Phonon-Limited Electron Mobility in a Polar Semiconductor Quantum Well

H. LEON et al.: Phonon-Limited Electron Mobility in a Semiconductor QW 449

phys. stat. sol. (b) 170, 449 (1992)

Subject classification: 72.10 and 73.40; 71.38; S7.12; S7.15

Institute de Ciencia de Materiales, CSIC, Madrid') ( a ) and Departamento de Fisica Teorica, Universidad de La Habana2) ( b )

Phonon-Limited Electron Mobility in a Polar Semiconductor Quantum Well

BY H. LEON (a), F. LEON (b), and F. COMAS (b)

Low-field drift mobility for transport parallel to the interfaces in a polar semiconductor quantum well is calculated following an iterative procedure. Calculations are carried out in the size-quantum-limit approximation for a degenerate electron gas. Screening of the electron-phonon interaction via acoustic deformation potential, piezoelectric of the accoustic branch, and longitudinal optical mode couplings is taken into account by means of a dielectric function deduced in an earlier work. Confinement of LO vibrations is also taken into account by means of an appropriate Hamiltonian successfully employed in previous works. The role of the electron-LO-phonon interaction is accurately established and the conclusion is drawn that this coupling becomes considerable at intermediate and very important at sufficiently high temperatures for quantum wells of the GaAs/AIGaAs type. Comparison with experiment gives an acceptable agreement between calculated and measured mobility values.

Mit einem Iterationsverfahren wird die Niedrigfeld-Driftbeweglichkeit fur den Transport parallel zu den Grenzflachen eines polaren Halbleiter-Quanten-Wells berechnet, und zwar in der Naherung des Size-Quantenlimits fur ein entartetes Elektronengas. Die Abschirmung der Elektron-Phonon-Wechsel- wirkung mittels Kopplung durch das akustische Deformationspotential, die akustisch-piezoelektrischen und die longitudinal-optischen Moden wird durch eine in einer friiheren Arbeit abgeleitete dielektrische Funktion berucksichtigt. Confinement der LO-Schwingungen wird ebenfalls durch einen in einer friiheren Arbeit verwendeten, geeigneten Hamilton-Operator einbezogen. Die Rolle der Elektron-LO- Phonon-Wechselwirkung wird exakt nachgewiesen, und es wird gefunden, daS diese Kopplung fiir Q W vom Typ GaAs/AlGaAs bei mittleren Temperaturen betrachtlich und bei hinreichend hohen Tem- peraturen sehr bedeutend ist. Vergleich mit dem Experiment ergibt eine annehmbare Ubereinstimmung zwischen theoretischen und gemessenen Beweglichkeitswerten.

1. Introduction

The investigation of the transport phenomena in quasi-two-dimensional (Q2D) semiconductor heterostructures, on the basis of one or more heterojunctions (HJ), still deserves the attention of many experimental and theoretical works. The possibility of fabricating high-speed semiconductor devices became an actual problem after the construction of the so-called TEGFET [l], where there are two ways of enhancing the carrier drift velocity: by increasing the electron mobility or by increasing the applied electric field. The latter presents a saturation effect, which can be clearly observed in the experimental results of [2]; on the other hand, another interesting effect appears for sufficiently high electric field, when the Hall mobility begins to decrease as the electric field increases [2]. Thus, the main efforts have been devoted to enhance the carrier mobility along the conduction channel. Most of

I) Serrano 123, E-28006 Madrid, Spain. ') San LBzaro y L, Habana, Cuba.

450 H. LEON, F. L E ~ N , and F. COMAS

the experiments have been performed in single heterostructures, while only a few of them have been carried out in superlattices (SL) or quantum wells (QW) [2 to lo]. However, the latter structures have at least two advantages over the former ones: the electron density can be larger and the conduction channel thickness (QW width) can be well defined, which allow the manipulation of these parameters for finding the optimal values of mobility.

