energy (eV)1.614

Figure 4.9: (a) Measured transient Kerr rotation signal with pump photon energy at 1.6075 eV and probe energy at 1.6062 eV. The time constants shown are
obtained from nonlinear ﬁtting. (b) The Kerr rotation amplitude of A1 and A2(described in the text) as a function of pump photon energy, with probe energy
at 1.6062 eV. B = 1 T and T = 4 K.

quantum wells (several ps). We therefore believe the hole recombination corresponds to the initial electronic spin relaxation observed in the Kerr rotation signal.
With the presence of spin unpolarized holes, the electron-hole spin exchange interaction contributes to a faster electronic spin relaxation rate. The electron spin
relaxation slows down after the recombination of the spin unpolarized holes, anda longer electronic spin relaxation time is observed (τ1 = 453 ps).

70

Figure 4.10: Low temperature (a) PL and (b) PLE spectra for the 120 Å CdTe
quantum wells with 2DEG densities of 5 × 109 , 8 × 1010 , 1.6 × 1011 and 2.4 ×
1011 cm−2 . T = 4 K.

4.6

Electron spin coherence vs. 2DEG density
We study the electron spin coherence at diﬀerent 2DEG density by using a

set of modulation-doped CdTe/Cd0.85 Mg0.15 Te quantum wells with diﬀerent doping densities (see Section 4.2 for sample characteristics). For this study, only thespin relaxation of the electron gas is concerned, with the short time spin relaxation(such as that of the excitons and trions) ignored. We measured the low temperature PL spectra [Figure 4.10(a)] of the quantum wells. For the nominally undoped
sample with 5 × 109 cm−2 both the trion and exciton lines are clearly visible as
the two peaks separated by ∼ 3 meV, which is the trion binding energy. With
increasing carrier density the exciton loses intensity, while the trion resonancebroadens and evolves into a broad emission band with a linewidth approximately
equal to the 2DEG Fermi energy [60]. Because all the quantum well samples are

71

3.5
11

2.4x10

-2

cm

9

5x10

-2

cm

-1

Spin dephasing rate 1/T* (ns )

3.0

11

1.6x10

-2

cm

2

2.5

2.0

10

8x10

1.5

-2

cm

6

1.0

5

0.0

4
3

2

*

T (ns)

0.5

2
1

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

11

-2

Electron density (x10 cm )

0

1

2

3

4

5

6

T)

Magnetic field (Figure 4.11: Spin dephasing rate 1/T2∗ as a function of the magnetic ﬁeld of
the 120 Å CdTe quantum wells with various electron densities. Lines are ﬁts toEquation 4.19. Inset: Spin dephasing time T2∗ vs. 2DEG density at zero magnetic
ﬁeld. T = 2 K.

grown on thick, opaque GaAs substrate, direct absorption measurement is notpossible. Instead, we measure the PLE spectra of the samples, which are shownin Figure 4.10(b). With increasing electron density, the excitonic absorption peaks(both exciton and trion) evolves into a broad, asymmetric peak, corresponding to
the absorption of a 2DEG [60].
The transient Kerr rotation signals are measured by using degenerate pumpand probe pulses with spectral width of 0.5 nm. The pump and probe photon

72
energy is tuned near the bandgap of the quantum wells. A transverse magnetic
ﬁeld up to 6 T is applied. After the initial polarization, the spin of the 2DEG
precesses under the magnetic ﬁeld, and decays on a nanosecond timescale. This
decay time corresponds to the electron spin dephasing time, T2∗ , under a transverse
magnetic ﬁeld. For measuring reliable values of T2∗ , it is important to analyze thecontribution to spin dephasing arising from a spread of the electron g factor, which
results in an inhomogeneous dephasing of the electron spin under magnetic ﬁeld.
This is shown as a linear dependence of the spin relaxation rate on the magneticﬁeld,
1/T2∗ (B) ≈ 1/T2∗ (0) + ∆gµB B/2h̄,(4.19)

with µB the Bohr magneton, h̄ the Planck’s constant, and ∆g a Gaussian varianceof the g factor [24]. Our goal is to compare the electron spin dephasing times in
the limit of zero magnetic ﬁeld, T2∗ (0). Figure 4.11 shows the magnetic ﬁeld
dependence of the electron spin dephasing rate for all four quantum wells. T2∗
values are derived by a single exponential ﬁt to the Kerr rotation transients atdelays longer than 1 ns to measure only 2DEG dynamics. The lines in Figure 4.11
are ﬁts to Equation 4.19. These ﬁts provide the zero ﬁeld spin dephasing time,
which is plotted as a function of the 2DEG density in the inset of Figure 4.11. A
non-monotonic behavior with increasing electron density can clearly be seen. T2∗
ﬁrst increases with electron density, then goes through a maximum of 5.5 ns at
8 × 1010 cm−2 , and ﬁnally decreases to 0.9 ns at high carrier densities.
Although a quantitative description of the non-monotonic behavior is notpossible due to the limited numbers of the available samples (2DEG densities) (asystematic study on electron density will be presented in Chapter 5), the density
dependence of the electron spin coherence can be nevertheless qualitatively understood: at low carrier densities most electrons in the quantum well are localized. In

73this regime the dominant spin dephasing mechanism is the hyperﬁne interaction
with lattice nuclei spins, which generate an eﬀective “hyperﬁne magnetic ﬁeld”[61, 62]. The T2∗ increases as the electron becomes more delocalized. When the2DEG density crosses the metal-insulator transition [63, 8], the spin dephasingtime decreases drastically due to the D’yakonov-Perel’ mechanism [64]. Similar tobulk material [8], this overall behavior leads to an optimum doping concentrationin quantum wells for achieving longest spin dephasing times.

Chapter 5
Effects of disorder on electron spin dynamics in semiconductor
quantum wells

Successful implementation of spin-based electronics requires preservation of
the electron spin coherence. Extending spin coherence times in semiconductors istherefore of great interest for future practical devices. In the previous chapters,
a nonmonotonic dependence of the electron spin coherence time on the doping
density in n-doped semiconductors has been established. A long spin coherencetime is observed for either lightly n-doped bulk or lightly n-doped quantum well
samples, which has also been observed in Reference [9] and [24]. It has been shownthat the spin dephasing rate of the electrons is suppressed close to the metalinsulator transition [8, 63]. Below and above the critical electron density, diﬀerent
spin dephasing mechanisms dominate and are detrimental to spin preservation[22]. A systematic study of spin dephasing on the electron density is therefore
very useful, not only for ﬁnding the maximum spin coherence time, but also
for a better physical understanding of the various spin dephasing mechanisms at
diﬀerent carrier densities.
Samples with diﬀerent doping levels can be used for this kind of study.
However, particularly at low temperature, the dephasing of the carrier spin isfound to vary widely between diﬀerent samples [2]. One reason for this is thatspin dephasing can be very sample dependent since it is strongly aﬀected by

