JOURNAL OF APPLIED PHYSICSVOLUME 93, NUMBER 3

1 FEBRUARY 2003

Intersubband absorption linewidth in GaAs quantum wells due toscattering by interface roughness, phonons, alloy disorder, and impurities
Takeya Unumaa) and Masahiro Yoshita
Institute for Solid State Physics, University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8581, Japan

Takeshi Noda and Hiroyuki SakakiInstitute of Industrial Science, University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan

Hidefumi Akiyama
Institute for Solid State Physics, University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8581, Japan

共Received 12 August 2002; accepted 13 November 2002兲
We calculate the intersubband absorption linewidth 2⌫ op in quantum wells 共QWs兲 due to scatteringby interface roughness, LO phonons, LA phonons, alloy disorder, and ionized impurities, andcompare it with the transport energy broadening 2⌫ tr⫽2ប/ ␶ tr , which corresponds to the transport
relaxation time ␶ tr related to the electron mobility ␮ . Numerical calculations for GaAs QWs clarify
the different contributions of each individual scattering mechanism to the absorption linewidth 2⌫ op
and transport broadening 2⌫ tr . Interface roughness scattering contributes about an order ofmagnitude more to the linewidth 2⌫ op than to the transport broadening 2⌫ tr , because thecontribution from the intrasubband scattering in the first excited subband is much larger than that in
the ground subband. On the other hand, LO phonon scattering 共at room temperature兲 and ionized
impurity scattering contribute much less to the linewidth 2⌫ op than to the transport broadening 2⌫ tr .
LA phonon scattering makes comparable contributions to the linewidth 2⌫ op and transport
broadening 2⌫ tr , and so does alloy disorder scattering. The combination of these contributions withsignificantly different characteristics makes the absolute values of the linewidth 2⌫ op and transport
broadening 2⌫ tr very different, and leads to the apparent lack of correlation between them when aparameter, such as temperature or alloy composition, is changed. Our numerical calculations canquantitatively explain the previously reported experimental results. © 2003 American Institute ofPhysics. 关DOI: 10.1063/1.1535733兴

I. INTRODUCTION

trasubband scattering in the first excited subband is muchlarger than that in the ground subband.11 Even in wide GaAs
QWs, where interface roughness scattering should be lesseffective, recent reports12,13 showed that interface roughness
scattering has a larger effect on linewidth than either
electron–electron scattering or bulk impurity scattering.
In the present article, we apply our theoretical method11
to scattering by LO phonons, LA phonons, alloy disorder,
and ionized impurities as well as interface roughness scattering, in order to compare their respective contributions to intersubband absorption linewidth and transport mobility. Numerical calculations for GaAs QWs confirm that the very
high sensitivity of linewidth to interface roughness scatteringis the key to quantitatively explaining the previously reported experimental results for linewidth in comparison withmobility.
The method presented here follows Ando’s theory,1 inwhich the intrasubband and intersubband energy-dependent
single-particle14 relaxation rates for various scattering
mechanisms are first calculated and then included in a formula for the two-dimensional 共2D兲 dynamical conductivity
Re ␴ zz ( ␻ ) to give the absorption line shape for photon frequency ␻ . This method is similar to a familiar method of
calculating transport mobility.15,16 It is important to note thatintersubband optical absorption is the collective excitation
among a confined electron gas. However, our present calcu-

The intersubband absorption linewidth in semiconductor
quantum wells 共QWs兲 closely relates to fundamental problems in the physics of optical transition, such as relaxation,1
many-body effects,2,3 and disorder.4,5 Furthermore, it is a key
factor in improving the performance of quantum cascadelasers6 and QW infrared photodetectors.7
To investigate the effects of various scattering processes,
intersubband absorption linewidths have been measured forvarious temperatures,8 well widths,9 alloy compositions,9 anddoping positions10 in GaAs and other QWs. These results
show that absorption linewidth has a weak dependence on
temperature and alloy composition and apparently has little
correlation with mobility. Its strong well-width dependence
suggests that the main contribution is from interface roughness scattering.
In a previous letter,11 we discussed the effect of interface
roughness scattering on linewidth by comparing calculations
based on a microscopic theory by Ando1 and experimental
data for modulation-doped GaAs/AlAs QWs with a wellwidth of 80 Å. The results made it clear that linewidth ismuch more sensitive to interface roughness scattering thantransport mobility is, because the contribution from the ina兲Electronic mail: unuma@issp.u-tokyo.ac.jp

0021-8979/2003/93(3)/1586/12/$20.00

1586© 2003 American Institute of Physics

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Unuma et al.

J. Appl. Phys., Vol. 93, No. 3, 1 February 2003

lation of single-particle relaxation rates and line shape isvery important and useful, because the absorption line shapeRe ˜␴ zz ( ␻ ) of collective excitation is given by the singleparticle dynamical conductivity ␴ zz ( ␻ ) via17

␴ zz 共 ␻ 兲
˜␴ zz 共 ␻ 兲 ⫽
i
1⫹
␴ 共␻兲
⑀ 0 ␬ 0 ␻ d eff zz

共1兲

in the crudest approximation. Here, ⑀ 0 is the vacuum permittivity, ␬ 0 is the static dielectric constant of the 2D material,and d eff is the effective thickness of the 2D electron gas.15
The collective excitation effects, or many-body interaction effects, on intersubband absorption linewidth have been
issues of recent interest in both theoretical and experimental
studies. In the limit of small band-nonparabolicity and constant single-particle relaxation rates, Nikonov et al. theoretically showed that many-body effects only cause blueshifts inabsorption spectra 共the depolarization shift兲 and that the linewidth is solely determined by the single-particle relaxation
rate.2In largely nonparabolic systems, the variation in energy
separation between the ground and first excited subbands
produces additional width in the single-particle excitationline shape Re ␴ zz ( ␻ ). However, many-body effects lead toredistribution of oscillator strength and collective excitationthat has a sharp resonance. As a result, the linewidth of thecollective excitation spectrum Re ˜␴ zz ( ␻ ) is significantly different from that of the single-particle excitation spectrum
Re ␴ zz ( ␻ ). 2,18,19 Furthermore, nonparabolicity causes difficulties in calculating single-particle relaxation rates andRe ␴ zz ( ␻ ) by Ando’s formalism.20 Experiments to elucidate
these effects were performed by Warburton et al. on InAs/
AlSb QWs.3
In the more popular systems of GaAs/AlGaAs and
InGaAs/InAlAs QWs, however, our present calculation,
which assumes small nonparabolicity, is applicable. The purposes of this article are 共1兲 to calculate intrasubband andintersubband single-particle relaxation rates for relevant scattering mechanisms as functions of in-plane kinetic energy
assuming small nonparabolicity and 共2兲 to quantitatively explain previously reported experimental data on linewidth andmobility in GaAs-based QWs, which appeared to have little
correlation.
In the next section, we summarize how linewidth andmobility are related to single-particle relaxation rates, andcalculate the single-particle relaxation rates for various scattering mechanisms as functions of the kinetic energy E. It isshown that linewidth and mobility have very different sensitivities to the same scattering mechanism. In Sec. III, previously reported experimental data for various temperatures,
well widths, alloy compositions, and doping positions arequantitatively explained by numerical calculations.
II. FORMULATION OF THE PROBLEM
A. General theory of intersubband absorption
lineshape and transport mobility in quantum wells

