APPLIED PHYSICS LETTERSVOLUME 78, NUMBER 22

28 MAY 2001

Effects of interface roughness and phonon scattering on intersubband
absorption linewidth in a GaAs quantum wellTakeya Unuma,a) Teruyuki Takahashi, Takeshi Noda,b) Masahiro Yoshita,Hiroyuki Sakaki,b) Motoyoshi Baba, and Hidefumi Akiyama
Institute for Solid State Physics, University of Tokyo, 5-1-5, Kashiwanoha, Kashiwa,
Chiba 277-8581, Japan

共Received 27 December 2000; accepted for publication 4 April 2001兲
We experimentally and theoretically study the effects of interface roughness and phonon scattering
on intersubband absorption linewidth in a modulation-doped GaAs/AlAs quantum well.
Quantitative comparisons between experimental results and theoretical calculations make it clearthat interface roughness scattering is the dominant scattering mechanism for absorption linewidth in
the temperature range below 300 K. Even at room temperature, phonon scattering processes
contribute little to linewidth, while polar-optical phonon scattering limits electron mobility.
© 2001 American Institute of Physics. 关DOI: 10.1063/1.1376154兴

Intersubband absorption linewidth in a quantum well
共QW兲 has been said to have little correlation with electron
mobility. Campman et al. reported that linewidth is sensitive
to interface roughness scattering, while it is not to alloy scattering, to which mobility is sensitive.1 In addition to ionized
impurity scattering2 and band nonparabolicity,3 the collective
nature of intersubband transition and its effect on linewidth
at high electron concentrations have been revealed by
Warburton et al.4 It is also interesting that quite narrow linewidths have been reported for a particular double QW
system.5 As a whole, systematic and quantitative understanding of absorption linewidth has not yet been achieved, whichis in contrast to the case of mobility.6,7
In this letter, we investigate the effects of interface
roughness and phonon scattering on absorption linewidth ina GaAs QW. We measure linewidth in comparison with mobility in a modulation-doped GaAs/AlAs QW at temperatures ranging from 4.5 to 300 K, which are indeed different
in terms of both absolute values and temperature dependence. Theoretical analysis shows that linewidth is one orderof magnitude more sensitive to interface roughness scatteringthan mobility, but that linewidth is less sensitive to phonon
scattering. We obtain good agreement between the experimental results and theoretical calculations for both linewidthand mobility.
The sample used in this study was a modulation-doped
GaAs single QW with AlAs barriers grown by molecularbeam epitaxy. The QW structure consisted of a Si-doped
Al0.3Ga0.7As layer, a 50 Å undoped Al0.3Ga0.7As spacer, a 50
Å undoped AlAs barrier, an 80 Å undoped GaAs QW layer,a 50 Å undoped AlAs barrier, a 50 Å undoped Al0.3Ga0.7As
spacer, and a Si-doped Al0.3Ga0.7As layer, as shown in theinset of Fig. 1. In the growth process, a growth interruption
was used at the AlAs-on-GaAs interface for smoothing. Themobility ␮ and the sheet electron concentration N s , deter-

mined by Hall measurement, were 2.9⫻104 cm2/V s and9.8⫻1011 cm⫺2 at 4.2 K, respectively.
The structure was chosen so that roughness scattering at
the GaAs-on-AlAs interface would dominate the electron
mobility at low temperatures,8 and that polar-optical 共LO兲phonon scattering would do so at room temperature.7 The
effect of ionized impurity scattering was reduced by the 100
Å spacer layers, and the influence of alloy scattering wasabsent because of the GaAs QW with AlAs barriers.
The sample was processed into a 3-mm-long waveguidestructure with a thickness of 0.3 mm. Both ends were polished at 45° so that the light incident to the face could bereflected at the top and bottom surfaces of the waveguide.
Aluminum 共100 nm thick兲 was evaporated onto the top surface to serve as a gate for modulating the electron concentration in the QW. Absorption spectra were measured at temperatures T⫽4.5—300 K with a Fourier transform infrared
spectrometer and microscope 共␮-FTIR兲. By modulating theelectron concentration between the accumulation and depletion conditions, and detecting signals with a lock-in technique, we obtained background-free absorption spectra withhigh signal-to-noise ratios to measure the linewidth precisely.

a兲FIG. 1. An intersubband absorption spectrum of a GaAs/AlAs single QWobserved at 4.5 K with a ␮-FTIR via a modulation technique. The inset
represents the structure of this sample.

