Spin coherence of holes in GaAs/AlGaAs quantum wells
M. Syperek1,2, D. R. Yakovlev1,†, A. Greilich1, J. Misiewicz2, M. Bayer1, D. Reuter3, and A. D. Wieck3
Experimentelle Physik II, Universität Dortmund, D-44221 Dortmund, Germany
Institute of Physics, Wroc law University of Technology, 50-370 Wroc law, Poland and
Angewandte Festkörperphysik, Ruhr-Universität Bochum, D-44780 Bochum, Germany
(Dated: August 2, 2021)
The carrier spin coherence in a p-doped GaAs/(Al,Ga)As quantum well with a diluted hole gas
has been studied by picosecond pump-probe Kerr rotation with an in-plane magnetic field. For
resonant optical excitation of the positively charged exciton the spin precession shows two types
of oscillations. Fast oscillating electron spin beats decay with the radiative lifetime of the charged
exciton of 50 ps. Long lived spin coherence of the holes with dephasing times up to 650 ps. The
spin dephasing time as well as the in-plane hole g factor show strong temperature dependence,
underlining the importance of hole localization at cryogenic temperatures.
PACS numbers: 42.25.Kb, 78.55.Cr, 78.67.De
Recently the investigation of the coherent spin dynam-
ics in semiconductor quantum wells (QW) and quantum
dots has attracted much attention, due to the possible
use of the spin degree of freedom in novel fields of solid
state research such as spin-based electronics or quantum
information processing [1, 2, 3]. Until now the inter-
est has been mostly focused on the spin coherence of
electrons, while experimental information about the spin
coherence of holes is limited [4]. The hole as a Luttinger
spinor has properties strongly different from the electron
spin, such as a strong spin-orbit coupling, a strong direc-
tional anisotropy, etc. It plays an important role also in
coherent control of electron spins, since in many optical
schemes charged electron-hole complexes are proposed as
intermediate manipulation states [5].
Earlier, the hole spin dynamics in GaAs-based QWs
has been measured by optical orientation detecting
photoluminescence (PL) either time-integrated or time-
resolved [4, 6, 7, 8, 9]. Experimental studies addressed
the longitudinal spin relaxation time T1 [6, 7, 8] and the
dephasing time T ∗
, exploiting the observation of hole spin
quantum beats [4]. The reported relaxation times vary
from 4 ps [6] up to ∼1 ns [4, 8] demonstrating strong de-
pendence on doping level, doping density and excitation
energy. A major drawback of PL techniques is, however,
that the spin coherence can be traced only as long as
both electrons and holes are present and photolumines-
cence can occur. Further, they work only for studying
the spin dynamics of minority carriers in a sea of ma-
jority carriers and are therefore restricted to undoped or
n-type doped QWs. However, then the holes can interact
with electrons, providing additional relaxation channels
via exchange or shake-up processes [8, 10]. These mech-
anisms can be excluded for p-doped structures if the hole
spin relaxation occurs on time scales longer than the ra-
diative annihilation of electrons. A pump-probe Kerr ro-
tation (KR) technique using resonant excitation allows to
realize such measurements, which up to now have been
reported only for bulk p-type GaN [11] and not yet for
low-dimensional systems.
The theoretical analysis of the hole spin dynamics in
QWs has been focused on free holes [10, 12, 13, 14]
by considering different relaxation mechanisms: (i) a
Dyakonov-Perel like mechanism, (ii) an acoustic phonon
assisted spin-flip due to spin mixing of valence band
states, (iii) an exchange induced spin-flip due to scatter-
ing on electrons, which resembles the Bir-Aronov-Pikus
mechanism, but for holes. Recently the attention has
been drawn on the role of hole localization and the de-
phasing caused by fluctuations of the in-plane g factor
has been calculated [15].
In this paper we use time-resolved pump-probe Kerr
rotation [16] to investigate the spin coherence of holes in
a p-doped GaAs/Al0.34Ga0.66As single QW with a low
hole density. We find spin dephasing times reaching al-
most the ns-range at a temperature T = 1.6 K with a
hole in-plane g factor of about 0.01. Both quantities de-
crease strongly with increasing temperature, suggesting
the important influence of hole localization. We discuss
also a mechanism that provides generation of spin co-
herence for the hole gas under resonant excitation of the
positively charged exciton.
The structure was fabricated by molecular-beam epi-
taxy on a (100) oriented GaAs substrate. A 15 nm-
wide GaAs QW was grown on top of a 380 nm-
thick Al0.34Ga0.66As barrier and overgrown by a 190
nm-thick Al0.34Ga0.66As layer. 21 nm-thick layers
with Al0.34Ga0.66As effective composition realized by
GaAs/AlAs short-period superlattices were deposited on
both sides of the QW in order to improve interface pla-
narity. Two δ-doped layers with Carbon acceptors were
positioned symmetrically at 110 nm distance from the
QW. The hole gas concentration and mobility in the QW
are 1.51× 1011 cm−2 and 1.2× 105 cm2/Vs, respectively,
as determined by Hall measurements at T = 4.2 K. It
was possible to deplete the hole density in the QW by
above barrier illumination and even invert the majority
carrier type, resulting in a diluted electron gas. The sam-
ple temperature was varied from 1.6 to 6 K.
