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Density-of-states of a two-dimensional electron gas in multiple quantum well structures due to electron-phonon interaction
Solid State Communications, Vol. 67, No. 6, pp. 625-627, 1988. Printed in Great Britain.
0038-1098/88 $3.00 + .00 Pergamon Press plc
DENSITY-OF-STATES OF A TWO-DIMENSIONAL ELECTRON GAS IN MULTIPLE QUANTUM WELL STRUCTURES DUE TO ELECTRON-PHONON INTERACTION M.P. Chaubey Department of Mathematics, Dawson College, 350 Selby Street, Montreal (Quebec) H3Z lW7, Canada
(Received 9 February 1988 by M.F. Collins) Density-of-states (DOS) of a quasi-two-dimensional electron gas subjected to a high magnetic field applied perpendicularly to the electronic layer as formed in multiple quantum well structures (MQWS) is calculated. The second moment (tr0) representing the broadening of the Gaussian density-of-states of the ground Landau level due to the electron-acoustic and electron-piezoelectric phonon interaction is theoretically and numerically computed for the MQWS in GaAs/AIAs. The contribution of the acoustic phonon scattering is found to be larger than that due to the electron-piezoelectric phonon interaction, however piezoelectric phonon scattering is significant in MQWS. Magnetic field (B) dependence of % due to the acoustic and piezoelectric scattering is given approximately a s B 1/2 and B 1/4 respectively. Obtained numerical values of a0 indicate that at low temperatures the combined effect of an acoustic and piezoelectric phonon scatterings is of considerable and competing importance as that of the impurity scattering effects in MQWS.
IN RECENT years, several theoretical and experimental investigations [1,2] have been made to observe the density-of-states of a two-dimensional electron gas subjected to a high magnetic field applied perpendicularly to the layers as realized in the single quantum wells, multiple quantum wells and heterolayers of GaAs/A1As sytems [3]. Energy spectrum of such a system is completely quantized and the usual optical and transport quantities diverge due to the presence of the delta function singularities in the density-of-states. This divergence can be removed either by considering the inelastic [4] scattering of the electrons with low energy acoustic phonons or by considering the multiple elastic scattering [5-7] of the electrons with disorder, acoustic [8] and piezoelectric phonons or surface roughness or any combination of the two. However, in the limit of high magnetic field and low phonon energy, the inelastic scattering by acoustical or piezoelectric phonons gives vanishing conductivity [4] in the absence of the mixing between the Landau levels. Thus, the broadening of the density-of-states due to the inelastic scattering mechanism is not expected to be significant at low temperatures and high magnetic fields. Multiple scattering by disorder (called the Self Consistent Born Approximation (SCBA)) as developed by Ando [5] and Gerhardts [6] broadens the delta function singularity into semi-elliptical and
Gaussian shapes respectively with width varying in both cases as x/~ where B is the applied magnetic field. This x/~ variation of the width due to the disorder scattering has been widely observed in the density-ofstates [2] and cyclotron resonance measurements [9] in agreement with the theory [5-7]. Tsui and co-workers [10] have recently observed the effect of acoustic and piezoelectric phonon scattering competing with the long range coulomb scattering in the heterojunctions at low temperatures. Chaubey and Gupta [1 l] subsequently investigated the broadening of the density-ofstates due to the acoustic phonon scattering using the form factor given by Stern Howard's [12] variational wave function. The broadening of the density-ofstates as calculated [l l] was found to vary as x/B and in agreement with the observed magnetic field dependence experimentally [2]. In the present Communication, an earlier analysis [11] of the effect of the electron phonon interaction on the density-of-states (DOS) of a two dimensional electron gas as formed in heterojunctions is extended to the multiple quantum well structures (MQWS). The second moment which represents the broadening of the density-of-states is explicitly calculated for the electron acoustic and electron piezoelectric phonon interaction in a multiple quantum well structure in GaAs/AIAs system. The magnitude of the broadening