Electronic mail: unuma@issp.u-tokyo.ac.jp
Also at: Institute of Industrial Science, University of Tokyo, 7-22-1 Roppongi, Minato-ku, Tokyo 106-8558, Japan.

b兲

0003-6951/2001/78(22)/3448/3/$18.00
3448© 2001 American Institute of Physics
Downloaded 24 May 2001 to 157.82.227.2. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/aplo/aplcr.jsp

Unuma et al.

Appl. Phys. Lett., Vol. 78, No. 22, 28 May 2001

Figure 1 shows an observed absorption spectrum at 4.5
K. The full width at half maximum of the absorption, denoted hereafter as 2⌫ op , was 11.1 meV. To compare thisvalue with mobility, we defined transport energy broadeningas 2⌫ tr⫽2ប/ 具 ␶ tr(E) 典 ⫽2បe/ ␮ m * . Here, e is the elementarycharge, ប is the reduced Plank constant, m * is the electron
effective mass, ␶ tr(E) is the momentum relaxation time ofelectrons, and the brackets denote an average over energy.8
From the low-temperature mobility of ␮ ⫽2.9⫻104 cm2/V s,
we obtained 2⌫ tr⫽1.2 meV. Note that the linewidth 2⌫ op
was about an order of magnitude larger than the transport
broadening 2⌫ tr .
To explain the low-temperature values of the linewidth2⌫ op as well as the transport broadening 2⌫ tr , or the mobility ␮, we numerically calculated the most probable contribution of interface roughness scattering using Ando’s
theory.6,9,10 Ando’s theory is based on the single-particle picture, which is applicable when the depolarization shift issmall compared with linewidth. This was the case in ourexperiment, where N s ⫽9.8⫻1011cm⫺2 gives a depolarization shift of about 5 meV.
According to Ando’s theory, the intersubband absorption
line shape for elastic scattering processes can be expressed asRe ␴ zz 共 ␻ 兲 ⬀冕

dE f 共 E 兲⌫ op共 E 兲,共 ប ␻ ⫺E 10兲 2 ⫹⌫ op共 E 兲 2⌫ op共 E 兲 ⫽ 21 关 ⌫ intra共 E 兲 ⫹⌫ inter共 E 兲兴 .

共1兲
共2兲

Here, ␻ is the photon frequency, f (E) is the Fermi distribution function, E 10 is the intersubband energy separation,⌫ intra(E) is the width due to the difference in intrasubband
scattering matrix elements for the ground and first excited
subbands, and ⌫ inter(E) is the width due to the intersubband
scattering.9
Using a roughness model that characterizes the GaAson-AlAs interface by the Gaussian autocorrelation function
with a mean height of ⌬ and a correlation length of ⌳,6,8 wehave⌫ intra共 E 兲 ⫽m *⌬ 2⌳ 2
共 F 00⫺F 11兲 2
ប2⌫ inter共 E 兲 ⫽m *⌬ 2⌳ 2
F 012
ប2

冕 ␪␲

冕 ␪␲

2022

d e ⫺q ⌳ /4,2

d e ⫺q̃ ⌳ /40共3兲
共4兲

with F mn ⫽ 冑( ⳵ E m / ⳵ L)( ⳵ E n / ⳵ L). Here, L is the well width,
E n is the quantization energy of the nth subband, ␪ is thescattering angle, and q and q̃ are the absolute values of thetwo-dimensional scattering vector in the ‘‘intra’’ and ‘‘inter’’ processes,9 given by
q 2⫽

4m *
E 共 1⫺cos ␪ 兲 ,ប2q̃ 2 ⫽4m *E 10⫺ 冑E 共 E⫹E 10兲 cos ␪ ,
E⫹
2
ប
2

冋

共5兲

册

共6兲

respectively. In Eq. 共3兲, a screening factor that appeared in
the original theory9 is omitted because its contribution isnegligible in the following analysis.
We have a similar expression for the transport energy
broadening 2ប/ ␶ tr(E), which is expressed as83449

FIG. 2. Calculated 2⌫ op and 2⌫ tr due to interface roughness scattering,plotted as functions of ⌳. The 2⌫ op,para represents the linewidth without thenonparabolicity effect.