A mode-locked Ti:Sapphire laser with a repetition rate
of 75.6 MHz and a pulse duration of ∼1.5 ps (∼1 meV full
width at half maximum) was used for optical excitation.
http://arxiv.org/abs/0704.0592v1
FIG. 1: (a) KR traces for a p-type 15 nm-wide
GaAs/Al0.34Ga0.66As QW vs time delay between pump and
probe pulses at B = 0 and 7 T with field tilted by ϑ = 4◦
out of QW plane. Laser at 1.5365 eV is resonant with T+
line. Power was set to 5 and 1 W/cm2 for pump and probe,
respectively. Bottom trace was recorded with additional laser
illumination at 2.33 eV. T = 1.6 K. (b) PL spectra for DHG
(excitation at 1.579 eV) and DEG regime (above barrier ex-
citation at 2.33 eV). (c) Comparison of KR traces for ϑ = 0
and 4◦.
The laser beam was split into a circularly polarized pump
and a linearly polarized probe beam. Both beams where
focused on the sample surface to a spot diameter of ∼100
µm. Magnetic fields B ≤ 10 T were applied about per-
pendicular to the structure growth z-axis (Voigt geome-
try). In a pump-probe KR experiment the pump pulse
coherently excites carriers with spins polarized along the
z axis. The subsequent coherent evolution of the spins
in form of a precession about the magnetic field is tested
by the probe pulse polarization. To detect the change
of the linear probe polarization plane (the KR angle), a
homodyne technique based on phase-sensitive balanced
detection was used.
Photoluminescence spectra excited above and below
the band gap of the Al0.34Ga0.66As barriers are shown in
Fig. 1(b) at B = 0 and 7 T. A single PL line correspond-
ing to the positively charged trion T+ (consisting of two
holes and one electron) is seen for the regime of diluted
hole gas (DHG) established for below-barrier excitation.
After inverting the type of majority carriers to the DEG
FIG. 2: Top trace: KR signal measured at B = 7 T for ϑ = 4◦.
Bottom traces are obtained by separating electron and hole
contributions (see text). Excitation conditions as in Fig. 1.
regime by above barrier illumination the PL spectra con-
sist of the exciton (X) and negatively charged trion (T−)
lines.
The type of majority carriers in the QW can be iden-
tified by the KR signals measured at B = 7 T, with the
laser energy tuned to the trion resonance. The bottom
trace in Fig. 1(a) was measured with additional above-
barrier illumination (DEG regime) and shows long-lived
electron spin beats with a dephasing time of 2.5 ns which
is considerably longer than the radiative decay time of
resonantly excited trions of about 50 ps. The precession
frequency corresponds to a g factor | ge |= 0.285± 0.005,
which is typical for electrons in GaAs-based QWs.
Without above-barrier illumination in the DHG
regime, fast electron precession is observed only during
∼ 200 ps after pump pulse arrival [see middle trace in
Fig. 1(a)]. This signal is caused by the coherent preces-
sion of the electron in T+ and disappears with the trion
recombination. The electron beats are superimposed on
the hole beats with a much longer precession period. The
hole beats decay with a time constant of about 100 ps and
can be followed up to 500 ps delay. At these long times
the KR signal is solely given by coherent hole precession.
Experimentally, it is difficult to observe the hole spin
quantum beats due to the very small in-plane hole g fac-
tor. To enhance the visibility we tilted the magnetic
field slightly out of the plane by an angle ϑ = 4◦ to
increase the hole g factor by mixing the in-plane com-
ponent (gh,⊥) with the one parallel to the QW growth
axis (gh,‖), which typically is much larger: gh(ϑ) =
sin2 ϑ+ g2
h,⊥ cos
2 ϑ [17]. The strong change of the
hole beat frequency with the tilt angle is seen in Fig. 1(c).
The precession frequency is analyzed by ωh = µB |
gh | B/~, where µB is the Bohr magneton, and gives
| gh,⊥ |= 0.012±0.005 for ϑ = 0
◦ and | gh |= 0.048±0.005
for ϑ = 4◦.