A general theory of intersubband absorption linewidth
due to elastic scatterers in 2D systems was formulated by

1587

Ando.1,20 According to Ando’s theory, the absorption line
shape of single-particle excitation between the two lowest
subbands can be expressed asRe ␴ zz 共 ␻ 兲 ⫽

*
冕
*
␲

e 2 f 10

m

2m

ប2

⫻

dE f 共 E 兲ប⌫ op共 E 兲
共 ប ␻ ⫺E 10兲 2 ⫹⌫ op共 E 兲 2,

共2兲

when all electrons are initially in the ground subband. Here,⌫ op共 E 兲 ⫽ 21 关 ⌫ intra共 E 兲 ⫹⌫ inter共 E 兲兴 ,⌫ intra共 E 兲 ⫽2 ␲共3兲具 兩 共 0k⬘ 兩 H 1 兩 0k兲兺k⬘⫺ 共 1k⬘ 兩 H 1 兩 1k兲 兩 2 典 ␦ 关 ␧ 共 k兲 ⫺␧ 共 k⬘ 兲兴 兩 E⫽␧(k) ,共4兲⌫ inter共 E 兲 ⫽2 ␲

具 兩 共 0k⬘ 兩 H 1 兩 1k兲 兩 2 典兺k⬘⫻ ␦ 关 ␧ 共 k兲 ⫺␧ 共 k⬘ 兲 ⫹E 10兴 兩 E⫽␧(k) ,共5兲

e is the elementary charge, ប is the reduced Planck constant,m * is the electron effective mass, f 10 is the oscillator
strength, E 10 (⫽E 1 ⫺E 0 ) is the intersubband energy separation, f (E) is the Fermi distribution function at temperature T,兩 nk) is the state vector of the electron with subband index n
and wave vector k, E n is the quantization energy, ␧(k)⫽ប 2 k 2 /2m * , H 1 is the scattering potential, and 具 ••• 典 denotes the average over distribution of scatterers. This theory
assumes a parabolic conduction band, or a constant effective
mass for different subbands; a modification for slightly nonparabolic systems like GaAs QWs will be described in a later
paragraph. In this article we denote the full width at half
maximum of the spectrum given by Eq. 共2兲 as 2⌫ op .
Note, on the other hand, that the transport relaxation
time ␶ tr(E), or the transport relaxation rate 2⌫ tr(E)⫽2ប/ ␶ tr(E) can be expressed as152ប
⫽4 ␲ 兺 具 兩 共 0k⬘ 兩 H 1 兩 0k兲 兩 2 典 ␦ 关 ␧ 共 k兲 ⫺␧ 共 k⬘ 兲兴␶ tr共 E 兲k⬘
⫻ 共 1⫺cos ␪ 兲 兩 E⫽␧(k) ,共6兲

where ␪ is the angle between k and k⬘ . The mobility is givenby ␮ ⫽e ␶ tr /m * with an average relaxation time of15,16

␶ tr⫽

冕 ␶

dE tr共 E 兲 E

⳵ f 共E兲
⳵E

冒冕

dEE

⳵ f 共E兲.
⳵E

共7兲

To enable quantitative comparison between the linewidth2⌫ op and mobility ␮ , we define the transport energy broadening as 2⌫ tr⫽2ប/ ␶ tr⫽2បe/m * ␮ . 11 In particular, lowtemperature transport broadening is given by 2⌫ tr⫽2⌫ tr(E F )⫽2ប/ ␶ tr(E F ), where E F is the 2D Fermi energy.
There are two relevant many-body effects: static and dynamic screening. The former screens the potentials of elastic
scatterers while the latter induces collective charge-density
excitation because of the incident optoelectric field.

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J. Appl. Phys., Vol. 93, No. 3, 1 February 2003

The static screening effect can be included by replacing
the scattering matrix element (mk⬘ 兩 H 1 兩 nk) with1

冋册

1
F (00)(mn) 共 q 兲⫺1.
共 mk⬘ 兩 H 1 兩 nk兲 ⫹ 共 0k⬘ 兩 H 1 兩 0k兲
⑀ 共 q,T 兲
F (00)(00) 共 q 兲
共8兲Here, qÄkÀk⬘ , ⑀ (q,T) is the static dielectric function,
and F (kl)(mn) (q) is a form factor defined by1

15,16

F (kl)(mn) 共 q 兲⫽冕 冕 ⬘␨ ␨ ␨ ⬘ ␨ ⬘dz

dz

k共 z 兲 l共 z 兲 m共 z

兲 n 共 z 兲 e ⫺q 兩 z⫺z ⬘ 兩 .

再

冋

册

冎

1
1
F (00)(11) 共 q 兲 ⫺1⫺
⫺1
, 共10兲
⑀ 共 q,T 兲 ⑀ 共 q,T 兲
F (00)(00) 共 q 兲and (0k⬘ 兩 H 1 兩 0k) in Eq. 共6兲 by ⑀ (q,T). In this article we
only treat symmetrical QWs, so there is no screening factor
in Eq. 共5兲. ⑀ (q,T) significantly increases mobility, while
S(q,T) hardly affects absorption linewidth because S(q,T)⬃1.
The dynamic screening effect is counted as a depolarization field, and the absorption line shape Re ˜␴ zz ( ␻ ) of theinduced collective charge-density excitation is given by
˜␴ zz 共 ␻ 兲 ⫽␴ zz 共 ␻ 兲⑀ zz 共 ␻ 兲

共11兲with the dynamical dielectric function of⑀ zz 共 ␻ 兲 ⫽1⫹i
␴ 共 ␻ 兲.
⑀ 0 ␬ 0 ␻ d eff zz

共12兲

The resonance energy Ẽ 10 of Re ˜␴ zz ( ␻ ) is blueshifted fromthe original resonance energy E 10 of Re ␴ zz ( ␻ ), andẼ 10⫽ 冑E 102 ⫹ 共 ប ␻ p 兲 2

共13兲with the plasma frequency of

␻ p⫽

冑

f 10N S e 2.
⑀ 0 ␬ 0 m * d eff

共14兲

The blueshift Ẽ 10⫺E 10⬇(ប ␻ p ) 2 /(2E 10) is called the depolarization shift. The linewidth of Re ˜␴ zz ( ␻ ) is the same asthat of Re ␴ zz ( ␻ ) if 2⌫ op(E) is independent of energy,2though they are different in general. When the depolarization
shift is sufficiently small, orẼ 10⫺E 10⬍2⌫ op共 0 兲 ,Band gap of Alx Ga1⫺x As (x⭐0.45) at 0 K
Band gap of AlAs at 0 K
Conduction-band discontinuity ratiofor GaAs/AlGaAs
Static dielectric constant
Optical dielectric constant
LO phonon energy
Deformation potential constantLongitudinal elastic constantSpin-orbit splitting
Kane energy(1.519⫹1.247x) eV
3.113 eV⬃0.65

␬ 0 ⫽12.91
␬ ⬁ ⫽10.92
ប ␻ LO⫽36.5 meV
D⫽13.5 eV
c l ⫽1.44⫻1011 N/m2
0.341 eV22.7 eV

共9兲

The z axis is set along the growth direction of samples, and␨ n (z) is the wave function for the nth subband electron motion in the z direction, which is chosen to be real. The screening correction only results in dividing (0k⬘ 兩 H 1 兩 0k) in Eq.
共4兲 by the factor
S 共 q,T 兲 ⫽TABLE I. Material constants of GaAs.