2ប
2m * ⌬ 2 ⌳ 2⫽
F 002
␶ tr共 E 兲ប2冕 ␪⑀␲d01⫺cos ␪ ⫺q 2 ⌳ 2 /4e.
共 q,T 兲 2

共7兲Here, ⑀ (q,T) is the static dielectric function.7,9
It is useful here to compare the similar expressions for⌫ intra(E), ⌫ inter(E), and 2ប/ ␶ tr(E) in Eqs. 共3兲–共7兲. First, all
the three are proportional to ⌬ 2 , and also to ⌳ 2 for small ⌳.
Second, ⌫ inter(E) is much smaller than ⌫ intra(E), because q̃
is larger than q owing to E 10 . Third, (1⫺cos ␪)/⑀(q,T)2 appearing only in 2ប/ ␶ tr(E) shows that forward scattering ( ␪⬃0) makes no contribution to the transport energy broadening, and that screening reduces the scattering rate. Finallyand most importantly, there are the factors (F 00⫺F 11) 2 ,F 012 , and 2F 002 in front of the integrals. Since E 1 is moresensitive to L than E 0 , F 11 is much larger than F 00 , whichmakes ⌫ intra(E) much larger than 2ប/ ␶ tr(E). 共In the infinitebarrier approximation, F 11 is four times larger than F 00 .兲
At low temperatures, Eq. 共7兲 with the substitution ofFermi energy E F into E directly gives 2⌫ tr . If the E dependence in Eqs. 共3兲 and 共4兲 and band nonparabolicity are negligible, the sum of Eqs. 共3兲 and 共4兲 is equal to 2⌫ op .
In practice, however, 2⌫ op is obtained by performing theintegral in Eq. 共1兲, where the nonparabolicity effect is to be
introduced by replacing E 10 with E 10⫺(1⫺m *0 /m 1* )E.
**Here, m 0 and m 1 are the electron effective masses in the
ground and first excited subbands, respectively, and are set at
0.069m and 0.076m according to the Kane model, where mis the free electron mass. In the numerical calculations, L⫽80 Å, N s ⫽9.8⫻1011 cm⫺2, and a barrier height of 1 eVwere used. The other material parameters were taken from
Table I in Ref. 7.
Figure 2 shows the calculation of 2⌫ op 共solid curve兲 incomparison with 2⌫ tr 共broken curve兲 as a function of ⌳ for⌬⫽4 Å. The dash-dotted curve shows 2⌫ op,para , the linewidth without the nonparabolicity effect. It is shown that2⌫ op,para and 2⌫ tr are proportional to ⌳ 2 for small ⌳ with adifference in values of about an order of magnitude. For
large ⌳, the insensitivity of 2⌫ tr to the forward scattering
causes its value to be smaller. With nonparabolicity, 2⌫ op
has a lower limit of (1⫺m *0 /m 1* )E F ⫽3.3 meV in addition to2⌫ op,para . As a result, 2⌫ op is one order of magnitude larger
than 2⌫ tr , which explains the higher sensitivity of linewidth
to interface roughness scattering.

Downloaded 24 May 2001 to 157.82.227.2. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/aplo/aplcr.jsp

3450

Unuma et al.

Appl. Phys. Lett., Vol. 78, No. 22, 28 May 2001

FIG. 3. Measured 2⌫ op and 2⌫ tr 共or ␮兲 are plotted as functions of temperature by solid and open circles, respectively. Calculated values with L⫽80 Å, N s ⫽9.8⫻1011 cm⫺2, ⌬⫽4 Å, and ⌳⫽43 Å are also plotted bylines. In the calculations, we considered interface roughness 共IFR兲, LO phonon, and LA phonon scattering.