The electron and hole contributions to the KR ampli-
FIG. 3: Hole component of KR signal at different magnetic
fields and ϑ = 4◦. Top inset: Magnetic field dependence of the
hole dephasing time T ∗2 . Solid line is 1/B fit to data. Bottom
inset: Hole spin precession frequency vs magnetic fields. Line
is guide to the eye. In inserts closed and open circles show the
data measured for pump to probe powers of 1 to 5 W/cm2
and 5 to 1 W/cm2, respectively. T = 1.6 K.
tude, ΘK, can be separated by fitting the experimental
data with a superposition form of exponentially damped
harmonic functions for electrons and holes:
i=e,h
Ai exp(−
) cos(ωi∆t). (1)
Ai are the corresponding signal amplitudes at pump-to-
probe delay ∆t = 0, and T ∗
2,i are the spin dephasing
times. An example for a decomposition of the KR signal
in the DHG regime is shown in Fig. 2.
Let us turn now to the hole coherence. Figure 3 shows
the hole contribution to the KR signal for different B at
T = 1.6 K. From the fit by Eq.(1) we have obtained the
dephasing time T ∗
, which is plotted versus B in the inset.
A very long lived hole spin coherence with T ∗2 = 650 ps is
found at B = 1 T. With increasing B up to 10 T it short-
ens to 70 ps. The field dependence can be well described
by a 1/B-form (see the line in the inset), from which
we conclude that the dephasing shortening arises from
the inhomogeneity of the hole g factor ∆gh = 0.0007 in
the QW, which is translated into a spread of the preces-
sion frequency: ∆ωh = ∆ghµBB/~. Since T
∝ 1/∆ωh,
this explains the 1/B-dependence of the dephasing time.
The magnetic field dependence of the hole precession fre-
quency in the lower inset of Fig. 3 shows an approximate
B-linear dependence up to 7 T. For higher fields a super
linear increase is seen which indicates a change of the
hole g factor due to mixing between heavy and light hole
states, induced by the field.
Two sets of experimental data measured for pump to
FIG. 4: Temperature dependence of the hole KR signal at
B = 7 T and ϑ = 5◦. Pump and probe powers are set to 1
and 5 W/cm2, respectively. Inset: Hole spin dephasing time
T ∗2 vs temperature.
probe powers 1 to 5 W/cm2 and 5 to 1 W/cm2 are com-
pared in the insets of Fig. 3. The very similar results
demonstrate performance of the experiment in the linear
regime for both pump and probe beams with power not
exceeding 5 W/cm2.
Insight into the origin of the long hole spin coherence
can be taken from KR at varying temperatures. The
data in Fig. 4 measured at ϑ = 5◦ show that (i) the
dephasing time T ⋆
decreases from 110 down to 60 ps
when increasing the temperature from 1.6 to 6 K (see also
inset), and (ii) simultaneously the precession frequency
decreases notably corresponding to a g factor decrease
from 0.057 to 0.030.
These results can be naturally explained by hole lo-
calization in the QW potential relief due to monolayer
well width fluctuations. The localization energy does not
exceed 0.5 meV, which is comparable with the thermal
energy at T = 6 K. Free holes are expected to have a
short spin coherence time T2 limited by the efficient spin
relaxation mechanisms due to the spin-orbit interaction
[10, 13, 14]. For localized holes these mechanisms are
mostly switched off and one can expect long T2 times.
However, in a KR experiment we do not measure the
T2 time, but rather the ensemble dephasing time T
(Ref. 15), as confirmed by the 1/B dependence in the
inset of Fig. 3. The T ∗2 time gives a lower boundary for
the spin coherence time T2. Therefore we can conclude,
that the T2 for localized holes is at least 650 ps. Thermal
delocalization of holes on the one hand decreases the role
of inhomogeneities and reduces ∆gh, which should lead
to longer dephasing times. On the other hand, then the
fast decoherence of free holes becomes the limiting factor
for the spin beats dynamics.
Mixing of heavy and light hole states in a QW is en-
hanced by localization effects. This should be detectable
by an increase of the in-plane hole g factor, which is close
to zero for free holes [4, 15, 18]. The decrease of the hole
g factor with increasing temperature shown in Fig. 4 is
consistent with a hole delocalization scenario.
We turn now to discussing the mechanism for opti-
cal generation of hole spin coherence in a QW with a
DHG. The generation mechanism is similar to the one
suggested for singly charged quantum dots [19, 20] and
QWs with a DEG [21]. In our experiment pump and
probe are resonant in energy with the positively charged
trion T+. Due to the considerable heavy-light hole split-
ting, the circularly polarized pump creates holes and elec-
trons with well-defined spin projections, Jh,z = ±3/2 and
Se,z = ±1/2, respectively, according to the optical selec-
tion rules [12]. Therefore, |⇑⇓↓〉 (|⇑⇓↑〉) trions can be
generated by a σ+ (σ−) polarized pump. Here the thick
and thin arrows give the spin states of holes and elec-
trons, respectively.