共15兲

˜␴ zz ( ␻ ) is approximately equal to ␴ zz ( ␻ ).
Although Eqs. 共2兲–共5兲 were derived assuming parabolic
bands, we may apply them to slightly nonparabolic systems
in which the additional width due to nonparabolicity is small
compared with the width due to scattering mechanisms. The
condition is expressed as (1⫺m 0* /m 1* )E F ⬍2⌫ op(0) at low

temperatures, where m n* is the electron effective mass in thenth subband. In this case, we can use the present theory by
19,21
replacing E 10 in Eq. 共2兲 with E 10(0)⫺(1⫺m *0 /m 1* )E,
which has a much larger influence on absorption linewidth than other corrections. Here, E 10(0) represents theintersubband energy separation at k⫽0. For consistency,
respective ␦ functions appearing with the squares of scattering matrix elements 兩 (0k⬘ 兩 H 1 兩 0k) 兩 2 , 兩 (1k⬘ 兩 H 1 兩 1k) 兩 2 ,兩 (0k⬘ 兩 H 1 兩 0k)(1k⬘ 兩 H 1 兩 1k) 兩 , and 兩 (0k⬘ 兩 H 1 兩 1k) 兩 2 in Eqs. 共4兲
and 共5兲 should be replaced by ␦ 关 ␧ 0 (k)⫺␧ 0 (k⬘ ) 兴 ,␦ 关␧1 (k)⫺␧1 (k⬘ ) 兴 , 21 兵 ␦ 关 ␧ 0 (k)⫺␧0 (k⬘ ) 兴 ⫹␦ 关 ␧ 1 (k)⫺␧1 (k⬘ ) 兴 其 ,and ␦ 关 ␧ 1 (k)⫺␧ 0 (k⬘ ) 兴 , where ␧ n (k)⫽E n ⫹ប 2 k 2 /2m n* . Values of m *n obtained from the Kane model are used in our
numerical calculations.
In most cases of GaAs QWs examined later in this article, the depolarization shift Ẽ 10⫺E 10 and the nonparabolicity effect (1⫺m *0 /m *1 )E F are small compared with 2⌫ op(0),so the absorption linewidth is estimated directly fromRe ␴ zz ( ␻ ) in Eq. 共2兲.
B. Scattering mechanisms

Here we calculate and compare 2⌫ op(E) and 2⌫ tr(E)due to scattering by interface roughness 共IFR兲, LO phonons,
LA phonons, alloy disorder 共AD兲, and ionized impurities
共ION兲. Furthermore, numerical calculations of each individual scattering mechanism are performed for modulationdoped GaAs 共or InGaAs兲/AlAs QWs. In actual samples, several scattering mechanisms coexist; the total scattering rate
can be obtained as the sum of their rates. Namely,
(IFR)
(LO)
(LA)
(AD)
⌫ op共 E 兲 ⫽⌫ op
共 E 兲 ⫹⌫ op
共 E 兲 ⫹⌫ op
共 E 兲 ⫹⌫ op
共E兲
(ION)
⫹⌫ op
共 E 兲 ⫹•••,

共16兲

⌫ tr共 E 兲 ⫽⌫ tr(IFR) 共 E 兲 ⫹⌫ tr(LO) 共 E 兲 ⫹⌫ tr(LA) 共 E 兲 ⫹⌫ tr(AD) 共 E 兲⫹⌫ tr(ION) 共 E 兲 ⫹•••.

共17兲

For simplicity, we perform numerical calculations insingle QWs with a finite barrier height of V 0 where band
bending due to doping is neglected. The origin of the z axis
is set at the center of the QWs. Material constants of GaAs
used in calculations are shown in Table I.
1. Interface roughness scattering

In GaAs/AlGaAs QWs, dominant monolayer 共ML兲 fluctuations are formed at the GaAs-on-AlGaAs interface 共Al-

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Unuma et al.

J. Appl. Phys., Vol. 93, No. 3, 1 February 2003

1589

GaAs surface covered by GaAs兲. We assume that the roughness height ⌬(r) at the in-plane position r⫽(x,y) has acorrelation function15,22

冉具 ⌬ 共 r兲 ⌬ 共 r⬘ 兲 典 ⫽⌬ 2 exp ⫺兩 rÀr⬘ 兩 2

⌳2

冊

共18兲

,

where ⌬ is the mean height of roughness and ⌳ is the correlation length. The scattering matrix element is given by
共 mk⬘ 兩 H 1 兩 nk兲 ⫽冕

d 2 rF mn ⌬ 共 r兲 e iq"r

共19兲

with
F mn ⫽V 0 ␨ m 共 ⫺L/2兲 ␨ n 共 ⫺L/2兲 ,共20兲

where L is the well width and ␨ n (⫺L/2) is the wave function
at the GaAs-on-AlGaAs interface. Because interface roughness is equivalent to local fluctuations in well width, F mn in
Eq. 共20兲 can also be expressed as
F mn ⫽ 冑共 ⳵ E m / ⳵ L 兲共 ⳵ E n / ⳵ L 兲 .

共21兲

In the case of the infinite-barrier approximation, Eq. 共20兲 canbe expressed in an alternative form as15,23
F mn ⫽

冏ប 2 d ␨ m共 z 兲 d ␨ n共 z 兲,
dz z⫽⫺L/2
2m * dz

共22兲

which is found to be proportional to L ⫺3 .
Substituting Eq. 共19兲 into Eqs. 共4兲 and 共5兲, we get(IFR)⌫ intra共 E 兲⫽(IFR)⌫ inter
共 E 兲⫽m *⌬ 2⌳ 2
ប2

␲

d

0

m *⌬ 2⌳ 2
ប

冕 ␪冋

2

F 012

册

2
F 00
2 2
⫺F 11 e ⫺q ⌳ /4,
S 共 q,T 兲
共23兲

冕 ␪
␲

2

2

d e ⫺q̃ ⌳ /4,

共24兲

0

where the absolute values of the 2D scattering vectors q andq̃ are given by11
q 2 ⫽2k 2 共 1⫺cos ␪ 兲 ,q̃ 2 ⫽2k 2 ⫹2m * E 10
ប

2

⫺2k

冑

共25兲
k 2⫹

2m * E 10ប2

cos ␪ .