75
impurities and defects. It is therefore desirable to use the same sample within astudy. However, circumstances may prevent this, as in studies of the dependenceof spin dephasing on background doping level (Section 2.4), where diﬀerent wafers
must be used. Even for samples grown on the same wafer (Section 4.6), complexityof spin dephasing could result from diﬀerent donor concentrations. In this work,
we avoid the use of multiple samples for varying carrier concentrations. Instead,a specially designed quantum well sample is used, in which the electron density inthe quantum well can be optically controlled in a continuous way. We are therefore
able to systematically study the spin dynamics as a function of two-dimensional
electron gas (2DEG) density by using the transient Kerr rotation technique.
We vary the density in a 2DEG, and show that spin coherence is lost becauseof the interplay between localization by disorder and dynamical scattering. By
measuring the electron Landé g factor dependence on density, we determine thedensity of states (DOS), which characterizes the disorder potential. Using ourknowledge of the DOS, a simple model estimates the temperature and excitation
intensity dependence of the g factor, qualitatively agreeing with experiments. Thisagreement conﬁrms the importance of disorder and provides predictive power fordesigning spin-based electronic devices. We are able to compare the measured
spin dephasing times to the degree of localization by concurrently probing the
DOS from the electron g factor measurements. Our results show that the longest
spin coherence time is obtained for weakly localized spins, which may dictate acompromise in the design of devices between increasing the spin coherence timeand improving transport properties [65].
5.1Electron Landé g factor
Electron spin dynamics probe fundamental physical processes in condensed

matter systems. One example is the electron Landé g factor, which has been

76
discussed brieﬂy in Section 1.2.3. It is of importance in quantum transport [66]as well as being a useful test of band-theory calculations [67]. Gate-voltagemediated control of electron coherent spin precession has also been demonstrated
by utilizing materials with diﬀerent g factors [68]. Under a magnetic ﬁeld B, theelectron spin will precess with frequency ωL = g ∗ µB B/h̄, where g ∗ is the eﬀective
Landé g factor, µB is the Bohr magneton and h̄ is the Planck’s constant. g ∗in semiconductors diﬀers from the free electron value, 2.0, due to the spin-orbit
interaction, and varies with the energy in the band. The variations of the electrong factor in quantum wells from the bulk values are shown to arise mainly fromthe change of quantum conﬁnement energy [69], with lesser contributions fromanisotropy of the quantum well potential, carrier penetration into the barriers
and strain [27]. This is conﬁrmed by measurements of g ∗ for conduction electrons
as a function of quantum well width [69, 27], where g ∗ is sampled at diﬀerent
energies determined by the electron conﬁnement energy for each quantum wellwidth. The energy dependence of the electron g factor for quantum wells has also
been studied with electron spin resonance [70] and cyclotron resonance [26], and it
has been found that within a small energy range the g factor can be approximatedbyg ∗ = g0 + βE,(5.1)

with β a constant and E the energy. The measured g ∗ through transient Kerr
rotation should correspond to that at the Fermi energy due to Fermi blocking.
Measurements of the electron g factor as a function of the electron density also provide information on the electron DOS. The electron DOS follows
D(E) = dn/dE, with n the electron density. The g factor, on the other hand,satisﬁes dg ∗ /dE = β within a small energy range (following Equation 5.1). Givena measurement of g ∗ as a function of electron density, the electron DOS at the

77
Fermi energy can therefore be determined byD(n) = β(dg ∗ /dn)−1 ,(5.2)

with n the electron density up to the Fermi energy. In an unperturbed 2DEG,
a linear dependence of g ∗ on the 2DEG density is expected, given that the DOSof the 2DEG is a constant as a function of energy. As will be shown later, thisexpectation does not agree with our experimental results, where a nonlinear dependence of g ∗ on the electron density is observed. We will show that the observed
nonlinearity of g ∗ vs. density is due to the presence of the localized states, which
modiﬁes the electron gas DOS. We characterize the disorder potential by determining the DOS based on Equation 5.2. The localized states have important
implications for the electron spin dynamics in quantum wells.
5.2

Sample characteristics and experiments

5.2.1

Mixed type I/type II quantum wells

The sample used for this study is an asymmetric GaAs/AlAs mixed typeI/type II double-quantum-well system [71, 72]. The double quantum well structure is schematically shown in Figure 5.1(a). It consists of a wide and narrow
GaAs quantum well, separated by a AlAs barrier layer. The thicknesses of theselayers are specially chosen (in particular, the thickness of the narrow GaAs layer
is less than 35 Å) to ensure that the lowest electronic subband in each layer forms
a “staircase” [see Figure 5.1(b)]. The lowest electron state is localized at Γ inthe wide GaAs quantum well, the second lowest at the X minimum in the AlAs
barrier,1 and the third at Γ in the narrow GaAs well. This arrangement necessitates the narrow quantum well being type II with respect to the barrier, while
1

The bandgap of Alx Ga1−x As is indirect for x > 0.4, with the energy at X minimum lower
than that at Γ for the conduction band.

78

Figure 5.1: (a) Schematic diagram of the conduction and valence band edges
showing the lowest conﬁned X and Γ electron states and hole states in a mixed
type I/type II quantum well. (b) Band diagram showing the conduction andvalence bands (in the in-plane direction of layers, kx and ky ) in the two quantum
wells and the barrier. The limiting electron and hole relaxation mechanisms areindicated by arrows.

79
the wider well is type I, hence the nomenclature “mixed type I/type II” for the
double quantum wells. Under photoexcitation with energy above the gap of thenarrow well (HeNe excitation for our experiments), electron-hole pairs are generated in the narrow well. The electrons in the narrow well can scatter into thewide GaAs well via the AlAs X states on a picosecond time scale. On the other
hand, the lowest hole states in the two quantum wells are essentially uncoupled,and holes in the narrow well can only tunnel through the AlAs barrier into thewide well (in tenth of microsecond for the sample we use) at low temperatures.
We therefore have spatially separated electrons and holes in the wide and narrow
quantum wells, respectively. The drastic diﬀerence between the electron scattering time and hole tunneling time means that the density of the carriers can bevery large (∼ 1012 cm−2 ) for optical excitation intensity of only a few mW/cm2[73]. By varying the optical excitation intensity, we are able to continuously vary
the electron density in the wide quantum well, in which we will study the spin
dynamics through transient Kerr rotation technique.
The sample is grown by molecular beam epitaxy on n+ GaAs substrate.
It consists of nominally a 6800 Å GaAs buﬀer layer, a 6000 Å Al0.6 Ga0.4 As
etch stop layer, a multiple quantum well consisting of 20 periods of 25- and 68Å GaAs double quantum wells separated by AlAs barriers with a width of 80 Å,a 6000 Å Al0.6 Ga0.4 As etch stop layer, and a 1000 Å GaAs capping layer. The
quantum wells are nominally undoped.
5.2.2