The cleanliness in the growth of modulation-doped QW and the inclusion of spacer layers have substantially reduced the impurity scattering mechanism in such a way that electron-phonon interaction is the main mechanism able to limit the mobility (excluding very low temperatures). In a polar semiconductor QW (e.g. of the GaAs/AlGaAs type) there are three concurrent scattering mechanisms due to electron-phonon interaction: via acoustic deformation potential (DP), piezoelectric (PZ) of the acoustic branch, and longitudinal optical (LO) mode couplings [ l l to 131. The role of the PZ coupling was established in our earliest work on this subject [ 121, differing from previous calculations of [ 14, 151 in the way we approximate the essentially anisotropic electron-PZ-phonon interaction by an isotropic version, but limiting ourselves to the low carrier concentration case neglecting screening and using non-degenerate carrier statistics. In the present paper we take into account screening by means of the static dielectric function deduced in the self-consistent field (SCF) approximation in [16], which differs from the one employed in [14], and we use the degenerate carrier statistics aiming at proceeding with the investigation of this scattering mechanism in limiting the mobility.

Electron-LO-phonon interaction has been widely considered by making use of the 3D Frohlich Hamiltonian [l 1,13, 17 to 191. However, Raman backscattering experiments have clearly shown that LO-phonons in layered semiconductor structures bear an essentially confined character [20]; and different models for confined LO-vibrations in a QW have been proposed [21 to 241. The effect of confinement of LO-phonons in the scattering of electrons and mobility values was considered in our previous works [25,26] on the basis of the model developed in [23], which is neither as simple as the one of [22, 241 nor as complicate as the one of [21]. The screening of this interaction has been ignored in earlier calculations [13, 191, so this is an interesting question to be investigated, and we shall use the dynamic dielectric function deduced in the SCF approximation in [16]. Different approaches of the continuum model for LO lattice vibrations in a double HJ or a SL, mainly differing by the boundary conditions at the interfaces, give rise to interface phonons besides the confined ones. According to [21,24] only two such modes can occur, but most of the works coincide in that four, two symmetric and two antisymmetric, such modes takes place [27 to 301. The main feature of these modes is the exponential decay of their amplitudes with increasing distance from the interface on both sides [22,27,29,30]. This suggests that their contribution to electron-LO-phonon interaction varies strongly with the active layer thickness and must be important for small QW widths. Indeed, this conclusion is supported by scattering rate calculations performed recently [24, 311. On the other hand, interface phonons are required for completeness of the set of LO lattice vibration eigenfunctions [28]. However, in the simplest models with boundary conditions assuming strictly [24] or almost strictly [234 rigid interfaces (in the latter case LO lattice vibrations are allowed in the plane of the interface) interface phonons do not appear. Their appearance requires some corrections of these simplest models in the line of those developed in [24]. Such kind of enhancement of the model of [23] stays an open problem to be investigated and could be the subject of a future work allowing further corrections to our mobility calculations. Therefore, interface phonons were left out of consideration in the present paper.

Phonon-Limited Electron Mobility in a Polar Semiconductor Quantum Well 451

The main purposes of our work are to calculate accurately the phonon-limited electron mobility in polar semiconductor QW and to establish the role of electron-LO-phonon interaction taking into account the confinement as well as the screening effect.

Section 2 is devoted to summarize the general procedure to calculate the low-field mobility in a Q2D system, limiting ourselves to the size-quantum-limit (SQL) approximation, i.e. considering only the first subband to be populated and neglecting intersubband transitions, in close analogy with the iterative method developed for a 3D system in [32]. Section 3 is concerned with mobility calculations in a QW assuming a simple model to describe electron states and energies. Discussion of the obtained results, comparison with experiment, and general conclusions are presented in Section 4.

2. General Theory

Assuming the effective mass approximation we deal with an electron gas confined by an appropriate potential. Each electron state is characterized by the subband index 1 and the 2D wave vector k parallel to the HJ. Electron-phonon interaction is assumed isotropic, an acceptable model for Q2D structures of the GaAs/AlGaAs type when transport phenomena are studied.