The pump pulse duration is much shorter than the spin
coherence and the electron-hole recombination times. If
in addition the pump duration is shorter than the charge
coherence time of the trion state the pulse creates a co-
herent superposition of a resident hole from the DHG and
a hole singlet trion T+. The spin state of the resident hole
with arbitrary spin orientation before excitation can be
described by α |⇑〉+ β |⇓〉, where |α|2 + |β|2 = 1. With-
out magnetic field and for fields oriented normal to the
z-axis, the net spin polarization of the hole ensemble is
zero, so that the ensemble averaged coefficients are equal:
α = β.
For σ+ polarized excitation, for which injection of an
|⇑↓〉 electron-hole pair is possible, the excited superposi-
tion is given by α |⇑〉+β cos(Θ/2) |⇓〉+iβ sin(Θ/2) |⇑⇓↓〉.
Here Θ =
d · E(t)dt/~ is the dimensionless pulse area
with the pump laser electric field E(t) and the dipole
transition matrix element d. In general, the hole-trion
superposition state may be driven coherently by varying
the pulse area, giving rise to Rabi-oscillations as reported
recently for (In,Ga)As quantum dots [20]. Such oscilla-
tions have not been found yet in QWs, most probably due
to the fast carrier dephasing, in particular for strong ex-
citation. Dephasing of the superposition occurs shortly
after the pulse on a time scale of a few ps, converting
the coherent polarization into a population consisting of
holes with original spins ⇑ and ⇓ and trions with ⇑⇓↓.
In a simplified picture, the spin coherence generation
can be described as follows: The σ+ polarized pump cre-
ates with certain efficiency trions T+ of spin configura-
tion |⇑⇓↓〉. By this process |⇓〉 holes are pumped out of
the DHG, leaving behind holes with opposite spin |⇑〉.
Right after the pump pulse the KR signal is contributed
by the |⇑〉 hole from the DHG and |↓〉 electron of the T+.
The further evolution of the coherent signal depends on
the strength of external magnetic field applied perpen-
dicular to the z-axis.
AtB=0, the carrier spins experience no Larmor preces-
sion. The electron spin relaxation time usually exceeds
the lifetime of trions, which is limited by radiative decay,
by one-two orders of magnitude. Trion recombination re-
turns the hole to the DHG with the same spin orientation
as it was pumped out, if no electron spin scattering oc-
curred in the meantime. This compensates the induced
spin polarization and nullifies the KR signal at delays
exceeding the trion lifetime. Indeed, the KR signal in
the top trace in Fig. 1(a) shows a fast decay with a time
constant of ∼50 ps, which is characteristic for radiative
trion recombination in GaAs/(Al,Ga)As QWs [22]. The
long-lived tail of the signal has a very small amplitude
and is due to hole coherence provided by weak spin re-
laxation of electrons in T+ and/or hole relaxation in the
DHG during the trion lifetime.
In finite magnetic fields, the carrier spins start to pre-
cess about B. Due to the electron spin precession in
T+, the hole spin returned to the DHG after trion re-
combination will not compensate the spin polarization of
the resident holes. Therefore, a long-lived hole coherence
with considerable amplitude will be induced. This co-
herence is observed in the KR signal as spin beats with
low frequency (see Figs. 1 and 3). Note that the Larmor
precession of the resident holes may also contribute to
generation of hole spin coherence, but the effect is pro-
portional to the ratio of the hole and electron Larmor
frequencies and therefore will be rather small.
Let us compare the spin coherence generation for QWs
with DHG and DEG resonantly excited in the T+ and
T− states, respectively. We are interested in a long-lived
spin coherence which goes beyond the trion lifetime, i.e.
in spin coherence induced for the resident carriers. In
both cases the amplitude of the KR signal is controlled
by the ratio of the electron spin beat period to the trion
lifetime. Nevertheless, the two cases are quite different as
for DHG the precessing electron is bound in the T+ trion,
while for DEG the background electron precesses. In the
latter case the electron precession in T− is blocked due
to the singlet spin character of the trion ground state.
To conclude, a long-lived spin coherence has been
found for localized holes in a GaAs/(Al,Ga)As QW with
a diluted hole gas. The spin coherence time exceeds 650
ps and is still masked by the spin dephasing due to g
factor inhomogeneities. Localization of holes suppresses
most spin relaxation mechanisms inherent for free carri-
ers. It is also worth to note, that due to the p-type Bloch
wave functions the holes do not interact with the nuclear
spins, which provides the most efficient spin relaxation
mechanism for localized electrons [23].
Acknowledgements. This work was supported by
the BMBF program ’nanoquit’.
[†] Also at Ioffe Physico-Technical institute, Russian
Academy of Sciences, St. Petersburg, Russia.
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  The carrier spin coherence in a p-doped GaAs/(Al,Ga)As quantum well with a
diluted hole gas has been studied by picosecond pump-probe Kerr rotation with
an in-plane magnetic field. For resonant optical excitation of the positively
charged exciton the spin precession shows two types of oscillations. Fast
oscillating electron spin beats decay with the radiative lifetime of the
charged exciton of 50 ps. Long lived spin coherence of the holes with dephasing
times up to 650 ps. The spin dephasing time as well as the in-plane hole g
factor show strong temperature dependence, underlining the importance of hole
localization at cryogenic temperatures.