Experimental setup

We perform transient Kerr rotation measurements in a magneto-opticalcryostat, in which a magnetic ﬁeld of 6 T is applied normal to the quantum well
growth direction (Voigt geometry) [30, 33]. Pump and probe pulses are produced
by a mode-locked Ti:sapphire laser, and are circularly and linearly polarized, re-80spectively. Both pulses are spectrally ﬁltered to enable independent tuning of
their energies. The energies of the pump and probe pulses are tuned to the trion
resonance of the wide quantum well for low energy excitation, except for spectral
study. The spectral width of the pulses is adjusted to be 0.2 nm (resulting in ∼5
ps pulses) for spectrally resolved measurements. Average powers are 100 µW in
the pump (except as speciﬁed in the text) and 20 µW in the probe beam. The
carrier density upon pump pulse excitation is estimated to be 3 × 108 cm−2 /well.
For time resolved photoluminescence measurement, we modulate the HeNe laser
beam with an acousto-optic modulator and detect the photoluminescence (fromHeNe excitation) by a 1/4-m grating spectrometer followed by a photomultiplierand lock-in ampliﬁer. The photoluminescence signal is processed with a digital
oscilloscope. All experiments were performed at 4 K except for the temperaturedependence measurements.
5.2.3

Photoluminescence

Photoluminescence (PL) spectra at low temperature as a function of HeNe
excitation intensity reveal electron accumulation in the wide quantum wells. We
measure the PL spectra of the sample at a temperature of 4 K as a function of
the HeNe excitation intensity, as shown in Figure 5.2. For this measurement, thesample is excited by both a CW Ti:sapphire laser at 1.676 eV (below the eﬀective
gap of the narrow quantum wells) and a HeNe laser (above the gap of the narrow
wells). The PL spectrum due to HeNe excitation alone is removed from the total
measured spectrum by subtraction. The resulting PL spectra shown in Figure 5.2
therefore originate from photoexcitation into the wide quantum wells only, withthe electron density in the wide wells varied by varying the HeNe intensity. Underno HeNe excitation, the spectrum is dominated by the heavy hole exciton emission (1.6072 eV) in the wide quantum wells. As the HeNe excitation intensity

81

X

H

eNe

Photoluminescence intensity (arb. unit)Tintensity:

0

4.7
9.3
1.9
4.7
9.3
1.9
4.7
7.0
9.3

x
x
x
x
x
x
x
x
x

-3

10

-3

10

-2

10

-2

10

-2

10

-1

10

-1

10

-1

10

-1

10

1.9
2

6.5 W/cm

1.595

1.600

1.605

1.610

1.615

Energy (eV)Figure 5.2: Low temperature (T = 4 K) PL spectra under photoexcitation into
the wide wells only for various HeNe excitation intensities. The spectra are oﬀset
for clarity. X and T stand for exciton and trion, respectively.

is increased, a low energy shoulder emerges at 1.6047 eV, which is ascribed torecombination involving negatively charged excitons, or trions. We measure thespin dynamics in the wide quantum wells with a HeNe excitation intensity up
to 1.9 × 10−1 W/cm2 . Above this intensity, the trion PL line gradually evolves
into a broader peak, associated with the PL of a two-dimensional electron gas.
The red shift of the peaks with increasing intensity is a result of the bandgap
renormalization [74].

82
As the HeNe beam passes through the sample, it is attenuated due to theabsorption by the quantum wells. This attenuation could result in a decrease in
the electron density as a function of depth. This electron density inhomogeneity
in the wide wells, if any, will result in diﬀerent PL spectra originating from thefront and back wide quantum wells (the PL spectrum is very sensitive to theelectron density by the relative strength between exciton and trion emissions, as
shown in Figure 5.2). We check the electron density inhomogeneity in the 20 wide
quantum wells by measuring the PL spectrum with co-propagating and counterpropagating CW Ti:sapphire beam and HeNe beam. For this study, the sample ismounted on a sapphire plate with the opaque GaAs substrate removed by etching,
to enable optical measurements from both sides of the sample. We measure the PL
spectra from photoexcitation into the wide wells only, with the CW Ti:sapphire
excitation at 1.624 eV (light hole exciton resonance in the wide wells, at which
the absorption strength is comparable to that at HeNe energy). The luminescenceis detected at the same side of the sample as the CW Ti:sapphire excitationenters. Assuming a signiﬁcant attenuation of the CW Ti:sapphire beam by thequantum wells, the PL spectrum detected will have more contributions from thebeam entrance side (if there is no signiﬁcant attenuation of the Ti:sapphire beam,
the HeNe beam intensity should also be fairly uniform in diﬀerent wells). By
directing HeNe excitation to the sample from either side, we are able to monitor
the electron density inhomogeneity. We adjust the HeNe excitation intensity toobtain a PL spectrum with equal peak amplitudes of the exciton and trion lines(similar to the PL spectrum for 9.3 mW/cm2 in Figure 5.2), and record the HeNe
intensity for excitation from either side. We obtain almost identical intensities for
excitation from both sides (10.6 mW/cm2 for co- and 10.8 mW/cm2 for counterpropagating), and therefore conclude that there is negligible diﬀerence of electron
densities in the front and back quantum wells. We also measure the absorption

83

H

hh

eNe

Photoluminescence excitation (normalized)

lh

intensity:

continuum

0

4.7

1.9

4.7

1.9

2.8
4.7

6.5
9.3

x
x
x
x
x
x
x
x

-3

10

-2

10

-2

10

-1

10

-1

10

-1

10

-1

10

-1

10

2

3.7 W/cm

1.60

1.61

1.62

1.63

1.64

1.65

1.66

Energy (eV)Figure 5.3: Low temperature (T = 4 K) PLE spectra of the wide wells for various
HeNe excitation intensities. The spectra are normalized and oﬀset for clarity. hh
and lh stand for heavy hole and light hole exciton, respectively.

of HeNe by the sample and observe no saturation for the whole HeNe intensity
range used.
5.2.4

Photoluminescence excitation and absorption spectra

The PL spectra shown in Figure 5.2 provide information on interband transitions near the wide quantum well bandgap. The energy position and strength ofthe electronic transition obtained from the PL spectrum are, in general, diﬀerent

84
from those obtained from the absorption spectrum. With the existence of localized states in quantum wells (especially in narrow wells), there is a shift in energy
between positions of the band maxima of the absorption and luminescence spectraof the same electronic transition (Stokes shift). It is therefore desirable to measure
the absorption spectrum of the sample. The mixed type I/type II quantum wells
is grown on thick GaAs substrate and a direct absorption measurement of thesample is not possible. We measure the photoluminescence excitation spectrum(PLE) instead. Figure 5.3 shows the low temperature PLE spectra of the wide
quantum wells for various HeNe excitation intensities. The PLE spectrum is measured by scanning the excitation photon energy (from a CW Ti:sapphire laser)and monitoring the PL signal strength at an energy of 1.6025 eV (below the trion
resonance, as shown in Figure 5.2).2

At low HeNe intensity, the PLE spectrum

shows the heavy hole and light hole excitonic peaks, as well as the absorption of
the continuum states, which are labeled in the ﬁgure. At high HeNe intensity,
the excitonic peaks are replaced by a broad, asymmetric peak called the “Fermiedge singularity” [60], with the absorption enhanced near the Fermi level. The
intensity of the absorption spectrum rises relatively quickly on the low energy side
but falls oﬀ slowly on the high energy side, to merge into the continuum absorption. The shift of resonance to higher energy with increasing HeNe intensity is adirect evidence of electron accumulation in the wide wells, due to the band-ﬁllingof electrons. By comparing the PLE spectrum with the PL spectrum shown in
Figure 5.2, we observe a Stokes shift of ∼1.5 meV between the exciton PLE peak
and the corresponding PL peak at low HeNe excitation intensity, indicating carrier
localization in the quantum wells.
2

The low energy PL signal monitored originates from CW Ti:sapphire excitation only. Thisis achieved by modulating the Ti:sapphire beam with an optical chopper and detecting the PL
signal with phase-sensitive lock-in detection. The HeNe excitation does not contribute to the
PL signal monitored, and is only used to vary electron density in the wide wells.