共26兲FIG. 1. 2⌫ op(E), ⌫ intra(E), ⌫ inter(E), and 2⌫ tr(E) due to interface roughness scattering, plotted as functions of the in-plane kinetic energy E.

ing rates. Finally, and most importantly, they include different factors 关 F 00 /S(q,T)⫺F 11兴 2 , F 012 , and 2F 002 . S(q,T)
can be neglected because S(q,T)⬃1. As is shown below, F 11
is much larger than F 00 , because E 1 is more sensitive to L
than E 0 . 共In the infinite-barrier approximation, F 11 is fourtimes larger than F 00 .) As a result, ⌫ intra(E) is much larger
than 2⌫ tr(E).
Figure 1 shows 2⌫ op(E), ⌫ intra(E), ⌫ inter(E), and2⌫ tr(E) in a modulation-doped GaAs/AlAs QW with L⫽80 Å, ⌬⫽3 Å, and ⌳⫽50 Å. These values of ⌬ and ⌳ aretypical for the GaAs-on-AlAs interface.11,22,24 Temperature
was set at T⫽0 K, and sheet electron concentration was chosen to be N S ⫽5⫻1011 cm⫺2 , which gives Fermi energy ofE F ⫽17.8 meV. The same values of L and N S are also used
for calculations of other scattering mechanisms in Sec. II.
In Fig. 1, ⌫ intra(E) decreases as E increases, and it has amaximum value of 8.3 meV at E⫽0 meV. ⌫ inter(E) is almost constant with respect to E, and its value of 0.6 meV ismuch smaller than that of ⌫ intra(E). The values of 2⌫ tr(E)are 0 meV at E⫽0 meV owing to the screening effect, and
0.6 meV at E⫽E F , which determines the low-temperaturetransport broadening. As a result, 2⌫ op(E), the sum of⌫ intra(E) and ⌫ inter(E), is found to be much larger than2⌫ tr(E).
Figure 2 shows 2⌫ op and 2⌫ tr as functions of correlation
length ⌳; they are calculated respectively from Eqs. 共2兲 andOn the other hand, we can express the transport relaxation time ␶ tr(IFR) (E), 26 or the transport relaxation rate2⌫ tr(IFR) (E)⫽2ប/ ␶ tr(IFR) (E) as
2⌫ tr(IFR) 共 E 兲 ⫽2m * ⌬ 2 ⌳ 2
ប

2

F 002

冕 ␪⑀
␲

d

0

1⫺cos ␪共 q,T 兲222e ⫺q ⌳ /4,
共27兲

which is similar to Eqs. 共23兲 and 共24兲.
It is useful here to comment on the similarities and differences in the equations for ⌫ intra(E), ⌫ inter(E), and2⌫ tr(E). First, all three are proportional to ⌬ 2 , and also to⌳ 2 for small ⌳. Second, ⌫ inter(E) is much smaller
than ⌫ intra(E), because q̃ is smaller than q. Third,(1⫺cos ␪)/⑀(q,T)2 appearing in 2⌫ tr(E) shows that the forward scattering ( ␪ ⬃0) does not contribute to transport
broadening, and that the screening effect reduces the scatter-

FIG. 2. Intersubband absorption linewidth 2⌫ op and the transport energy
broadening 2⌫ tr due to interface roughness scattering, plotted as functionsof the correlation length ⌳. The 2⌫ op,para represents the absorption linewidthwithout the additional width due to band-nonparabolicity.

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共6兲 in a modulation-doped GaAs/AlAs QW with L⫽80 Å,
N S ⫽5⫻1011 cm⫺2 , T⫽0 K, and ⌬⫽3 Å. 2⌫ op,para represents the linewidth calculated without changing E 10 into
E 10(0)⫺(1⫺m 0* /m *1 )E in Eq. 共2兲.
In Fig. 2, 2⌫ op,para and 2⌫ tr are both proportional to ⌳ 2
for small ⌳ with the difference in absolute values being
about one order of magnitude. With nonparabolicity, 2⌫ op
has a lower limit of (1⫺m *0 /m 1* )E F ⫽1.35 meV in addition
to 2⌫ op,para . For large ⌳, the insensitivity of 2⌫ tr to forward
scattering causes its value to be smaller. This shows that thecorrelation length of ⌳⬃1/k F contributes most to 2⌫ tr ,where k F is the Fermi wave number. In principle, values of
the roughness parameters ⌬ and ⌳ can be uniquely determined if linewidth and mobility are both measured at low
temperatures.
As shown earlier, the main characteristic of interface
roughness scattering is its order-of-magnitude different contributions to linewidth and transport broadening 共and hence,
to mobility兲. This is the key point for understanding the apparent lack of correlation between them.

Since LO phonon scattering is an inelastic process, Eqs. 共4兲
and 共5兲 are not applicable in their original forms. However,by modifying the ␦ functions in Eqs. 共4兲 and 共5兲:␦ 关 ␧ 共 k兲 ⫺␧ 共 k⬘ 兲兴 → ␦ 关 ␧ 共 k兲 ⫺␧ 共 k⬘ 兲 ⫾ប ␻ LO兴 ,␦ 关 ␧ 共 k兲 ⫺␧ 共 k⬘ 兲 ⫹E 10兴 → ␦ 关 ␧ 共 k兲 ⫺␧ 共 k⬘ 兲 ⫹E 10⫾ប ␻ LO兴 ,共34兲

such that total energy is conserved, we can estimate width of
the zero-phonon band in an approximation that neglects phonon sidebands. Here, ⫾ indicates phonon absorption (⫹)
and emission (⫺). Thus, we have(LO)⌫ intra共 E 兲⫽

兩 M 2D兩 2 ⫽

再

⫹

(LO)
⌫ inter
共 E 兲⫽冕 ␨

m共 z 兲 ␨ n共 z 兲 eiq z z.

共29兲

Since the method of calculating the 3D scattering matrix
element is well established, the 2D scattering matrix element
can be easily obtained.
In polar optical phonon scattering, or simply LO phonon
scattering, the 3D scattering matrix element is given by25

具 兩 M 3D兩 典 ⫽e 2 ប ␻ LO共 N LO⫹1 兲
2 ⑀ 0Q 2

冉

1
1
⫺
␬⬁ ␬0

冊

冉

e 2 ប ␻ LON LO 1
1
⫺
2
␬
␬
⬁
0
2 ⑀ 0Qd␪

具 N LO⫹1 典
qe关 F (00)(00) 共 q e兲

关 F (00)(00) 共 q a兲 ⫺2F (00)(11) 共 q a兲

qa

冎

冊

共31兲

共32兲

冉

共35兲

m * e 2 ␻ LO 11⫺
4 ␲ ⑀ 0ប ␬ ⬁ ␬ 0

⫹

冊冕

␲

0

具 N LO⫹1 典
q̃ e册具 N LO典d␪F (01)(10) 共 q̃ e兲
共36兲F (01)(10) 共 q̃ a兲 ,q̃ awhere ⌰(E) is the Heaviside step function. Absolute valuesof scattering vectors are given by
q 2e ⫽2k 2 ⫺2m * ␻ LO⫺2kប

q a2 ⫽2k 2 ⫹q̃ 2e ⫽2k 2 ⫹⫺2kfor phonon absorption processes. Here, Q is the absolutevalue of the 3D scattering vector, ␬ ⬁ is the optical dielectric
constant, ␻ LO is the LO phonon frequency, and N LO is theLO phonon occupation given by1.
N LO⫽ ប ␻ /k T
e LO B ⫺1

具 N LO典冋

共30兲

for phonon emission processes, and by

具 兩 M 3D兩 2 典 ⫽0⫻ ⌰ 共 E⫹E 10⫺ប ␻ LO兲

共28兲

Here, M 2D and M 3D are the 2D and 3D scattering matrix
elements, respectively, and I(q z ) is given by2␲⫺2F (00)(11) 共 q e兲 ⫹F (11)(11) 共 q e兲兴

z

dz

冊冕

⫹F (11)(11) 共 q a兲兴 ,兩 M 3DI 共 q z 兲 兩 2 .
兺
q

I 共 q z 兲 ⫽I mn 共 q z 兲 ⫽冉m * e 2 ␻ LO 11⫺
4 ␲ ⑀ 0ប ␬ ⬁ ␬ 0⫻ ⌰ 共 E⫺ប ␻ LO兲

2. LO phonon scattering

In considering phonon scattering processes, it should be
noted that phonons have approximately three-dimensional
共3D兲 properties, since they are hardly confined to QWs. The
z-component momentum conservation in 3D systems requires the scattering matrix element of 2D electrons to be
calculated from25