The value of 10.4 meV for 2⌫ op without the contributionof phonon scattering, which will be shown below, agrees
with the calculation for ⌳⫽43 Å. This is a reasonable valueof ⌳ for the GaAs-on-AlAs interface.8 The corresponding
calculated value of 2⌫ tr is 0.73 meV, which partly explains
the experimental value of 1.2 meV. In the following, we use
the pair of ⌬⫽4 Å and ⌳⫽43 Å as representative; anotherpair in the range of ⌬⫽3.5—5 Å and ⌳⫽55—30 Å is allowable.
Figure 3 shows the plots of the linewidth 2⌫ op 共solidcircles兲 measured as a function of temperature. For comparison, the transport broadening 2⌫ tr , or the mobility ␮ 共rightvertical axis兲, is also plotted by open circles. At a low temperature (T⫽4.5 K), as already seen in Fig. 1, the 2⌫ op of
11.1 meV is much larger than the 2⌫ tr of 1.2 meV. Note now
that the linewidth 2⌫ op slowly increases as the temperature
increases up to 300 K, which is in contrast to the rapid increase of the transport broadening 2⌫ tr .
To explain the experimental results, we also applied Ando’s theory to LO phonon scattering and acoustic 共LA兲 phonon scattering via deformation potential coupling. Since thistheory is for elastic scattering processes, our analysis is acrude approximation for deriving the width of a zero-phononband while ignoring phonon sidebands. The formulations weobtained are more complex; they will be presented elsewhere. With regard to transport, the formulations are thesame as those in Ref. 7, except for a difference in wave
functions.
The calculated results, obtained by including interface
roughness 共IFR兲, LO phonon, and LA phonon scattering, arealso shown in Fig. 3 by broken 共IFR兲, dash-dotted 共IFR
⫹LO兲, and solid 共IFR⫹LO⫹LA兲 curves, respectively. Fine
agreement between the experimental results and theoretical
calculations was obtained by the solid curves for both linewidth and mobility.
At low temperatures, interface roughness scattering
dominates absorption linewidth, though the intersubband LOphonon emission process adds a little to the width. Even atroom temperature, interface roughness scattering is still
dominant, whereas the effects of LO and LA phonon scattering are small. This is because the difference in intrasubband
scattering matrix elements for the two subbands is smaller inphonon scattering than in interface roughness scattering. In
contrast, LO phonon scattering rapidly lowers electron mobility with increasing temperature, and the contribution ofinterface roughness scattering remains small, as is known
very well. In this way, the same scattering mechanisms havevery different influences on linewidth and mobility, which
explains the apparent lack of correlation between the two.
It should be noted that the high sensitivity of linewidth
to interface roughness scattering is due to the larger contribution of the first excited state than that of the ground statein Eq. 共3兲, or F 11⫺F 00⫽ ⳵ (E 1 ⫺E 0 )/ ⳵ L. If one makes E 1⫺E 0 ⫽E 10 insensitive to the change of L in double QWs orother structures, the dominant term ⌫ intra(E) should vanish,
and narrow linewidths should be obtained.
In summary, we have quantitatively studied the effects
of interface roughness and phonon scattering on intersubband absorption linewidth in a GaAs/AlAs QW. We haveobtained significant agreement between experimental data
and theoretical calculations for both linewidth and mobility
at temperatures ranging from 4.5 to 300 K. In this temperature range, interface roughness scattering is a very strong
scattering mechanism for linewidth, because the contributionof the first excited subband is much larger than that of theground subband. Phonon scattering processes have little influence on linewidth even at room temperature, while mobility is strongly affected by LO phonon scattering.
The authors are grateful to Professor T. Ando for his
helpful discussions about the theory of absorption linewidth.
The authors also thank Dr. R. J. Warburton, Dr. C. Sirtori,and Professor J. Faist for their valuable instruction and discussions. This work was partly supported by a Grant-in-Aid
from the Ministry of Education, Science, Sports, and Culture, Japan.
1

K. L. Campman, H. Schmidt, A. Imamoglu, and A. C. Gossard, Appl.
Phys. Lett. 69, 2554 共1996兲.
2
E. B. Dupont, D. Delacourt, D. Papillon, J. P. Schnell, and M. Papuchon,Appl. Phys. Lett. 60, 2121 共1992兲.
3
P. von Allmen, M. Berz, G. Petrocelli, F. K. Reinhart, and G. Harbeke,Semicond. Sci. Technol. 3, 1211 共1988兲.
4
R. J. Warburton, K. Weilhammer, J. P. Kotthaus, M. Thomas, and H.
Kroemer, Phys. Rev. Lett. 80, 2185 共1998兲; R. J. Warburton, K.
Weilhammer, C. Jabs, J. P. Kotthaus, M. Thomas, and H. Kroemer,
Physica E 共Amsterdam兲 7, 191 共2000兲.
5J. Faist, C. Sirtori, F. Capasso, L. N. Pfeiffer, and K. W. West, Appl. Phys.
Lett. 64, 872 共1994兲.
6T. Ando, A. B. Fowler, and F. Stern, Rev. Mod. Phys. 54, 437
共1982兲.
7K. Hirakawa and H. Sakaki, Phys. Rev. B 33, 8291 共1986兲.
8
H. Sakaki, T. Noda, K. Hirakawa, M. Tanaka, and T. Matsusue, Appl.
Phys. Lett. 51, 1934 共1987兲.
9T. Ando, J. Phys. Soc. Jpn. 54, 2671 共1985兲.
10T. Ando, Z. Phys. B 24, 33 共1976兲.

Downloaded 24 May 2001 to 157.82.227.2. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/aplo/aplcr.jsp