85

Figure 5.4: Low temperature (T = 10 K) absorption spectra near the wide quantum well bandgap for various HeNe excitation intensities. The spectra are takenwith a sample with its substrate etched away. The spectra are oﬀset for clarity.
hh and lh stand for heavy hole and light hole exciton, respectively. Data provided
by P. Dawson.

We measure the absorption spectrum by mounting the sample on a sapphire plate with the opaque GaAs substrate removed by etching. The process ofmounting and etching, however, strains the sample, which shifts and broadens theresonances as compared to the unetched sample used in our study. Therefore, theabsorption spectra are only measured as a reference. The absorption spectra near
the wide quantum well bandgap are plotted in Figure 5.4 at diﬀerent HeNe intensity. They are very similar to the corresponding PLE spectra. We again observe
the heavy and light hole exciton peaks, and observe a blue-shift of the resonances

86with increasing HeNe intensity. The resonances are also somewhat broadened byan underlying Fabry-Pérot fringe (resulting from the interference eﬀect of the thinsample) which can be clearly seen on a wider range spectrum.
5.2.5

Electron density estimation

The optically excited 2DEG density (through HeNe excitation) in the wide
quantum well can be calibrated through time-resolved PL (from HeNe excitationonly) [73]. The basic idea behind this work is that if the decay of the PL from thewide quantum well is monitored following pulsed HeNe excitation, not only will
we observe the fast (<ns) recombination of carriers excited directly in the wide
well but also we should observe a much slower transient. This slow decay shouldarise from the sequential process of hole tunneling and subsequent recombinationof holes with electrons in the wide well. The time constant associated with thissequential process will be governed by the hole tunneling rate, which is expected
to be much less than the rates associated with scattering and recombination. In
Figure 5.5 we show the photoluminescence transient after pulsed HeNe excitation
for several HeNe intensities. The PL is measured by sampling all the light emitted
by the sample. The slow decays, corresponding to the hole tunneling process fromthe narrow well to the wide well, can be clearly seen after the fast decay of thedirectly excited electron hole pairs. The measured decay of the initial fast transient
is limited by the system response. In the inset of Figure 5.5, the hole tunnelingtime is plotted as a function of HeNe excitation intensity. The time constants
are obtained from exponential ﬁt of the photoluminescence transients. The holetunneling time remains fairly constant (∼ 50 µs) over the whole HeNe intensity
range, expect at low intensities, where the measured PL signals becomes noisy toprovide reliable time constants.
As discussed in Reference [73], this method can only give estimated values ofPL (arb. units)

Fast transient

Hole tunneling time (

s)

87

100

80

60

40
0

100

200

300

400

500
2

HeNe intensity (mW/cm )Slow transient
HeNe intensity:
375

150
2

75 mW/cm

0

100

200

Time (

300

400

s)

Figure 5.5: Time-resolved photoluminescence from pulsed HeNe excitation (T =4 K). The transients are normalized and oﬀset for clarity. Inset: Hole tunnelingtime (from narrow well to wide well) as a function of HeNe excitation intensity.
The time constants are obtained from exponential ﬁt of the photoluminescencetransients.

the 2DEG density due to multiple processes in the PL decay. Therefore, the valuesof the excitation intensity instead of the equivalent excited electron densities arequoted in the text. However, given that our measured hole tunneling time remains
fairly constant over the whole HeNe intensity range used in our measurements,
the 2DEG density should be proportional to the HeNe intensity. At the relatively
low HeNe intensities used in this experiment, saturation of electron accumulation
due to the space-charge ﬁeld set up by the spatially-separated electrons and holes
is negligible. The electron density in the wide quantum well is estimated to be∼ 1 × 109 cm−2 per mW/cm2 per well.

88
5.3

Electron spin dynamics

5.3.1

Data analysis

The transient Kerr rotation signal oscillates under the transverse magnetic
ﬁeld due to Larmor precession of the electron spin. The oscillatory dynamics
reveal the electron spin dephasing time and the precession frequency (which givesg ∗ ). CW HeNe excitation is used to continuously vary the electron density in thewide quantum well, with the electron density proportional to the HeNe intensity
over the intensity range used in our measurements, as detailed in Section 5.2.5.
Only the data after 400 ps are analyzed, so that our results are not aﬀectedby short time delay properties such as exciton spin dephasing [transient grating
measurements yield a carrier lifetime of ∼100 ps (not shown)]. With the opticallycreated electrons and holes recombined already, only the spin precession of excess
electrons is measured, and the resulting spin dephasing time and g factor shouldbe that of the electrons without exchange interaction with the holes [47].
The transient signal measured normally consists of oscillations with two different frequencies, and their relative strengths depend on where the pump andprobe photon energies are set. Figure 5.6 shows the transient signals with one
or two frequency components at diﬀerent pump and probe energies, with a HeNe
intensity of 4.7mW/cm2 . As will be discussed in Section 5.3.2, the low frequency
component originates from the main quantum well width, while the high frequency
component is believed to correspond to several quantum wells with slightly narrower well width, which will not be considered in this study. To retrieve information only related to the main quantum well width, we perform least-squares
ﬁtting on the data using a formula with two oscillation frequencies and decay
times, θK (t) = A1 e−t/τ1 cos(ωL1 t) + A2 e−t/τ2 cos(ωL2 t), which give the electron spindephasing times. The precession frequency (g ∗ ) is obtained from fast Fourier89

err rotation (arb. units.)

pump: 1.6091, probe: 1.6087 eV

Kpump: 1.6104, probe: 1.6054 eVexperimental datanonlinear fit

0

1000

2000

3000

Time (ps)Figure 5.6: Transient Kerr rotation signals measured at two diﬀerent pump andprobe photon energies, with a HeNe intensity of 4.7mW/cm2 . The transient signal
for pump energy of 1.6091 eV is ﬁt with a single-frequency oscillation decay, while
the signal for pump energy of 1.6104 eV is ﬁt with a formula with two oscillation
frequencies and decay times.

analysis, with the value being the peak position of the Fourier spectra.
5.3.2

Spectral study

In general, the transient Kerr rotation signal measured consists of beating
signals from two diﬀerent precession frequencies under magnetic ﬁelds. We checkthe origin of the two frequencies by a spectral study of the pump excitation energy. We plot the spectrum of Kerr rotation amplitude for the two frequency
components in Figure 5.7(a), with the amplitude obtained from least-square ﬁtof the transient signal. The probe photon energy is set at 1.6108 eV for thismeasurement. The amplitude of the low frequency component peaks at the heavy
hole exciton resonance, while that of the high frequency component peaks at ahigher energy (∼3 meV above). The PLE spectrum of the wide quantum wellO