Spin coherence of holes in GaAs/AlGaAs quantum wells
M. Syperek1,2, D. R. Yakovlev1,†, A. Greilich1, J. Misiewicz2, M. Bayer1, D. Reuter3, and A. D. Wieck3
Experimentelle Physik II, Universität Dortmund, D-44221 Dortmund, Germany
Institute of Physics, Wroc law University of Technology, 50-370 Wroc law, Poland and
Angewandte Festkörperphysik, Ruhr-Universität Bochum, D-44780 Bochum, Germany
(Dated: August 2, 2021)
The carrier spin coherence in a p-doped GaAs/(Al,Ga)As quantum well with a diluted hole gas
has been studied by picosecond pump-probe Kerr rotation with an in-plane magnetic field. For
resonant optical excitation of the positively charged exciton the spin precession shows two types
of oscillations. Fast oscillating electron spin beats decay with the radiative lifetime of the charged
exciton of 50 ps. Long lived spin coherence of the holes with dephasing times up to 650 ps. The
spin dephasing time as well as the in-plane hole g factor show strong temperature dependence,
underlining the importance of hole localization at cryogenic temperatures.
PACS numbers: 42.25.Kb, 78.55.Cr, 78.67.De
Recently the investigation of the coherent spin dynam-
ics in semiconductor quantum wells (QW) and quantum
dots has attracted much attention, due to the possible
use of the spin degree of freedom in novel fields of solid
state research such as spin-based electronics or quantum
information processing [1, 2, 3]. Until now the inter-
est has been mostly focused on the spin coherence of
electrons, while experimental information about the spin
coherence of holes is limited [4]. The hole as a Luttinger
spinor has properties strongly different from the electron
spin, such as a strong spin-orbit coupling, a strong direc-
tional anisotropy, etc. It plays an important role also in
coherent control of electron spins, since in many optical
schemes charged electron-hole complexes are proposed as
intermediate manipulation states [5].
Earlier, the hole spin dynamics in GaAs-based QWs
has been measured by optical orientation detecting
photoluminescence (PL) either time-integrated or time-
resolved [4, 6, 7, 8, 9]. Experimental studies addressed
the longitudinal spin relaxation time T1 [6, 7, 8] and the
dephasing time T ∗
, exploiting the observation of hole spin
quantum beats [4]. The reported relaxation times vary
from 4 ps [6] up to ∼1 ns [4, 8] demonstrating strong de-
pendence on doping level, doping density and excitation
energy. A major drawback of PL techniques is, however,
that the spin coherence can be traced only as long as
both electrons and holes are present and photolumines-
cence can occur. Further, they work only for studying
the spin dynamics of minority carriers in a sea of ma-
jority carriers and are therefore restricted to undoped or
n-type doped QWs. However, then the holes can interact
with electrons, providing additional relaxation channels
via exchange or shake-up processes [8, 10]. These mech-
anisms can be excluded for p-doped structures if the hole
spin relaxation occurs on time scales longer than the ra-
diative annihilation of electrons. A pump-probe Kerr ro-
tation (KR) technique using resonant excitation allows to
realize such measurements, which up to now have been
reported only for bulk p-type GaN [11] and not yet for
low-dimensional systems.
The theoretical analysis of the hole spin dynamics in
QWs has been focused on free holes [10, 12, 13, 14]
by considering different relaxation mechanisms: (i) a
Dyakonov-Perel like mechanism, (ii) an acoustic phonon
assisted spin-flip due to spin mixing of valence band
states, (iii) an exchange induced spin-flip due to scatter-
ing on electrons, which resembles the Bir-Aronov-Pikus
mechanism, but for holes. Recently the attention has
been drawn on the role of hole localization and the de-
phasing caused by fluctuations of the in-plane g factor
has been calculated [15].
In this paper we use time-resolved pump-probe Kerr
rotation [16] to investigate the spin coherence of holes in
a p-doped GaAs/Al0.34Ga0.66As single QW with a low
hole density. We find spin dephasing times reaching al-
most the ns-range at a temperature T = 1.6 K with a
hole in-plane g factor of about 0.01. Both quantities de-
crease strongly with increasing temperature, suggesting
the important influence of hole localization. We discuss
also a mechanism that provides generation of spin co-
herence for the hole gas under resonant excitation of the
positively charged exciton.