共33兲

冑

⫺2k

k 2⫺

2m * ␻ LO⫺2k
ប

2m * E 10q̃ a2 ⫽2k 2 ⫹

冑
冑

ប

2

k 2⫹

⫺

冑

ប

2

k 2⫹

共37兲

2m * ␻ LOcos ␪ ,
ប

共38兲

k 2⫹

2m * ␻ LO
ប2m * E 102m * E 102m * ␻ LOcos ␪ ,
ប⫺

2m * ␻ LOcos ␪ ,បប2⫹2m * ␻ LO
ប2m * E 10
ប2⫹2m * ␻ LOcos ␪ ,
ប

共39兲

共40兲

and the subscripts ‘‘e’’ and ‘‘a’’ represent emission and absorption of LO phonons, respectively.
On the other hand, the transport relaxation rate can be
expressed as

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Unuma et al.

J. Appl. Phys., Vol. 93, No. 3, 1 February 2003

FIG. 3. Dependence of the form factors F (00)(00) (q), F (00)(11) (q),F (11)(11) (q), and F (01)(10) (q) on the absolute value of the two-dimensional
scattering vector q.

2⌫ tr(LO) 共 E 兲 ⫽冉冊

m * e 2 ␻ LO 111⫺
2 ␲ ⑀ 0 ប ␬ ⬁ ␬ 0 1⫺ f 共 E 兲

再

冕 ␪
␲

⫻

具 N LO⫹1 典
qe

具 N LO典

Acoustic phonon scattering via deformation potential
coupling, or simply LA phonon scattering, is virtually elastic.
The 3D scattering matrix element is given by25

qa

F (00)(00) 共 q e兲 ⫹ 关 1⫺ f 共 E⫹ប ␻ LO兲兴F (00)(00) 共 q a兲冎2⌫ tr(E) due to LO phonon scattering at T⫽300 K. First,⌫ intra(E) is much smaller than 2⌫ tr(E), because the difference in intrasubband scattering matrix elements for the two
subband is small in LO phonon scattering as already shownin Fig. 3. Second, ⌫ inter(E) is much smaller than 2⌫ tr(E)owing to the small form factor of F (01)(10) (q) and the large
absolute value of scattering vector q̃. Third, when the kinetic
energy E is larger than the LO phonon energy of E LO
⫽36.5 meV, intrasubband LO phonon emission is allowed,
which makes 2⌫ tr(E) and ⌫ intra(E) larger. As a result,2⌫ op(E) is much smaller than 2⌫ tr(E) at room temperature.
When systems are cooled down to 0 K, only intersubband LO phonon spontaneous emission is allowed in the caseof E 10⬎E LO . Therefore, ⌫ intra(E) and 2⌫ tr(E) vanish, andonly ⌫ inter(E) has a finite value of about 1 meV.

3. LA phonon scatteringd0⫻ ⌰ 共 E⫺ប ␻ LO兲关 1⫺ f 共 E⫺ប ␻ LO兲兴
⫻

1591

共41兲

in the approximation that neglects the ‘‘in-scattering term.’’16
Since LO phonon frequency is high, the screening effect canbe neglected.
The four form factors F (00)(00) (q), F (00)(11) (q),F (11)(11) (q), and F (01)(10) (q) appearing in Eqs. 共35兲, 共36兲,and 共41兲 are plotted as functions of q in Fig. 3. First,
F (00)(00) (q), F (00)(11) (q), and F (11)(11) (q) are very close,
which makes F (00)(00) (q)⫺2F (00)(11) (q)⫹F (11)(11) (q) in⌫ intra(E) much smaller than F (00)(00) (q) in 2⌫ tr(E). In otherwords, the difference in intrasubband scattering matrix elements for the two subbands is small in LO phonon scattering.
Second, F (01)(10) (q) in ⌫ inter(E) is much smaller thanF (00)(00) (q).
Figure 4 shows 2⌫ op(E), ⌫ intra(E), ⌫ inter(E), and具 兩 M 3D兩 2 典 ⫽k B TD 2
2c l

for both LA phonon emission and absorption processes,
where D is the deformation potential constant and c l is thelongitudinal elastic constant. Note that Eq. 共42兲 is independent of the scattering vector as a result of the linear dispersion relation of LA phonons. Therefore, we have(LA)⌫ intra共 E 兲⫽(LA)⌫ inter
共 E 兲⫽

m * k B TD 2

␲ ប 2c lm * k B TD 2

␲ប cl
2

冕 ␪冕 冋 ␨
␲

d

dz

0

冕 ␪冕
␲

d

0

0共 z 兲

册

2

S 共 q,T 兲2⫺ ␨ 1共 z 兲 2 ,dz 关 ␨ 0 共 z 兲 ␨ 1 共 z 兲兴 2 .

共43兲
共44兲

(LA)
⌫ inter(E) is independent of the kinetic energy E, and(LA)(E)is also almost independent of it because S(q,T)⌫ intra⬃1.
On the other hand, the transport relaxation rate is given
by15,252⌫ tr(LA) 共 E 兲 ⫽FIG. 4. 2⌫ op(E), ⌫ intra(E), ⌫ inter(E), and 2⌫ tr(E) due to LO phonon scattering, plotted as functions of the in-plane kinetic energy E. LO phonon
energy is ប ␻ LO⫽36.5 meV.

共42兲

2m * k B D 2 T

␲ប cl
2

冕 ␪⑀
␲

d

0

1⫺cos ␪共 q,T 兲 2

冕 ␨

dz 0 共 z 兲 4 ,共45兲

which has an energy dependence due to the screening effect.
Note here that the z-integrals of ␨ 0 (z) 4 , ␨ 1 (z) 4 , and关 ␨ 0 (z) ␨ 1 (z) 兴 2 have comparable values (3/2L, 3/2L, and1/L, respectively, in the infinite-barrier approximation兲; thus
⌫ intra(E) and ⌫ inter(E) are nearly equal, and 2⌫ op(E), thesum of them, is comparable with 2⌫ tr(E).
Figure 5 shows 2⌫ op(E), ⌫ intra(E), ⌫ inter(E), and2⌫ tr(E) at T⫽300 K. ⌫ intra(E) and ⌫ inter(E) have almost thesame constant values of about 0.5 meV. 2⌫ tr(E) vanishes at
E⫽0 meV owing to the screening effect, and approaches aconstant value of about 1.5 meV as E increases.

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1592

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J. Appl. Phys., Vol. 93, No. 3, 1 February 2003

FIG. 5. 2⌫ op(E), ⌫ intra(E), ⌫ inter(E), and 2⌫ tr(E) due to LA phonon scattering, plotted as functions of the in-plane kinetic energy E.