Low Frequency

(a)
6

(b)

1.609 eVHigh Frequency
B=2T

K

T=4

4

2

PLE (normalized)

scillation amplitude (arb.unit)

90

1.612 eV

0
1.605

1.610

1.615

Excitation Energy (eV)

1.605

1.610

1.615

Excitation Energy (eV)Figure 5.7: (a) Kerr rotation amplitude for the two frequency components as afunction of pump excitation energy, with the probe photon energy set at 1.6108 eV.
(b) Circles: PLE spectrum of the wide quantum well under no HeNe excitation;
full lines: Fit of the spectrum with two Gaussian lines centered at diﬀerent photon
energies.

under no HeNe excitation is plotted in Figure 5.7(b), which can be well ﬁt withtwo Gaussian lines centered at 1.609 eV and 1.612 eV, respectively. The similarity between the spectra in Figure 5.7(a) and (b) suggests that the two frequency
components correspond to the electron spin precession in quantum wells with two
diﬀerent well widths (supported by the dependence of the electron g factor on
quantum well width). While the low frequency components originates from themain quantum well width, the high frequency component is assigned to electron
spin precession in several quantum wells with slightly narrower width, which exciton peak is also visible in the PLE spectrum (centered at 1.612 eV). In thefollowing experiments, we will be using low energy pump and probe excitation(near trion resonance), so that the narrower quantum well contribution to theKerr rotation signal is minimized.

91
5.3.3

Electron localization

In quantum wells, especially narrow wells in our sample, ﬂuctuations of thewell width may result in the formation of localized electronic tail states. This isdue to the dependence of the quantization energy on the well width. As shownby the diagram in Figure 5.8(a), the conduction band electrons tend to localize
at areas with slightly wider well width. Electrons in the quantum well will ﬁrst
occupy these localized states, since they have a lower energy. As the electron
density in the quantum well is increased, electrons become more delocalized andcan ﬁnally move freely in the quantum well plane after the localized states arecompletely ﬁlled. We therefore have a transition from localized to delocalized
electrons with increasing electron density. The electrons can also be localized
by random potential ﬂuctuations produced by the laterally localized holes in the
narrow quantum in a mixed type I/type II structure [75]. For simplicity, we
model the electron localization by only considering the eﬀects from well width
ﬂuctuations.
The unperturbed DOS (per unite area) of a two-dimensional system, i.e.
without considering electron localization, is expressed as
D(E) =m∗
Θ(E − Ec ),
πh̄2

(5.3)

with m∗ the eﬀective mass of the conduction band electron, Θ(E) the Heaviside
step function, and Ec the cutoﬀ energy. This steplike DOS is plotted in Figure 5.8(b) as dashed line. Now consider a quantum well with a distribution of
the well widths. The DOS for each well width still follows Equation 5.3, with
Ec for each width diﬀers due to quantum conﬁnement. The overall DOS of theperturbed two-dimensional system is then obtained by integrating the DOS foreach width over the well width distribution [P (E)], which follows,D̄(E) =

Z ∞

−∞

′

′

′

D(E − E )P (E )dE .

(5.4)

92

Figure 5.8: (a) Schematic diagram of electron localization in a quantum well withwell width ﬂuctuation. (b) DOS of a two-dimensional system with a localization
tail (full line) and that of the unperturbed two-dimensional system (dashed line).

We model the DOS with well width ﬂuctuations by assuming a Gaussian distribution of the well width


√
1
−1
2
P (E) = ( 2π∆E) exp − (E/∆E) ,
2

(5.5)

with ∆E the ﬂuctuation width. The DOS of the perturbed two-dimensional system is then carried out by inserting Equation 5.5 in Equation 5.4:!

E − Ecm∗√) .
D̄(E) =
2 1 + erf(
2πh̄
2∆E

(5.6)

The calculated DOS is plotted in Figure 5.8(b) as a full line. The electron localization is represented by the tail states at low energies, which have smaller DOS

93

(a)

(b)

0.112

0.108

S

0.106

D

O

g factor

0.110

0.104

0.102

Energy

0

50

100

150

200
2

HeNe intensity (mW/cm )

1x10

10

2x10

10

3x10

10

10

4x10
-2

Electron density (cm )Figure 5.9: (a) Measured g ∗ dependence on the HeNe excitation intensity. (b)Calculated g ∗ dependence on electron density, as described in the text. The
arrows indicate the electron density and energy of the mobility edge. Inset: DOSof a two-dimensional system with a localization tail (full line) and that of theunperturbed two-dimensional system (dashed line).

than the unperturbed two-dimensional system. The mobility edge, the criticalpoint at which a transition from localized to extended electronic states, is at Ec .
5.3.4

Electron g factor nonlinearity

The g factor dependence on HeNe excitation intensity is plotted in Figure 5.9(a). The sign of the g factor is determined to be positive for our sample.3
The value of g ∗ increases nonlinearly with increasing HeNe excitation intensity, incontrast with a linear dependence expected for an unperturbed two-dimensional
system (see Section 5.1). The g ∗ data in Figure 5.9(a) have a bigger increase3

The sign of the g factor is not available from transient Kerr rotation measurements. We
measure the anisotropy of the g factor and obtain a larger absolute value of g for field perpendicular to the growth axis than that for field parallel. According to Reference [76], this indicatesa positive sign of g factor for our sample.

94
at low HeNe intensity (electron density) than at high HeNe intensity, indicatingsmaller densities of states for low electron densities (refer to Equation 5.2). We
therefore propose that the nonlinearity of the electron g factor is associated witha transition from localized to delocalized electrons. We model the g ∗ dependence
on electron density by using a DOS following Equation 5.6, which is for a disordered quantum well system resulting from a Gaussian distribution of the wellwidth ﬂuctuation. We also assume a linear dependence of g ∗ on electron energy,g ∗ = g0 + βE. The linear coeﬃcient β in the energy dependence of g ∗ has been
calculated using ﬁve-band k · p perturbation theory [14]. Based on the calculated
DOS and the energy dependence of g factor, we calculate the electron g factor
dependence on electron density, which is shown in Figure 5.9(b). The calculation
assumes zero temperature and equal spin-up and spin-down electron populations,with the width of the Gaussian distribution set at 1.5 meV to account for the
Stokes shift and g0 as the only free parameter. The calculation represents theform of the experimental data well except at low HeNe intensity. The deviation
is probably due to increased electron temperature resulting from pump excitation, which has a bigger eﬀect on g ∗ at low carrier density (discussed in detaillater). We therefore conﬁrm the presence of localized states, which have a smaller
DOS, and thus produce a more rapid change in the g factor with electron density(Equation 5.2).
5.3.5

Spin dephasing time vs. electron density

Figure 5.10 shows the electron spin dephasing time as a function of the
HeNe intensity. The electron spin dephasing time shows a nonmonotonic dependence on carrier concentration, with a maximum time constant at an intensityof 25 mW/cm2 (see Section 5.2.5 for electron density). The presence of localized
and delocalized states accounts for the opposite trends in the spin dephasing time

2500

Spin

Spin dephasing time (ps)

3000

2000

1500

dephasing rate (ns

-1

)