When a constant and uniform electric field F is applied along the conduction channel, neglecting concentration and temperature gradients, the non-equilibrium distribution functions fi(k) must be determined from a system of Boltzmann transport equations (BTE) coupled through the collision integrals in their right-hand sides. The difference between the scattering-in and scattering-out fluxes per unit time, related to the state 11, k ) , gives the corresponding collision integral taking into account the intra- and intersubband transitions as well as the Pauli principle. Using polar coordinates we expand f , (k) in Legendre polynomials up to the first-order term, which is a linear approximation in the applied field F as well as the relaxation time (RT) approximation. Bearing in mind the Fermi golden rule to calculate the required transition probabilities, and making use of the detailed balance principle, one obtains a system of algebraic equations for finding the RT z,(k). All of this can be clearly understood from our previous work [26].

The SQL approximation works well when the Fermi level W, lies below the second subband, and the thermal energy plus the phonon energy is not enough for an electron to escape from the first subband. Then the mobility is given by [26]

where N is the number of electrons in the conduction channel, E the quasiclassical kinetic energy associated with k , u = VkE, and fY(E) is the equilibrium distribution function with the Fermi level E,, = W, - El related to the bottom of the first subband E l .

We shall consider only the first subband to be populated and neglect intersubband transitions, thus straightforward calculations lead to

z(k) (SDP(k, k) - gDP(k, k) + SPZ(k, k) - gPz(k, k )

+ ~ ' ( k , ) SLo(k, k,) + u - ( k - ) SLo(k, k,))

= 1 + z(k,) V ( k , k,) u"+(k) S""O(k, k,) + z (k - ) V(k , k - ) u"-(k) gLo(k, k - ) , (2)

452 H. LEON, F. LEON, and F. COMAS

where (formally CP = D P , P Z , LO)

SCP(k, k,) = C Wcp' (k, k') , k'

S""(k, k,) = C WCP*(k, k') cos cp It'

(3)

(4)

with k, the wave vector associated with energy E+ = E ha, (hw, is the long-wave length phonon energy), Wcp' (k, k') the transition rate corresponding to phonon absorption (+) or emission (-), and cp the scattering angle between k and k'. We remind that for electron-acoustic-phonon interaction the scattering processes can be considered elastic, since hw, --f 0 and k, = k, hence (formally EL = D P , PZ)

SEL(k, k) = SEL(k, k + ) + SEL(k, k-) = 2SEL(k, k,),

gEL(k, k) = gEL(k, k,) + gEL(k, k-) = 2gEL(k, k,).

The following quantities have been also introduced:

Notice that in (2) there are three unknown quantities z(k), z(k+), and z(k-) in close analogy with (16) of [32] for a 3D case. This difficulty can be overcome by means of an iterative method to solve this equation. Really, starting from z(')(k+) = 0 to find z(')(k), one sets z("-')(k,) in the right-hand side of (2) to find z('")(k); the latter is placed in (1) to calculate p('"). This procedure is repeated as many times as is necessary to yield convergent values of the mobility within a convenient relative error.

The transition rates for a given coupling are calculated as usualIy (see, for instance, [6, 7, 14l),

where H Y * (Q; k, k') is the unscreened matrix element and ~ ( q ) an appropriate dielectric functions in the SQL approximation. The phonon 3D wave vector Q = q + qz, with 2 the electron confinement direction. Subscript j denotes the polarization mode, and au reminds us that one takes the statistical average over the equilibrium phonon system.

Let us finally notice that in this section the character of the subband structure (dependence of E on k ) and the form of the confining potential (dependence of E, on I ) have not been specified, neither the concrete expression of the dielectric function ~ ( q ) . The only restriction concerns the isotropy of the electron-phonon interaction.