The structure was fabricated by molecular-beam epi-
taxy on a (100) oriented GaAs substrate. A 15 nm-
wide GaAs QW was grown on top of a 380 nm-
thick Al0.34Ga0.66As barrier and overgrown by a 190
nm-thick Al0.34Ga0.66As layer. 21 nm-thick layers
with Al0.34Ga0.66As effective composition realized by
GaAs/AlAs short-period superlattices were deposited on
both sides of the QW in order to improve interface pla-
narity. Two δ-doped layers with Carbon acceptors were
positioned symmetrically at 110 nm distance from the
QW. The hole gas concentration and mobility in the QW
are 1.51× 1011 cm−2 and 1.2× 105 cm2/Vs, respectively,
as determined by Hall measurements at T = 4.2 K. It
was possible to deplete the hole density in the QW by
above barrier illumination and even invert the majority
carrier type, resulting in a diluted electron gas. The sam-
ple temperature was varied from 1.6 to 6 K.
A mode-locked Ti:Sapphire laser with a repetition rate
of 75.6 MHz and a pulse duration of ∼1.5 ps (∼1 meV full
width at half maximum) was used for optical excitation.
http://arxiv.org/abs/0704.0592v1
FIG. 1: (a) KR traces for a p-type 15 nm-wide
GaAs/Al0.34Ga0.66As QW vs time delay between pump and
probe pulses at B = 0 and 7 T with field tilted by ϑ = 4◦
out of QW plane. Laser at 1.5365 eV is resonant with T+
line. Power was set to 5 and 1 W/cm2 for pump and probe,
respectively. Bottom trace was recorded with additional laser
illumination at 2.33 eV. T = 1.6 K. (b) PL spectra for DHG
(excitation at 1.579 eV) and DEG regime (above barrier ex-
citation at 2.33 eV). (c) Comparison of KR traces for ϑ = 0
and 4◦.
The laser beam was split into a circularly polarized pump
and a linearly polarized probe beam. Both beams where
focused on the sample surface to a spot diameter of ∼100
µm. Magnetic fields B ≤ 10 T were applied about per-
pendicular to the structure growth z-axis (Voigt geome-
try). In a pump-probe KR experiment the pump pulse
coherently excites carriers with spins polarized along the
z axis. The subsequent coherent evolution of the spins
in form of a precession about the magnetic field is tested
by the probe pulse polarization. To detect the change
of the linear probe polarization plane (the KR angle), a
homodyne technique based on phase-sensitive balanced
detection was used.
Photoluminescence spectra excited above and below
the band gap of the Al0.34Ga0.66As barriers are shown in
Fig. 1(b) at B = 0 and 7 T. A single PL line correspond-
ing to the positively charged trion T+ (consisting of two
holes and one electron) is seen for the regime of diluted
hole gas (DHG) established for below-barrier excitation.
After inverting the type of majority carriers to the DEG
FIG. 2: Top trace: KR signal measured at B = 7 T for ϑ = 4◦.
Bottom traces are obtained by separating electron and hole
contributions (see text). Excitation conditions as in Fig. 1.
regime by above barrier illumination the PL spectra con-
sist of the exciton (X) and negatively charged trion (T−)
lines.
The type of majority carriers in the QW can be iden-
tified by the KR signals measured at B = 7 T, with the
laser energy tuned to the trion resonance. The bottom
trace in Fig. 1(a) was measured with additional above-
barrier illumination (DEG regime) and shows long-lived
electron spin beats with a dephasing time of 2.5 ns which
is considerably longer than the radiative decay time of
resonantly excited trions of about 50 ps. The precession
frequency corresponds to a g factor | ge |= 0.285± 0.005,
which is typical for electrons in GaAs-based QWs.
Without above-barrier illumination in the DHG
regime, fast electron precession is observed only during
∼ 200 ps after pump pulse arrival [see middle trace in
Fig. 1(a)]. This signal is caused by the coherent preces-
sion of the electron in T+ and disappears with the trion
recombination. The electron beats are superimposed on
the hole beats with a much longer precession period. The
hole beats decay with a time constant of about 100 ps and
can be followed up to 500 ps delay. At these long times
the KR signal is solely given by coherent hole precession.
Experimentally, it is difficult to observe the hole spin
quantum beats due to the very small in-plane hole g fac-
tor. To enhance the visibility we tilted the magnetic
field slightly out of the plane by an angle ϑ = 4◦ to
increase the hole g factor by mixing the in-plane com-
ponent (gh,⊥) with the one parallel to the QW growth
axis (gh,‖), which typically is much larger: gh(ϑ) =
sin2 ϑ+ g2
h,⊥ cos
2 ϑ [17]. The strong change of the
hole beat frequency with the tilt angle is seen in Fig. 1(c).
The precession frequency is analyzed by ωh = µB |
gh | B/~, where µB is the Bohr magneton, and gives
| gh,⊥ |= 0.012±0.005 for ϑ = 0
◦ and | gh |= 0.048±0.005
for ϑ = 4◦.