95

3.0

2

93 mW/cm
2.5
2.0

2

1.5

4.7 mW/cm

1.0
2

0.5

47 mW/cm
3

1000

4

5

6

Field (T)

500

0

0

50

100

150

200

2

HeNe intensity (mW/cm )Figure 5.10: Electron spin dephasing time vs. HeNe excitation intensity. The
inset shows the measured spin dephasing rate dependence on magnetic ﬁeld forseveral HeNe intensities, with solid lines the linear ﬁts to the data. Error bars areplotted except where they are smaller than the symbol size.

at low and high electron densities, as displayed in Figure 5.10. At high electron
concentrations, the decrease of spin dephasing time with increasing density suggests a D’yakonov-Perel’ (DP) mechanism [64], in which the spin dephasing rate isinversely related to the momentum scattering rate. The 2DEG eﬃciently screens
the potential ﬂuctuations, and suppresses the momentum scattering rate, resulting in a decrease of the spin dephasing time with carrier density. Below the criticalelectron density, the electrons become localized in the potential ﬂuctuations, andthe electron spins dephase from the inhomogeneous environment, which is often
attributed to strong hyperﬁne interactions with the nuclear magnetic moments.
The critical density should correspond to the crossover between the localized anddelocalized states, where diﬀerent spin dephasing mechanisms dominate.
An additional contribution to the spin dephasing may arise from a spread in

96
the electron g factor (due to distribution of electron energies over localized states),
which results in an inhomogeneous dephasing of the electron spin under magnetic
ﬁeld. This is shown as a linear dependence of the spin dephasing rate on themagnetic ﬁeld, 1/T2∗ (B) ≈ 1/T2∗ (0) + ∆gµB B/2h̄, with ∆g a Gaussian varianceof the g factor [24]. In the inset of Figure 5.10, spin dephasing rate is measured
as a function of the magnetic ﬁeld for several HeNe intensities. For the two lower
HeNe intensities, the spin dephasing rate increases linearly with ﬁeld. A ∆g of
0.0026 (0.0014) for 4.7 (47) mW/cm2 is retrieved from a linear ﬁt, which supports
an inhomogeneous spin dephasing for the localized electrons. The decrease of ∆gwith increasing electron density suggests a decrease of the g factor inhomogeneity
as the electrons become less localized, which is evidenced by the measured g factor
dependence on electron density, as shown in Figure 5.9. At a HeNe intensity of 93
mW/cm2 , where electrons become delocalized, we observe a small decrease of thedephasing rate with ﬁeld, which suggests a DP mechanism under magnetic ﬁeld[24]. Thus, as the electron density is increased, the electron spin relaxation undergoes a transition from being limited by inhomogeneous dephasing to homogeneous
decoherence.
5.3.6

Electron g factor vs. temperature and excitation intensity

The temperature dependence of the electron g factor provides further insight into the electron localization. Figure 5.11(a) shows the measured g ∗ as afunction of temperature for a series of HeNe intensities. Drastic temperature dependence diﬀerences are observed for diﬀerent intensities. Two aspects need to beconsidered to account for the variation of g factor on temperature: the variation
due to lattice temperature changes, and the ensemble average of the electron g
factor due to thermal distribution of electrons. The former comes mainly fromthe temperature dependence of the fundamental energy gap, but predictions from

97

0.116

(a)

(c)

Electron density

4.7

0.5

14

1.2

28

0.112

g factor(b)

HeNe intensity

0.114

2.3

47

10

3.4x10-2cm

293 mW/cm

0.110
0.108

Electron density

0.106

0.5
1.2

0.104

2.3
10

3.4x10

0.102
4

6

8

10

12

Temperature (

K

14

)

4

6

8

10

12

-2

cm

14

Electron temperature (

K

4

)

6

8

10

12

Temperature (

14

K

)

Figure 5.11: (a) Measured g ∗ dependence on temperature for a series of HeNe
intensities. (b) Calculated carrier temperature dependence of electron hg ∗ i forseveral 2DEG densities (lattice temperature eﬀect not included). (c) Calculated
temperature dependence of hg ∗ i with contributions from both the lattice andcarrier temperature.

k · p perturbation theory fail to describe the experimental results [23]. The latter contribution can be accounted for by calculating the average g factor fromits dependence on the energy, weighted by the DOS, along with a Fermi-Dirac
distribution of the electrons:∗

hg i =

R∞
0g(E)D(E)(f + (E) − f − (E))dE,+−
0 D(E)(f (E) − f (E))dE

R∞

(5.7)

with D(E) the modiﬁed DOS of the 2DEG [Figure 5.8(b)] and f + (f − ) the FermiDirac distribution for the spin-up (spin-down) electrons. The calculated results forthe carrier temperature dependence are shown in Figure 5.11(b) (lattice temperature eﬀect not included). Increasing carrier temperature will transfer electrons
from below to above the Fermi energy. For a degenerate electron gas which has aconstant DOS with energy, hg ∗ i should be that at the Fermi energy since g factor
increases linearly with energy. This is however not the case for localized electrons,

98

0.110
0.110

g factor

0.108

0.105

0.106

Electron density

0.104

0.5

0.100

HeNe intensity

1.2

14

0.102

2.3

47

(a)

0

2

3.4x1093 mW/cm

2

4

6

8
2

Pump intensity (W/cm )

D

0.0

0.2

10

0.4

-2

(b)

0.6

0.8

cm

egree of spin polarization

Figure 5.12: (a) Measured hg ∗ i vs. the pump excitation intensity for diﬀerent
HeNe intensities. (b) Calculated hg ∗ i as a function of degree of the electron spinpolarization.

which have a rapid change of the DOS. The average g factor will increase withcarrier temperature due to the greater DOS above the Fermi energy. This eﬀect
is well represented in Figure 5.11(b) for localized and delocalized electrons. In
Figure 5.11(c) we calculate hg ∗ i by including contributions from both the latticeand carrier temperature. Given the lack of theoretical results that match experimental lattice temperature dependence [23], we empirically represent the lattice
temperature contribution by adding a linear term to the carrier temperature dependence, so that the combined dependence at high density matches the measured
results at the highest HeNe intensity. The calculation represents the trend of theexperimental data.
Increasing the pump excitation intensity has previously been shown to primarily increase the temperature of the electron gas rather than the crystal lattice[77]. The excitation intensity dependence should be a good check to our theory on