4. Alloy disorder scattering

When there are alloy layers composed of Ax B1⫺x C, suchas Alx Ga1⫺x As and Inx Ga1⫺x As, electrons are scattered byconduction band disorder. The scattering matrix element due
to alloy disorder is given by26,27
a 3 共 ␦ E c 兲 2 x 共 1⫺x 兲
4

具 兩 共 mk⬘ 兩 H 1 兩 nk兲 兩 2 典 ⫽

⫻

冕

alloy

dz 关 ␨ m 共 z 兲 ␨ n 共 z 兲兴 2 ,共46兲

where a is the lattice constant and ␦ E C is the difference in
conduction band minima of crystals AC and BC 共AlAs and
GaAs in the case of Alx Ga1⫺x As). Note that Eq. 共46兲 isindependent of the scattering vector owing to the short-range
nature of the scatterers. Therefore, we have(AD)
⌫ intra共 E 兲⫽

m * a 3 共 ␦ E c 兲 2 x 共 1⫺x 兲⫻(AD)⌫ inter
共 E 兲⫽

冋

␲ប

2

冕 ␪冕
␲

d

0

册

alloy

共47兲

m * a 3 共 ␦ E c 兲 2 x 共 1⫺x 兲⫻

dz 关 ␨ 0 共 z 兲 ␨ 1 共 z 兲兴 .
2

d

alloy

共48兲

⌫ inter(E) is independent of E, and ⌫ intra(E) is also almost
independent of it because S(q,T)⬃1.
On the other hand, the transport relaxation rate is given
by26,272⌫ tr(AD) 共 E 兲 ⫽(ION)⌫ intra共 E 兲⫽

␲ប2

冕 ␪⑀
␲

d

0

冕 ␨
dz

m *e 4
4␲⑀0 ␬0 ប
2⫻

m共 z 兲 ␨ n共 z 兲 e⫺q 兩 z⫺Z 兩. 共50兲

1⫺cos ␪共 q,T 兲 2

冕

alloy

(ION)
⌫ inter
共 E 兲⫽dz ␨ 0 共 z 兲 4 ,共49兲

which has an energy dependence due to the screening effect.
Since alloy disorder scattering is due to ␦ E C , it can be
regarded as a kind of roughness scattering. If one substitutes V 0 ⫽ ␦ E C , ⌬ 2 ⫽a 2 x(1⫺x)/4, and ⌳ 2 ⫽a 2 /2␲ into

2 2冕

再 冕 冋␨
1qdz

dZN 共 Z 兲

0共 z 兲

冕 ␪
␲

d

0

册

2

S 共 q,T 兲⫺ ␨ 1 共 z 兲 2 e ⫺q 兩 z⫺Z 兩

冎

2

,
共51兲

2m * a 3 共 ␦ E C 兲 2 x 共 1⫺x 兲⫻e2
2 ⑀ 0␬ 0q

Therefore, we have␲0

When dopant donors of Si are ionized, electrons supplied to QWs suffer from scattering by the Coulomb potential of the donors. The scattering matrix element due to an
ionized impurity at position Z is given by1
共 mk⬘ 兩 H 1 兩 nk兲 ⫽

2

冕 ␪冕

Eqs. 共23兲, 共24兲, and 共27兲, one can recognize that the alloy
disorder scattering of Eqs. 共47兲–共49兲 is expressed as the sumof the ‘‘roughness scattering’’ rates due to the alloy layer at
position z.
Note here that ⌫ intra(E), ⌫ inter(E), and 2⌫ tr(E) for alloy
disorder scattering are similar in form to those for LA phonon scattering; thus 2⌫ op(E) is comparable with 2⌫ tr(E), as
in LA phonon scattering.
Figure 6 shows 2⌫ op(E), ⌫ intra(E), ⌫ inter(E), and2⌫ tr(E) in In0.1Ga0.9As/AlAs QWs (x⫽0.1) at T⫽0 K. We
set the lattice constant and conduction band offset to approximately a⫽5.66 Å and ␦ E C ⫽700 meV, respectively.
⌫ intra(E) and ⌫ inter(E) have almost the same constant valuesof about 0.2 meV. 2⌫ tr(E) vanishes at E⫽0 meV owing to
the screening effect, and approaches a constant value ofabout 0.57 meV as E increases.
5. Ionized impurity scatteringdz

2
␨ 0共 z 兲 2⫺ ␨ 1共 z 兲 2 ,S 共 q,T 兲␲បFIG. 6. 2⌫ op(E), ⌫ intra(E), ⌫ inter(E), and 2⌫ tr(E) due to alloy disorderscattering, plotted as functions of the in-plane kinetic energy E.

m *e 4
4␲⑀0 ␬0 ប
2

⫻

冋冕
1

q̃

2 2

冕

dZN 共 Z 兲

dz ␨ 0 共 z 兲 ␨ 1 共 z 兲 e

冕 ␪
␲

d

0

⫺q̃ 兩 z⫺Z 兩

册

2

,

共52兲

where N(Z) is the 3D impurity concentration at position Z.
The transport relaxation rate, on the other hand, is given
by15,16

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Unuma et al.

J. Appl. Phys., Vol. 93, No. 3, 1 February 2003

FIG. 7. 2⌫ op(E), ⌫ intra(E), ⌫ inter(E), and 2⌫ tr(E) due to ionized impurity
共ION兲 scattering, plotted as functions of the in-plane kinetic energy E.

2⌫ tr(ION) 共 E 兲 ⫽m *e 42␲⑀0 ␬0 ប

冋冕dZN 共 Z 兲

冕 ␪ ⑀
␲

d

q 2 共 q,T 兲 2

dz ␨ 0 共 z 兲 2 e ⫺q 兩 z⫺Z 兩 .

共53兲

2 2

册

FIG. 8. Intersubband absorption spectrum observed at 4.5 K in a GaAs/AlAs
single QW. The linewidth 2⌫ op was 11.1 meV and the transport energy
broadening 2⌫ tr was 1.2 meV, which was calculated from the mobility ␮ of
2.9⫻104 cm2/V s.

1⫺cos ␪

0

2

⫻

冕

1593

2

Figure 7 shows 2⌫ op(E), ⌫ intra(E), ⌫ inter(E), and2⌫ tr(E) in a ␦ -doped GaAs/AlAs QW with 60 Å spacer layers (Z⫽100 Å) at T⫽0 K. First, ⌫ intra(E) is much smaller
than 2⌫ tr(E), because the difference in intrasubband scattering matrix elements for the two subbands is small in ionized
impurity scattering. Second, ⌫ inter(E) is much smaller than2⌫ tr(E) owing to the large absolute value of scattering vector q̃. As a result, 2⌫ op(E) is much smaller than 2⌫ tr(E).
III. COMPARISON WITH EXPERIMENTAL RESULTS

In this section, the absorption linewidth 2⌫ op and transport energy broadening 2⌫ tr are calculated for some GaAs
共or InGaAs兲/AlAs 共or AlGaAs兲 QWs as functions of temperature, well width, alloy composition, and donor doping
position. These results are compared with previously reported experimental data.
A. Temperature dependence