The electron and hole contributions to the KR ampli-
FIG. 3: Hole component of KR signal at different magnetic
fields and ϑ = 4◦. Top inset: Magnetic field dependence of the
hole dephasing time T ∗2 . Solid line is 1/B fit to data. Bottom
inset: Hole spin precession frequency vs magnetic fields. Line
is guide to the eye. In inserts closed and open circles show the
data measured for pump to probe powers of 1 to 5 W/cm2
and 5 to 1 W/cm2, respectively. T = 1.6 K.
tude, ΘK, can be separated by fitting the experimental
data with a superposition form of exponentially damped
harmonic functions for electrons and holes:
i=e,h
Ai exp(−
) cos(ωi∆t). (1)
Ai are the corresponding signal amplitudes at pump-to-
probe delay ∆t = 0, and T ∗
2,i are the spin dephasing
times. An example for a decomposition of the KR signal
in the DHG regime is shown in Fig. 2.
Let us turn now to the hole coherence. Figure 3 shows
the hole contribution to the KR signal for different B at
T = 1.6 K. From the fit by Eq.(1) we have obtained the
dephasing time T ∗
, which is plotted versus B in the inset.
A very long lived hole spin coherence with T ∗2 = 650 ps is
found at B = 1 T. With increasing B up to 10 T it short-
ens to 70 ps. The field dependence can be well described
by a 1/B-form (see the line in the inset), from which
we conclude that the dephasing shortening arises from
the inhomogeneity of the hole g factor ∆gh = 0.0007 in
the QW, which is translated into a spread of the preces-
sion frequency: ∆ωh = ∆ghµBB/~. Since T
∝ 1/∆ωh,
this explains the 1/B-dependence of the dephasing time.
The magnetic field dependence of the hole precession fre-
quency in the lower inset of Fig. 3 shows an approximate
B-linear dependence up to 7 T. For higher fields a super
linear increase is seen which indicates a change of the
hole g factor due to mixing between heavy and light hole
states, induced by the field.
Two sets of experimental data measured for pump to
FIG. 4: Temperature dependence of the hole KR signal at
B = 7 T and ϑ = 5◦. Pump and probe powers are set to 1
and 5 W/cm2, respectively. Inset: Hole spin dephasing time
T ∗2 vs temperature.
probe powers 1 to 5 W/cm2 and 5 to 1 W/cm2 are com-
pared in the insets of Fig. 3. The very similar results
demonstrate performance of the experiment in the linear
regime for both pump and probe beams with power not
exceeding 5 W/cm2.
Insight into the origin of the long hole spin coherence
can be taken from KR at varying temperatures. The
data in Fig. 4 measured at ϑ = 5◦ show that (i) the
dephasing time T ⋆
decreases from 110 down to 60 ps
when increasing the temperature from 1.6 to 6 K (see also
inset), and (ii) simultaneously the precession frequency
decreases notably corresponding to a g factor decrease
from 0.057 to 0.030.
These results can be naturally explained by hole lo-
calization in the QW potential relief due to monolayer
well width fluctuations. The localization energy does not
exceed 0.5 meV, which is comparable with the thermal
energy at T = 6 K. Free holes are expected to have a
short spin coherence time T2 limited by the efficient spin
relaxation mechanisms due to the spin-orbit interaction
[10, 13, 14]. For localized holes these mechanisms are
mostly switched off and one can expect long T2 times.
However, in a KR experiment we do not measure the
T2 time, but rather the ensemble dephasing time T
(Ref. 15), as confirmed by the 1/B dependence in the
inset of Fig. 3. The T ∗2 time gives a lower boundary for
the spin coherence time T2. Therefore we can conclude,
that the T2 for localized holes is at least 650 ps. Thermal
delocalization of holes on the one hand decreases the role
of inhomogeneities and reduces ∆gh, which should lead
to longer dephasing times. On the other hand, then the
fast decoherence of free holes becomes the limiting factor
for the spin beats dynamics.
Mixing of heavy and light hole states in a QW is en-
hanced by localization effects. This should be detectable
by an increase of the in-plane hole g factor, which is close
to zero for free holes [4, 15, 18]. The decrease of the hole
g factor with increasing temperature shown in Fig. 4 is
consistent with a hole delocalization scenario.
We turn now to discussing the mechanism for opti-
cal generation of hole spin coherence in a QW with a
DHG. The generation mechanism is similar to the one
suggested for singly charged quantum dots [19, 20] and
QWs with a DEG [21]. In our experiment pump and
probe are resonant in energy with the positively charged
trion T+. Due to the considerable heavy-light hole split-
ting, the circularly polarized pump creates holes and elec-
trons with well-defined spin projections, Jh,z = ±3/2 and
Se,z = ±1/2, respectively, according to the optical selec-
tion rules [12]. Therefore, |⇑⇓↓〉 (|⇑⇓↑〉) trions can be
generated by a σ+ (σ−) polarized pump. Here the thick
and thin arrows give the spin states of holes and elec-
trons, respectively.