99
the electron g factor, since the carrier and lattice temperature have diﬀerent eﬀects
on the hg ∗ i [as seen in Figure 5.11(a) and (b)]. In Figure 5.12(a), the measured
hg ∗ i is plotted against the pump excitation intensity for several HeNe intensities.
The carrier density due to pump excitation is estimated to be 3 × 108 cm−2 per
1 W/cm2 . The hg ∗ i shows strong variation with excitation intensity at low 2DEG
density, but has little variation at high density. The ﬁgure also shows a similar
trend as the calculated carrier temperature dependence [Figure 5.11(b)]. The lattice temperature dependence of the hg ∗ i [Figure 5.11(a)], however, behaves quite
diﬀerently. This further establishes that increasing the excitation intensity does
heat up the carriers rather than the lattice. By increasing the excitation intensity,
we also increase the degree of spin polarization of the electron gas, which alone
should also aﬀect the g factor. We calculate the electron hg ∗ i dependence on the
degree of spin polarization of the 2DEG, as shown in Figure 5.12(b). The degreeof spin polarization is deﬁned as (n+ − n− )/(n+ + n− ), with n+ (n− ) the densityof the spin-up (spin-down) electrons. The calculation is based on Equation 5.7,
with the degree of spin polarization represented by a diﬀerence between the Fermi
energy for the spin-up electrons (f + ) and that for the spin-down electrons (f − ).
The results show that spin polarization of the 2DEG does not aﬀect the hg ∗ i as
much as increasing the carrier temperature, and g factor variation with excitation
intensity is mainly related to the change of the electron temperature.
Transient Kerr rotation measurements provide reliable values of the g factor
of bare electrons, as opposed to those obtained from magnetoquantum oscillations, which can be strongly enhanced by many-body exchange interactions [78].
We therefore ignore the exchange interactions between carriers in our model. The
variation of electron g factors in quantum wells arises mainly from the change inelectron energy, where the assumption of a linear dependence on energy should be
precise for the energy range (several meV) present in the measurements. Other

100
contributions, such as anisotropy of the quantum well potential, strain and electron wavefunction penetrating into barriers, should have minimal eﬀects, becauseof the use of the same sample, and the small change of 2DEG density in the measurements. The non-parabolicity of the conduction band has negligible eﬀect on
the variation of the g factor over the change in Fermi energy studied.
In summary, we systematically study the spin dynamics of the electron gas
in a GaAs quantum well sample, where the electron density in the quantum well
can be varied through optical excitation. We show that the electron spin coherence is limited by the interplay between localization by disorder and dynamical
scattering processes. For localized electrons, disorder limits the spin dephasingtime due to inhomogeneous g factor distributions. The measured electron g factor
dependence on electron density directly characterizes the disorder potential. Using
our knowledge of the DOS, a simple model estimates the temperature and excitation intensity dependence of the g factor, in agreement with our experimental
results. This work conﬁrms the importance of disorder in spin coherence in semiconductors, which provides predictive power for designing spin-based electronics
devices.

Chapter 6
Conclusion

In this thesis we focus on the optical manipulation of spin coherence insemiconductors. We use transient Faraday/Kerr rotation and other optical spectroscopy techniques (such as absorption, reﬂection, PL and PLE) to study spin
dynamics in various semiconductor quantum well systems. In Chapter 3, we
experimentally study the spectra of the transient Faraday/Kerr rotations near
the exciton resonances of an undoped GaAs multiple quantum well. While thetransient Faraday/Kerr techniques have been widely used for spin study, spectral studies can reveal the underlying spin dependent nonlinearities. They arealso useful in selectively generating and detecting spin polarizations of diﬀerent
species. A spectrum calculation with a multilayer characteristic matrix model isperformed, with the results quantitatively agree with the measured linear spectraand Faraday/Kerr spectra. We show that the Kerr spectra depend not only on the
optical nonlinearities but also on linear optical eﬀects such as interference fromlayer structures. In Chapter 4, we study the interplay of spin dynamics between
excitons, negatively-charged excitons (trions) and the two-dimensional electron
gas with lightly modulation-doped CdTe quantum wells. In these systems, theinterband properties are dominated by excitons and trions at low temperatures,
but the electron spin can persist after recombination of optically excited carriers. Coherent optical generation and manipulation of the electron spins require

102
an understanding of the connection between electron spin and optical transitions
through which the spin polarization is initiated and controlled. By a spectral
study of Kerr rotation, spin relaxation of individual species is resolved. The spin
dynamics is quantitatively described by solving rate equations involving the spin
populations of excitons, trions and the electron gas. Under resonant excitation of
excitons, spin polarization of the electron gas is generated through trion formation
process, with the spin coherence partially lost through exciton spin relaxation. In
Chapter 6, we study the eﬀects of disorder on electron spin dynamics in a GaAs
quantum well. A mixed type I/type II GaAs/AlAs quantum well is used, where
the carrier density in the quantum wells can be continuously varied with CW
optical excitation. We vary the electron density in a two-dimensional electrongas, and show that the electron spin coherence is lost due to the interplay between localization by disorder and dynamical scattering processes. By measuring
the electron Landé g factor dependence on electron density, we are able to determine the density of states, which characterizes the disorder potential. Using
our knowledge of the density of states, a simple model estimates the temperature and excitation intensity dependence of the g factor, in agreement with ourexperimental measurements. Our results show that the longest spin coherence isobtained for weakly localized spins, which may dictate a compromise in the designof spintronic devices between increasing the spin coherence time and improving
transport properties.

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Appendix A
Birefringence of cryostat windows

The windows in our magneto-optical cryostat could carry strain-induced
birefringence, which could be largely enhanced at low temperatures. In a transient
Faraday/Kerr rotation measurement, a circularly polarized pump beam is needed
to optically create spin polarized carriers in the samples, while a linearly polarizedprobe beam is required to measure the photo-induced Faraday/Kerr rotation. The
birefringence in the cryostat windows, if any, will vary the polarization states ofthe otherwise circularly polarized pump and linearly polarized probe beams. Thisin turn aﬀects both the degree of spin polarization in the samples after pumpexcitation [due to optical selection rules (Chapter 2)], and the detection sensitivity
due to the ellipticity in the linear probe. It is therefore necessary to check any
birefringence in the cryostat windows, and if any, compensate the birefringence.
A.1

Birefringence detected by Kerr rotation measurements
Circular polarization of the pump pulse is generated by inserting a quarter-

wave plate in the pump beam path, with the azimuth angle of the quarter-wave
plate at 45◦ with respect to the initial linear polarization of the laser output. If
the pump polarization is not pure circular, there is a reduction of the observed
Faraday/Kerr rotation amplitude as a result of optical selection rules. We measure
the transient Kerr rotation in an undoped GaAs multiple quantum well sample

data
fit

0

K

err rotation amplitude (arb. units)

111

0

25

50

75

100

125

/4 rotation angle (degree)

Figure A.1: Kerr rotation amplitude (absolute value) as a function of the azimuth
angle of the quarter-wave plate (λ/4). Data are taken with an undoped GaAs
multiple quantum well. Data are ﬁtted with a Jones matrices calculation, which
gives a window birefringence with a phase retardation of ∼ 90◦ (close to a quarterwave plate).

(Section 3.2). To check the birefringence in the cryostat windows, we vary thepolarization state of the pump pulse by rotating the azimuth angle of the quarterwave plate in the pump beam path, and record the absolute value of the Kerr
rotation amplitude. The Kerr rotation amplitude as a function of azimuth angleof the quarter-wave plate is plotted in Figure A.1. Without any birefringence inthe cryostat windows, the Kerr rotation amplitude should have a 4-fold rotation
symmetry, because the pump polarization varies from linear to circular to linear
polarization as the quarter-wave plate is rotated from 0◦ to 45◦ to 90◦ . As shownin Figure A.1, the measured Kerr rotation amplitude does not show a 4-foldsymmetry. This is due to the additional birefringence in the cryostat windows(front windows through which the pump beam passes before it hits the sample).