3. Mobility Calculations in a Quantum Well

Mobility calculations require specification of the subband and confining potential structures. We assume the model of parabolic and isotropic subbands (E = h2k2/(2m*)) and infinite- depth rectangular QW of width a (E, = l2E0; E, = h2n2/(2rn*a2)) as in [ll to 14, 16 to 19, 24 to 261. According to this model (1) can be written in the following way:

a,

where the electron concentration n must be low enough for the first subband to be the only one populated (n < nk,T/(2a2)), and k,T < 3E0 for acoustic phonons, while k,T + ho, < 3E, (leading to a < 4(h/2m*00)li2 [19]) for polar optical phonons

The dielectric function must be also specified. We make use of the SCF result of [16] obtained for the same model we assumed for electrons and, after some manipulations of (23) of [16], we write it as follows:

(wo = WLO).

where E, = e2/c,a (z0 is the static permittivity of the QW) and

&, = E - 2(EE,)’” cos cp + E i ( E , = E ho,). We also introduce

with cob the static permitivity of the barrier and m

d E f W ) s {[S, + (-1)’ho0l2 - 4E’&,}1’2 P”’(&,) =

0

(14) [d, + ( - l ) j h ~ o ] ~ d€’

46* cos2 0 I- cos2 0 &* + (-l)jhwo

2 8 f + (-1)ji

0

with j = 0, 1. Notice that for acoustic phonons one sets ho, = 0 and the imaginary terms in (12) disappear.

3.1 Acoustic-phonon-limited mobility

It is a well known fact that acoustic phonons are not confined at all in layered structures, giving rise to the so-called phonon folding in the case of SL [20]. In the frame of the assumed model the electron-phonon unscreened interaction is described by the usual 3D Hamiltonian


for both D P and PZ couplings (see, for instance, our earlier work [12]). Presently, we do not limit ourselves to the momentum-conservation approximation as in [12J for the form factor, but we use the exact expression given by (11) of [17]. The isotropic approximation for the essentially anisotropic PZ coupling is given by (8) to (12) of [12].

The RT is easily obtained from (2),

zAC(E) = {SDP(E, E ) - fD'(E, E) + SPZ(E, E ) - fPZ(E, E)}-' (15)

after making SLo(E, E,) = 0 and gL0(E, E,) = 0.

expressions for DP are obtained: According to (3) to (10) and the equipartition approximation (18) of [12] the following

where

with D,, the deformation potential energy, e the material density, and uL the velocity of the longitudinal polarization mode of lattice vibrations. In a similar way the following expressions for PZ coupling are obtained:

where

32n e2y2(2m*k,T)'I2 [i 2 1 w y = ___ 15 h4&& UL + u:

with y thc piezoelectric modulus and uT the velocity of the transverse polarization mode of lattice vibrations. Here we have introduced

x {exp [ --n (')"'] - I}] ,


where 6, is the same parameter as in (12), notice that for acoustic phonons 8 , = 2E(1 - cosy), since E , = E.

Substitution of (15) in (1 1) leads to the acoustic-phonon-limited mobility (the iteration procedure is not needed in this case). Obviously, by making SPZ(E, E ) = 0 and gPZ(E, E ) = 0 one obtains the DP phonon-limited mobility; in a similar way one obtains the PZ phonon-limited mobility.

3.2 LO-phonon-limited mobility

It is a well-known fact that LO-phonons are confined in layered structures with the appearance of new allowed phonon modes [20 to 241. In the frame of the assumed model the electron-phonon unscreened interaction is described by the Q2D Hamiltonian deduced in [23] and briefly summarized in an earlier work [25], which considers a total confinement of the LO-phonons in a QW. Now we employ the exact expression (20) of [25] for the form factor and make no use of (24) in [25], which is in error as we pointed out in our previous paper [26].