Experimental measurements of the temperature dependence of absorption linewidth are expected to clarify the effects of phonon scattering. We previously reported absorption linewidths in comparison with transport mobilities in amodulation-doped GaAs/AlAs single QW with a well widthof L⫽80 Å and a sheet electron concentration of N S ⫽9.8
⫻1011 cm⫺2 , at temperatures ranging from 4.5 to 300 K. 11
The absorption spectrum observed at 4.5 K is shown in Fig.
8. The low-temperature linewidth 2⌫ op was 11.1 meV andthe low-temperature transport broadening 2⌫ tr⫽2បe/m 0* ␮
was 1.2 meV, which was calculated from the mobility ␮ of
2.9⫻104 cm2/V s. Note that linewidth was one order of magnitude larger than transport broadening at low temperatures.
The temperature dependences of linewidth and transport
broadening are plotted in Fig. 9 by solid and open circles,
respectively.
We performed the corresponding calculations of linewidth and transport broadening by considering interface
roughness scattering with ⌬⫽4 Å and ⌳⫽43 Å, LO phonon

scattering, and LA phonon scattering. The contribution ofalloy disorder scattering was absent because the GaAs QW
had AlAs barriers, and the influence of ionized impurity scattering was sufficiently reduced by the spacer layers.
The calculated results for the linewidth 2⌫ op versus temperature are also shown in Fig. 9 by dashed 共IFR兲, dashdotted 共IFR⫹LO兲, and solid 共IFR⫹LO⫹LA兲 curves, in comparison with the transport broadening 2⌫ tr . Additional width
due to nonparabolicity is already included in these three
curves, making small corrections compared with the contribution of interface roughness scattering, as seen in Fig. 2. At
low temperatures, interface roughness scattering contributes
10.4 meV to linewidth, and LO-phonon spontaneous emission contributes 0.7 meV. Phonon scattering processes become more active as temperature rises, and LO and LA phonon scattering contribute 1.8 and 0.7 meV, respectively, tolinewidth at room temperature. These calculated results arein good agreement with the experimental data shown bysolid circles. Note that the increase in dashed line 共IFR兲 withincreasing temperature is due to the nonparabolicity effect;
the contribution of interface roughness scattering itself
slightly decreases with increasing temperature, as expectedfrom the energy dependence in Fig. 1.
For the transport broadening 2⌫ tr , interface roughness
scattering makes a dominant contribution of 0.73 meV at low

FIG. 9. Temperature dependence of the intersubband absorption linewidth2⌫ op and transport energy broadening 2⌫ tr 共or mobility ␮ ). Circles show
experimental values, and lines show numerical results calculated by considering interface roughness, LO phonon, and LA phonon scattering.

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1594

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FIG. 10. Well-width dependence of intersubband absorption linewidth, calculated at 0 K by considering interface roughness and LO phonon scattering. Solid circles show experimental results measured at low temperatures
by Campman et al. 共see Ref. 9兲.

temperatures, which nearly explains the experimental valueof 1.2 meV. As already pointed out in Sec. II, this value of0.73 meV is an order of magnitude smaller than the contribution of 10.4 meV to linewidth. In the temperature range
above 80 K, the contribution of LO phonon scattering totransport broadening rapidly increases as temperature rises,
and reaches a dominant value of 9.3 meV at 300 K, as is wellknown. Such an effect of LO phonon scattering on transport
broadening is very different from that on linewidth. The contribution of LA phonon scattering to transport broadening is
1.2 meV, which is comparable with that to linewidth.
As a result, linewidth and transport broadening havevery different dependences on temperature. Similar behaviorof linewidth versus temperature was also reported for
GaAs/Al0.3Ga0.7As QWs by Allmen et al.8

B. Well-width dependenceInterface roughness scattering is expected to give absorption linewidth considerably strong well-width dependence. Campman et al. reported low-temperature linewidths
and mobilities in modulation-doped GaAs/Al0.3Ga0.7As QWs
with N S ⬃6⫻1011 cm⫺2 for various well widths in the rangeL⫽75⫺110 Å. 9 Here, we calculate linewidth and transport
broadening for the same structures. As scattering mechanisms, interface roughness scattering with ⌬⫽3 Å and ⌳⫽85 Å and LO phonon scattering are included one by one.
Figure 10 shows the calculated results for lowtemperature linewidth versus well width in the range L⫽75⫺110 Å. First, the well-width dependence of linewidth due
to interface roughness scattering, shown by the dashed curve
共IFR兲, is not so strong for small L, because the confinementof the first excited state is weaker and thus F 11⫽ ⳵ E 1 / ⳵ L is
considerably smaller in GaAs/Al0.3Ga0.7As QWs than in
infinite-barrier QWs. Second, the contribution of LO phonon
scattering slowly increases as QWs become wider, whichmakes the well-width dependence of linewidth slightly
weaker. The solid curve 共IFR⫹LO兲 is in good agreementwith experimental results shown by solid circles.9 If barriers
are higher, as in GaAs/AlAs QWs, the first excited state ismore strongly confined and the interface roughness scatter-

FIG. 11. Well-width dependence of transport energy broadening, calculated
at 0 K by considering interface roughness scattering. Open circles show
experimental results measured at low temperatures by Campman et al. 共see
Ref. 9兲.

ing contributes much more to linewidth than LO phonon
scattering does, which will lead to a much stronger wellwidth dependence of linewidth.
On the other hand, the well-width dependence of lowtemperature transport broadening is shown in Fig. 11. The
transport broadening considered here is determined only byinterface roughness scattering, because intrasubband LOphonon emission and absorption are impossible at low temperatures. F 002 ⫽( ⳵ E 0 / ⳵ L) 2 in Eq. 共27兲 is proportional toL ⫺6 in the infinite-barrier approximation, and this leads to astrong well-width dependence of transport broadening evenin finite-barrier QWs. The calculated curve explains the experimental results plotted by open circles9 very well.
C. Alloy composition dependence

Experimental measurements of the alloy composition dependence of linewidth are expected to show the effects ofalloy disorder scattering. Campman et al. reported lowtemperature linewidths and mobilities in modulation-doped
Inx Ga1⫺x As/Al0.3Ga0.7As QWs with L⫽100 Å and N S ⬃8⫻1011 cm⫺2 for various compositions in the range x⫽0 – 0.1. 9 We calculate linewidth and transport broadening
for the same structures. As scattering mechanisms, interface
roughness scattering with ⌬⫽3.5 Å and ⌳⫽40 Å, LO phonon scattering, and alloy disorder scattering are included one
by one.
Figure 12 shows the calculated results for lowtemperature linewidth versus alloy composition x in therange x⫽0 – 0.1. The contribution of interface roughness
scattering is 1.6 meV at x⫽0, and slowly increases as x increases because QWs become deeper. LO phonon scattering
contributes approximately 1 meV to linewidth, almost independently of x. Although the contribution of alloy disorderscattering is proportional to x for small x, it is as small as
0.24 meV even at x⫽0.1. Our calculations explain the experimental observation of linewidth being insensitive to alloy
composition, plotted by solid circles.9
On the other hand, transport broadening is shown in Fig.
13 as a function of x. Interface roughness scattering contributes 0.1 meV to transport broadening, while alloy disorderscattering makes the larger contribution of 0.27 meV at x

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J. Appl. Phys., Vol. 93, No. 3, 1 February 2003

FIG. 12. Alloy composition dependence of intersubband absorption linewidth, calculated at 0 K by considering interface roughness, LO phonon,
and alloy disorder 共AD兲 scattering. Solid circles show experimental results
measured at low temperatures by Campman et al. 共see Ref. 9兲.