112

Figure A.2: Schematic setup for calibrating birefringence in the cryostat windows.
The polarization state after the polarizer and birefringence windows is detectedby measuring the output power as a function of the analyzer angle α. s.a., slow
axis.

The dependence of Kerr rotation amplitude on the quarter-wave plate angle canbe calculated by Jones matrices calculation [79], with the birefringence in thewindows represented by an arbitrary phase retardation. We ﬁt the data with
Jones matrices calculation (discussed in detail below). This is shown as a solidline in Figure A.1, which corresponds to a phase retardation of ∼ 90◦ for thewindows. The front cryostat windows work almost as a quarter-wave plate.
A.2

Calibration of birefringence
The birefringence in the cryostat windows is checked above through the mea-

surements of Kerr rotation amplitude. This method is however not an ideal way
for calibration of the window birefringence, because it requires accurate knowledgeof optical selection rules in the sample used. A general method for birefringencecalibration would be through the use of linear optical components such as polarizers and waveplates. Speciﬁcally, a polarizer (Newport Polarcor linear polarizer)
is inserted into the sample chamber in the cryostat (where the samples normally113sit). The laser beam after the polarizer is then directed through the back birefringence windows and another polarizer (analyzer). The polarization state after thebirefringence windows can be detected by measuring the output power (througha photodiode) as a function of the analyzer rotation angle (Figure A.2).
In order to calibrate the window birefringence, we ﬁrst go through Jones
calculus for the system shown in Figure A.2. The Jones matrix for a polarizer at
a rotation angle φ is






 cos φ − sin φ   1 0  





Mp = 


sin φ

0 0

cos φ





cos φ

sin φ − sin φ cos φ

2

cos φ


=



cos φ sin φ

cos φ sin φ 
2

sin φ
(A.1)

An initial arbitrary beam polarization
into a pure linear polarization
 is converted cos φ 
. Similarly the Jones matrix for theafter the polarizer, in the state of analyzer at an angle α is



sin φ



2

cos α



Ma = 
cos α sin α

cos α sin α 

.


2

sin α

(A.2)

The window birefringence is represented by a Jones matrix for a birefringencemedium with a phase retardation Γ and a slow axis at an angle θ,− sin θ   e

 cos θMw = 


sin θ


cos θ

 e

=


iΓ

2




iΓ

0

2

cos θ + sin θiΓ(e − 1) cos θ sin θ0   cos θ
1






sin θ − sin θ cos θ

iΓ



iΓ

2

2

cos θ + e sin θ cos φ  Ex 
.
 = Ma Mw 






Ey



(e − 1) cos θ sin θ 

The polarization state of the output ﬁeld is then in the form of





.


(A.3)

(A.4)

sin φ

The output power at the detector can be expressed as
P ∝ |Ex |2 + |Ey |2 =1
+ A cos(2α − 2Ψ),
2

(A.5)

.


114
80

Power (arb. units)

70

60

50

data

40
-250

fit

=780 nm

-200

-150

A

-100

nalyzer angle-500(degree)Figure A.3: Output power as a function of the analyzer rotation angle α formeasuring front window birefringence. The data are ﬁtted by using Equation A.5.

with
1
1
A cos 2Ψ = − (cos Γ − 1) cos 4θ + (cos Γ + 1),
4
4
1
A sin 2Ψ = − (cos Γ − 1) sin 4θ.
4

(A.6)

For simplicity, α, θ and Ψ in Equation A.5 and A.6 are expressed with respect toφ (which is known experimentally), so that φ can be omitted in the equations.
Experimentally, we set the polarizer at φ = −36◦ from the horizontal direction. The output power is recorded as a function of the analyzer rotation angle
α, for the back and front windows (by measuring in the reverse direction), andfor several laser wavelengths. Figure A.3 shows the result for a measurement of
the front window birefringence at a wavelength of λ = 780 nm. A nonlinear ﬁt
to the data using Equation A.5 gives A = 0.148 and Ψ = −33.7◦ (solid line in
Figure A.3). The parameters A and Ψ are then used to solve Equations A.6,

115
Back windows

Front windows(degree)

B=0 T, T=4.5

8875
B=4 T, T=4.5

K

86

70
B=0 T, T=100

K

84

65

R

etardation

K

80

(a)

82

(b)

60
760

770

780

790

800

Wavelength [nm]

810

820

760

770

780

790

800

810

820

Wavelength [nm]Figure A.4: Birefringence retardation as a function of wavelength for the (a) backand (b) front cryostat windows. Retardation for the back windows at a diﬀerent
temperature and magnetic ﬁeld is also shown in (a).

which gives a phase retardation of Γ = 88.3◦ and θ = 0.45◦ (from horizontal
direction). The results indicate that at λ = 780 nm the front windows in thecryostat have a birefringence similar to a quarter-wave plate (Γ = 90◦ ), and havethe fast or slow axis almost in the horizontal direction. The above method does
not distinguish the fast and slow birefringence axis, because both Γ and −Γ aresolutions to Equations A.6. To identify whether the fast or slow axis is associatedwith θ, an additional quarter-wave plate is inserted between the windows and theanalyzer [if the fast axis of the quarter-wave plate is parallel to the fast (slow)
axis of the windows, the polarization before the analyzer is almost linear, and isparallel (perpendicular) to the linear polarization immediately after the polarizer].
We ﬁnd that it is the fast axis that is close to the horizontal direction.
The birefringence retardation for the back and front cryostat windows areshown in Figure A.4 for diﬀerent wavelength. The fast axis is found to be at
horizontal directions for all wavelengths. We also check the eﬀect of temperature

116
and magnetic ﬁeld on the birefringence [shown in Figure A.4(a)]. The retardation
reduces drastically at a higher temperature (T = 100 K) in the sample chamber.
This conﬁrms a strain-induced birefringence in the cryostat windows, which becomes signiﬁcant at low temperatures. The magnetic ﬁeld is shown to have little
eﬀect on the window birefringence.
A.3

Birefringence compensation
In order to get pure circular polarization for the pump beam and linear

polarization for the probe beam (before they hit the samples), the birefringence inthe cryostat windows needs to be compensated. By using a pair of half-wave plateand quarter-wave plate and adjusting their rotation angles, any polarization state
can be generated. In the special case of a wavelength around λ = 760 − 780 nm,
the front windows work roughly as a quarter-wave plate with its fast axis in thehorizontal direction (as shown above). A circularly polarized pump beam canbe easily generated by ﬁrst rotating the linear polarization 45◦ away from thehorizontal direction through a half-wave plate. The birefringence windows willthen convert the linear polarization to an almost pure circular polarization. For
birefringence other than Γ = 90◦ (for example, at other wavelengths or for theback windows), a careful birefringence compensation using both the quarter- andhalf-wave plates is needed. The rotation angles for the waveplates can be decided
through Jones matrix calculus, or by analysis using the Poincaré sphere, which
provides a convenient way of predicting how a series of waveplates and retarders
will change the polarization form. The probe beam will remain linear as long as its
polarization axis is parallel to the fast or slow axis of the birefringence windows.