The iterative formula for the RT is easily obtained from (2) ,

z k S ( E ) (u+(E+) SLo(E, E,) + u-(E-) SLo(E, E - ) }

after making SDP(E, E ) = gDP(E, E) = SPZ(E, E) = gPz(E, E ) = 0.

are obtained: According to (3) to (10) and calculations of [26] the following expressions for LO coupling

n n

x 1 d 3 ’ ( E , E,, cos 9) E-’(E, E,, cos 9) dy , 0

n n

x 5 d 3 ) ( E , E,, cos y ) C 2 ( E , E+, cos cp) cos y dy , 0

where

and

456 H. LEI~N, F. LEON, and F. COMAS

with E ~ , the high-frequency permittivity of the QW, here we have introduced.

4 E,, cos cp) = o'*'(E, E,, cos cp) - -

7c3

where &* is the same as appearing in the dielectric function (12). We want to remark that if the LO-phonon confinement is ignored and the usual 3D Frohlich Hamiltonian is employed as in [17, 181, the second term in (27) does not appear. This means that the effect of confinement comes out as a reduction of the electron-LO-phonon interaction, giving rise to an increase of LO-phonon-limited mobility as was clearly established in our previous work [26].

Substitution of (23) in (11) leads to the LO-phonon-limited mobility by means of the iterative procedure described in Section 1.

3.3 Phonon-limited mobility

We have already said that in polar semiconductor Qw's there are three concurrent phonon scattering mechanisms. Now it is clear that the RT due to the simultaneous action of the

I O 7 r \ a = 1 0 nm

10

. t A v)

I

E

3-

N I10 0

W

10

\'\, LO

total \

1 Fig. 1. Temperature dependence of mobility

Phonon-Limited Electron Mobility in a Polar Semiconductor Quantum Well -

Id l 1 10

457

l 2 10 l 1 10 l 2

DP, PZ, and LO coupIings is given by the following iterative formula:

ym) ( E ) { [ T ' ~ ( E ) ] - ~ + u+(E+) SLo(E, E , ) + u - ( E - ) SLo(E, E - ) ) = 1 + T?:- (E,) V ( E , E,) U"+SLo(E, E , )

+ zt2- ( E - ) V(E, E - ) I T - S " ~ O ( E , E - ) (28)

obtained from (2) and (15). Substitution of (29) in (11) leads to the phonon-limited electron mobility.

4. Discussion and Conclusions

The main original result of this work is the calculation of low-field drift mobility considering the screening of the electron-phonon interaction by means of the dielectric function (12) and, on the other hand, taking into account the LO-phonon confinement through a model Hamiltonian, which leads to (27). We consider a QW of GaAs with the following parameter values: m* = 0.067m0 (mo the free electron mass), eu i = 14.03 x 10" N/m2, & = 4.864 x 10" N/rn2, D,, = 13.5eV, y = 1.44 x lo9 V/m, E, = 12.53, E , = 10.82, and ho, = 35.5 meV (in this material the SQL approximation works when a 5 16 nm and n 5 10" cm-'); while for the AlGaAs barrier cob = 11.6. When the iterative procedure is

10

T=100 K

a=10 nm -I

_ - - - - - N ?IO - LA;

_ - - - - - - _ / -

E

3 0

W

total i

' O 1

10 -1

10 7 2

T=300 K 1 a=10 nm 4

1 10 3,

30 physica (b) 170/2


10

10

t n

Lo I

E N G o 0

W

3

10

10

T=100 K

- t / . , , /

- A /

cn . /

1 , / / -I / /

pz/ /

10

10

10

10

10

a(nm) --- Fig. 3. Mobility as a function of the quantum well width

required we choose 1010 relative error. The AC mobility includes both D P and PZ couplings while the total mobility includes all of them: DP, PZ, and LO couplings.