⫽0.1; this shows that transport mobility drops remarkably as
x increases. Our calculations explain the experimental resultsplotted by open circles.9 The small disagreement may be due
to clustering in alloy layers, where the effective correlation
length of alloy disorder in terms of roughness scattering may
be larger than a/ 冑2 ␲ in actual samples grown by molecularbeam epitaxy 共MBE兲 or metalorganic chemical vapor deposition 共MOCVD兲.
It should be noted that the one-order-of-magnitude different contributions of interface roughness scattering to linewidth and transport broadening are important in explaining
their different behaviors versus alloy composition. Alloy disorder scattering in itself contributes fairly equally to linewidth and transport broadening, as mentioned in Sec. II.
D. Doping position dependence

If the donor doping position is varied, then the contribution of ionized impurity scattering to linewidth shouldchange. Dupont et al. measured low-temperature linewidths
in ␦ -doped GaAs/Al0.25Ga0.75As QWs with L⫽76 Å andN S ⬃1⫻1012 cm⫺2 for two different doping positions: Z⫽0
and 112 Å. 10 We calculate linewidth and transport broadening for the same structures. As scattering mechanisms, inter-

FIG. 13. Alloy composition dependence of transport energy broadening,calculated at 0 K by considering interface roughness and alloy disorder scattering. Open circles show experimental results measured at low temperatures by Campman et al. 共see Ref. 9兲.

Unuma et al.

1595

FIG. 14. Doping position dependence of intersubband absorption linewidth,
calculated at 0 K by considering interface roughness, LO phonon, and ionized impurity scattering. Solid circles show experimental results measured at
low temperatures by Dupont et al. 共see Ref. 10兲.

face roughness scattering with ⌬⫽5.66 Å 共2 MLs兲 and ⌳⫽70 Å, LO phonon scattering, and ion impurity scattering
are included one by one.
Figure 14 shows the calculated results for lowtemperature linewidth versus doping position Z in the rangeZ⫽0⫺120 Å. Interface roughness scattering and LO phonon
scattering contribute 5.8 and 0.8 meV to linewidth, respectively. When donors are doped in barriers, at Z⫽100 Å forexample, the contribution of ionized impurity scattering is as
small as 0.3 meV. When donors are doped in QWs, at thecenter Z⫽0 Å for instance, the contribution of ionized impurity scattering is 2.8 meV, which is smaller than that ofinterface roughness scattering. Our calculations explain theexperimental results plotted by solid circles.10
Note that the wave function ␨ 1 (z) of the first excitedstate penetrates largely into the low barriers in these narrow
GaAs/Al0.25Ga0.75As QWs, so the effect of ionized impurity
scattering is greatly enhanced even in barrier-doped QWs. If
wave functions are more strongly confined, for example, as
in the narrow GaAs/AlAs QWs used in our experiment, thecontribution of ionized impurity scattering to linewidth isless than 0.1 meV for barrier doping.
On the other hand, low-temperature transport broadening
is shown in Fig. 15 as a function of Z. Interface roughness
scattering contributes 0.44 meV to transport broadening,

FIG. 15. Doping position dependence of transport energy broadening, calculated at 0 K by considering interface roughness and ionized impurity scattering.

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1596

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J. Appl. Phys., Vol. 93, No. 3, 1 February 2003

while ionized impurity scattering contributes 12.2 meV at Z
⫽0 Å and 0.33 meV at Z⫽100 Å. Therefore, mobility
greatly decreases when donors are doped in or near QWs;
rather thick spacer layers, more than 150 Å in this case, arenecessary to completely remove the influence of ionized impurity scattering on mobility.
IV. NOTE

Here we add a note on interface roughness and inhomogeneous width. In general, interface roughness is responsible
for both homogeneous and inhomogeneous widths. The
qualitative criterion for distinguishing the two effects isgiven by comparing the correlation length ⌳ with the electron mean free path l: short-scale roughness (⌳⬍l) leads tohomogeneous width, while long-scale roughness (⌳⬎l)
causes inhomogeneous width.
We have presented, in this article, the theory of homogeneous width due to scattering by short-scale interface
roughness. This is expected to be a reasonable treatment for
actual GaAs/AlGaAs QW samples grown by MBE or
MOCVD, because an AlGaAs surface has short-scale roughness due to the short migration distance of Al atoms and a
GaAs surface can be made virtually flat using a growthinterruption technique. Numerical calculations based on thetheory are in good agreement with experimental results.
With regard to long-scale interface roughness, on theother hand, the condition ⌳⬎l means that it does not work
as scatterers. Instead, an appropriate picture is to regard eachflat region (⬃⌳⫻⌳) as independent. As a result, Eq. 共2兲
should be averaged over roughness distribution and the obtained line shape includes inhomogeneous width. Naturally, achange in layer thickness or alloy composition over a macroscopic region also causes inhomogeneous width. These
kinds of inhomogeneous widths may be important for other
QW systems.
V. SUMMARY

We have formulated the microscopic energy-dependent
relaxation rate 2⌫ op(E) of intersubband optical transition in
QWs due to scattering by interface roughness, LO phonons,
LA phonons, alloy disorder, and ionized impurities, and havenumerically calculated the absorption linewidth 2⌫ op for
GaAs-based QWs in comparison with the transport energy
broadening 2⌫ tr⫽2បe/m * ␮ related to the mobility ␮ .
The sensitivity of linewidth to interface roughness scattering is about one order of magnitude higher than that oftransport broadening, because the contribution from the intrasubband scattering in the first excited subband is larger
than that in the ground subband. This provides an essential
insight for understanding experimental values for linewidth
and the apparent lack of correlation between linewidth andmobility.
The contribution of LO phonon scattering to linewidth issmall, about 2 meV in narrow GaAs-based QWs even atroom temperature, because the difference in intrasubband
scattering matrix elements for the two subbands is small owing to the cancellation of form factors. In addition, intersubband LO-phonon spontaneous emission contributes approxi-

mately 1 meV to linewidth at low temperatures. Therefore,
linewidth has a very weak temperature dependence, whilemobility is greatly lowered by LO phonon scattering in thetemperature range above 80 K.
LA phonon scattering and alloy disorder scattering give
matrix elements that are independent of scattering vectors,
and lead to a linewidth comparable with transport broadening. The contribution of LA phonon scattering is, for example, about 1 meV at room temperature in narrow GaAsbased QWs, and this is small for linewidth compared withthe contribution of interface roughness scattering. Alloy disorder scattering contributes, for instance, about 0.3 meV in
Inx Ga1⫺x As QWs with x⫽0.1. This is negligible for linewidth but predominant for transport broadening, causing aremarkable drop in mobility as x increases.
Ionized impurity scattering contributes little to linewidth
in modulation-doped QWs, because the difference in intrasubband scattering matrix elements for the two subbands issmall. On the other hand, rather thick spacer layers, morethan 150 Å in narrow GaAs/Al0.3Ga0.7As QWs for example,
are necessary to remove the influence of ionized impurity
scattering on mobility.
ACKNOWLEDGMENTS

The authors are grateful to Professor T. Ando for helpful
discussions and showing them his unpublished formulationof intersubband optical transition. This work was partly supported by a Grant-in-Aid from the Ministry of Education,
Culture, Sports, Science and Technology, Japan. One of theauthors 共T.U.兲 also thanks the Japan Society for the Promotion of Science for financial support.
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