The pump pulse duration is much shorter than the spin
coherence and the electron-hole recombination times. If
in addition the pump duration is shorter than the charge
coherence time of the trion state the pulse creates a co-
herent superposition of a resident hole from the DHG and
a hole singlet trion T+. The spin state of the resident hole
with arbitrary spin orientation before excitation can be
described by α |⇑〉+ β |⇓〉, where |α|2 + |β|2 = 1. With-
out magnetic field and for fields oriented normal to the
z-axis, the net spin polarization of the hole ensemble is
zero, so that the ensemble averaged coefficients are equal:
α = β.
For σ+ polarized excitation, for which injection of an
|⇑↓〉 electron-hole pair is possible, the excited superposi-
tion is given by α |⇑〉+β cos(Θ/2) |⇓〉+iβ sin(Θ/2) |⇑⇓↓〉.
Here Θ =
d · E(t)dt/~ is the dimensionless pulse area
with the pump laser electric field E(t) and the dipole
transition matrix element d. In general, the hole-trion
superposition state may be driven coherently by varying
the pulse area, giving rise to Rabi-oscillations as reported
recently for (In,Ga)As quantum dots [20]. Such oscilla-
tions have not been found yet in QWs, most probably due
to the fast carrier dephasing, in particular for strong ex-
citation. Dephasing of the superposition occurs shortly
after the pulse on a time scale of a few ps, converting
the coherent polarization into a population consisting of
holes with original spins ⇑ and ⇓ and trions with ⇑⇓↓.
In a simplified picture, the spin coherence generation
can be described as follows: The σ+ polarized pump cre-
ates with certain efficiency trions T+ of spin configura-
tion |⇑⇓↓〉. By this process |⇓〉 holes are pumped out of
the DHG, leaving behind holes with opposite spin |⇑〉.
Right after the pump pulse the KR signal is contributed
by the |⇑〉 hole from the DHG and |↓〉 electron of the T+.
The further evolution of the coherent signal depends on
the strength of external magnetic field applied perpen-
dicular to the z-axis.
AtB=0, the carrier spins experience no Larmor preces-
sion. The electron spin relaxation time usually exceeds
the lifetime of trions, which is limited by radiative decay,
by one-two orders of magnitude. Trion recombination re-
turns the hole to the DHG with the same spin orientation
as it was pumped out, if no electron spin scattering oc-
curred in the meantime. This compensates the induced
spin polarization and nullifies the KR signal at delays
exceeding the trion lifetime. Indeed, the KR signal in
the top trace in Fig. 1(a) shows a fast decay with a time
constant of ∼50 ps, which is characteristic for radiative
trion recombination in GaAs/(Al,Ga)As QWs [22]. The
long-lived tail of the signal has a very small amplitude
and is due to hole coherence provided by weak spin re-
laxation of electrons in T+ and/or hole relaxation in the
DHG during the trion lifetime.
In finite magnetic fields, the carrier spins start to pre-
cess about B. Due to the electron spin precession in
T+, the hole spin returned to the DHG after trion re-
combination will not compensate the spin polarization of
the resident holes. Therefore, a long-lived hole coherence
with considerable amplitude will be induced. This co-
herence is observed in the KR signal as spin beats with
low frequency (see Figs. 1 and 3). Note that the Larmor
precession of the resident holes may also contribute to
generation of hole spin coherence, but the effect is pro-
portional to the ratio of the hole and electron Larmor
frequencies and therefore will be rather small.
Let us compare the spin coherence generation for QWs
with DHG and DEG resonantly excited in the T+ and
T− states, respectively. We are interested in a long-lived
spin coherence which goes beyond the trion lifetime, i.e.
in spin coherence induced for the resident carriers. In
both cases the amplitude of the KR signal is controlled
by the ratio of the electron spin beat period to the trion
lifetime. Nevertheless, the two cases are quite different as
for DHG the precessing electron is bound in the T+ trion,
while for DEG the background electron precesses. In the
latter case the electron precession in T− is blocked due
to the singlet spin character of the trion ground state.
To conclude, a long-lived spin coherence has been
found for localized holes in a GaAs/(Al,Ga)As QW with
a diluted hole gas. The spin coherence time exceeds 650
ps and is still masked by the spin dephasing due to g
factor inhomogeneities. Localization of holes suppresses
most spin relaxation mechanisms inherent for free carri-
ers. It is also worth to note, that due to the p-type Bloch
wave functions the holes do not interact with the nuclear
spins, which provides the most efficient spin relaxation
mechanism for localized electrons [23].
Acknowledgements. This work was supported by
the BMBF program ’nanoquit’.
[†] Also at Ioffe Physico-Technical institute, Russian
Academy of Sciences, St. Petersburg, Russia.
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