Fig. 1 shows the mobility as a function of temperature, when different couplings are considered, for typical values of the Q W width a = 10 nm and the electron concentration n = 2 x 10' cm-'. The temperature dependence of AC mobility qualitatively coincides with the one of D P mobility; however, the PZ coupling gives a finite contribution. Indeed, 11 - pDP/pACJ % 38,23, 13, and 9% for T = 20, 100, 200, and 300K, respectively. The temperature dependence of total mobility coincides with the one of AC mobility at low temperatures; the role of LO-phonons appears at intermediate temperatures and becomes very important at high temperatures. Really, (1 - pAC/,ufo'a'( = 41, 145, and 167% for T = 100, 200, and 300 K, respectively. Numerical results indicate that the effect of confinement of LO-phonons comes out as an increase of LO mobility values; thus 11 - p~~f/pt(nocconf~ % 25% all over the considered range of temperatures. On the other hand, the screening effect leads to a remarkable enhancement of LO mobility values; indeed, (1 - p~&, /p~~creen l = 81, 43, and 31% for T = 100, 200, and 300 K, respectively.

Fig. 2 presents the mobility as a function of the electron concentration when different couplings are considered for a typical value of QW width a = 10 nm and important temperatures T = 100 and 300 K. In analogy with Fig. 1 similar comments can be understood with respect to the role of each scattering mechanism as well as with respect


n=l.64xlO" cm-'

a=16.1 nm

r total

a 10 31

' 0 7 y a=15 nm

10

f n

0)

I

E

3

5 1 0

0 W

1

b 1 0 3 ! 1 I 1 I I I l l / 1 I 1 1 1 1 1 1

Fig. 4. Temperature dependence of the total mobility for a comparison with experimental data

to LO-phonon confinement and screening effects. The total mobility varies very slowly with n; numerical results indicate that at T = 100 K there is a maximum near n = 4 x 10" cm-' while at T = 300 K values increase monotonically.

Fig. 3 shows the mobility as a function of the QW width, when different couplings are considered, for a typical value of electron concentration n = 2 x 10" cm-' and important temperatures T = 100 and 300K. In analogy with Fig. 1 similar comments can be understood with respect to the role of each scattering mechanism as well as with respect to the LO-phonon confinement and screening effects; numerical results indicate that the effect of LO-phonon confinement decreases with increasing a, an expected result according to (27) and also on physical grounds. The total mobility rapidly varies with u showing a monotonous increase in its value.

Fig. 4 shows the temperature dependence of the total mobility for a comparison with available data when the SQL approximation can be accepted. For Fig. 4a calculations were carried out with the experimental parameter values: a = 16.1 nm and n = 1.64 x 10" cm-2; the stars represent the measured values of [6, 71. Notice that for T 2 50 K there is a quite good agreement between calculated and experimental values. The discrepancy for T < 50 K is due to different kinds of impurity-scattering mechanism, which cannot be ignored at low temperatures in the considered experiment. For Fig. 4b calculations were done with the


experimental parameter values: a = 15 nm and n = 2 x 10l1 cm-2; the stars represent the measured values of [9, 101. Similar comments to Fig. 4a hold here. A better fitting of the experiment requires to take into account different kinds of impurity-scattering mechanisms, which are certainly present and, on the other hand, more realistic electron wave functions ( e g a self-consistent calculation of envelope functions in the confinement direction).

Comparison with other experimental data such as the ones of [2,3, 5, 81 cannot be made since the SQL approximation calculations do not work well for too wide QW, or because of some peculiarities of the experiment (8-doping inside the QW, etc.).

In this work we have carried out a systematic investigation of phonon-limited electron mobility in a polar semiconductor QW taking into account two facts up to now ignored in mobility calculations in a QW: the LO-phonon confinement and the screening of the electron-LO-phonon interaction.

Acknowledgements

This work was initiated at the Universidad de LaHabana (Cuba) and completed at the Instituto de Ciencia de Materiales, CSIC (Madrid) while one of the authors (H. L.) was working under the auspices of the Direccion General de Investigacion Cientifica y Tecnica (Spain) with a Postdoctoral Fellowship. The hospitality and stimulating discussions provided by Prof. F. Garcia-Moliner and V. R. Velasco are gratefully acknowledged.

References

[l] B. VINTER, Heterojunctions and Semiconductor Superlattices, Springer-Verlag, Berlin 1986

[2] K. INOUE, H. SAKAKI, and J. YOSHINO, Appl. Phys. Letters 47, 614 (1985). [3] K. INOUE and H. SAKAKI, Japan. J . appl. Phys. 23, L61 (1984). [4] K. INOUE, H. SAKAKI, and J. YOSHINO, Japan. J. appl. Phys. 23, L767 (1984). [5] S. SASA, J. SAITO, K. NARIBU, T. ISHIKAWA, S. HIYAMIZU, and M. INOUE, Japan. J. appl. Phys. 24,

[6] C. GUILLEMOT, M. BAUDET, M. GAUNEAU, A. REGRENY, and J.-C. PORTAL, Superlattices and

[7] C. GUILLEMOT, M. BAUDET, M. GAUNEAU, A. REGRENY, and J.-C. PORTAL, Phys. Rev. B 35,2799

[8] Y. HORIKOSHI, A. FISCHER, E. F. SCHUBERT, and K. PLOOG, Japan. J. appl. Phys. 26,263 (1987). [9] G. WEIMANN and W. SCHLAPP, Appl. Phys. A 37, 139 (1985).

(p. 238).

L281 (1985).

Microstructures 2, 445 (1986).

(1987).

[lo] H. BURKHARD, W. SCHLAPP, and G. WEIMANN, Surface Sci. 174, 387 (1986). [l 11 V. K. ARORA and A. NAEEM, Phys. Rev. B 31, 3887 (1985). [12] F. COMAS, C. TRALLERO, H. LEON, and J. TUROR, Physica (Utrecht) B152, 352 (1988). [I31 J. LEE and M. 0. VASSELL, Japan. J. appl. Phys. 23, 1086 (1984). [I41 D. CHATTOPADHYAY, phys. stat. sol. (b) 135, 409 (1986). [15] P. J. PRICE, Ann. Phys. (USA) 133, 217 (1981). [16] J. LEE and H. N. SPECTOR, J. appl. Phys. 54, 6889 (1983). [17] B. K. RIDLEY, J. Phys. C 15, 5889 (1982). [IS] F. A. RIDDOCH and B. K. RIDLEY, J. Phys. C 16, 6971 (1983). [19] D. CHATTOPADHYAY, Phys. Rev. B 33, 7288 (1986). [20] B. JUSSERAND and D. PAQUET, Heterojunctions and Semiconductor Superlattices, Springer-Verlag,

[21] M. BABIKER, J. Phys. C 19, 683 (1986). 1221 L. WENDLER and R. PECHSTEDT, phys. stat. sol. (b) 141, 129 (1987). [23] C. TRALLERO and F. COMAS, Phys. Rev. B 37, 4583 (1988). [24] B. K. RIDLEY, Phys. Rev. B 39, 5282 (1989).

Berlin 1986 (p. 108).


[25] H. LEON and F. COMAS, phys. stat. sol. (b) 149, 533 (1988). [26] H. LEON and F. COMAS, phys. stat. sol. (b) 160, 105 (1990). [27] F. BECHSTEDT and R. ENDERLEIN, phys. stat. sol. (b) 31, 53 (1985). [28] R. ENDERLEIN, F. BECHSTEDT, and G. GERECKE, phys. stat. sol. (b) 148, 173 (1988). [29] R. CHEN, D. L. LIN, and T. F. GEORGE, Phys. Rev. B 41, 1435 (1990). [30] D. L. LIN, R. CHEN, and T. F. GEORGE, Solid State Commun. 73, 799 (1990). [31] L. WENDLER, R. HAUPT, F. BECHSTEDT, H. RUCKER, and R. ENDERLEIN, Superlattices and

[32] D. L. RODE, Phys. Rev. B 3, 3287 (1971). Microstructures 4, 577 (1988).

(Received February 5, 1992)

Phonon-Limited Electron Mobility in a Polar Semiconductor Quantum Well

Documents

Transcript of Phonon-Limited Electron Mobility in a Polar Semiconductor Quantum Well