625
DENSITY-OF-STATES OF A TWO-DIMENSIONAL ELECTRON GAS Vol. 67, No. 6
626
(a0) of the density-of-states of the ground Landau level due to the combined scattering from the acoustic and piezoelectric phonons is of the same order of magnitude as that obtained from the short range scattering theory [5], with the magnetic field dependence given as v/B in agreement with the experimental data [2]. Density-of-states Do(E ) of the ground Landau level is found to be given by the following expression [6, 11]
Do(E) =
l
1
2x12 ~ 0 . 0 exp
[
(E ~Eo)2.7 2og
_]'
and ep, the subband energy is found by solving the following transcendental equation
kpa+ 2 t a n - l ( ~ )
=
0.2
0.ac +
o.P~,
No(Q) =
0.U,Phin the limit of high magnetic field and Self Consistent Born Approximation is given by [8, 11, 13]
a (°2 = 4 j(2~)3 ( d3Q [v~°(Q)12[1 + 2N0(O)] e -½t2q~ N,Ph (3)
where the index (i) stands for the acoustic and piezoelectric phonon contributions to the width
Iv~(Q)I: =
fie 2 ~
2c, (q~ + q2),/2,
(4)
e2p2fi V~
IVP~(Q)I2 = 2~(q~ + q2),/2,
(5)
E, P, V~, cL and e~ are the deformation potential, piezoelectric constant, velocity of sound, elastic constant and static dielectric constant of GaAs respectively. The form factor for the multiple quantum well is given by [13, 14]
[ IFp~(q=)l z =
1 6 c o s 4 ( ~ -~)
-]2 7""W - -
+
~
--
+
'
/
[e~ ' ° ~ - 1]-',fl
-
ksT'
(9)
ks and T being the Boltzmann's constant and temperature respectively. Equation (3) is numerically evaluated by making use of the equations (4)-(7) and (8) for the ground Landau level (N = 0) and the ground subband (p = 1) for the acoustic and piezoelectric phonon interactions separately. The parameters are deformation potential E = 13.5eV, the static permittivity ~ = 12.9 e0, ~ = 5.37 x 103msec, m*/m = 0.0667 dimensionless piezoelectric constant [16] P = 0.064, c L = 2 × 1 0 1 ° J m 3. The quantum well parameters are a = b = 50 A, w = 0.1 eV. Variation of a0.phwith magnetic field is presented in Table 1 for acoustic as well as for piezoelectric phonon interaction. The magnetic field dependence of the width of the Gaussian density due to the deformation potential and piezoelectric interactions are approximately given by B ~/2 and B ~/4respectively. The value of a0.Phwhich lies between 0.2 meV and 0.5 meV with the increase of the magnetic field is of the same order of magnitude as obtained using the impurity scattering theory in the Self Consistent Born Approximation. It is further noteworthy that as compared to a single electron layer [11] the electron phonon interaction effects in multilayers are reduced by about a factor of 2. The contribution of the acous-
(2)
x [Z~)(½12q~)]EIFpp(q~)[2,
(8)
1
(1)
0.2
p~,p = 1,2 . . . . .
L°N(X) in equation (3) is Laguerre polynomials. Q describes the wave vector of a phonon of frequency ~°o, Q = (q~l,qz), qtt being the two-dimensional vector in the plane of the layer.
where 0.o is the full width at half the maximum of the above Gaussian function, E0 = h~,/2 scaled by the ground subband energy and l = (cfi/eB) aj2. The width 0.0.Phdue to the electron phonon interaction is decomposed as follows [15], 2 0.0.Ph
=
. e-
2zpb
2Zp(X2p+ kp2)COS(~-~)+ q-.sin(~-~)(k2p-3Z2p-q2-_)] z, x I sin (~ b/2)
+
- - 7 - -+ - - 21,;-
a is the width of the quantum well, b is the interquanturn well separation and w stands for the height of the quantum well. Also d = a + b k, =
, \,/2 (~2eP);Xp
=
[2m*(w -- ep).-]'/2 L ~2 ] ,
(7)
-
q2)2 :
+
(6)
tic phonon scattering is larger than that due to the electron piezoelectric phonon interaction, however the piezoelectric phonon scattering is significant in GaAs/ AlAs MQWS. Obtained numerical values of a0 indicate that at low temperatures, the combined effect of the acoustic and piezoelectric scattering mechanisms is
Vol. 67, No. 6 DENSITY-OF-STATES OF A TWO-DIMENSIONAL ELECTRON GAS
627
Table 1. The variation of ao.ph with magnetic fieM (B) in GaAs/A1As multiple quantum well structures at T = 10 K Magnetic field B
Second moment of the density D0(E)
O'o,ph = [O'o,ac2 + O.O.pz2 ] 1/2
5 Tesla 10 Tesla 15 Tesla 20 Tesla 25 Tesla
Deformation potential
Piezoelectric interaction
(~o,.c)
(~o,Pz)
0.175 meV 0.325 meV 0.379 meV 0.420 meV 0.432 meV
0.092 meV 0.102 meV 0.108 meV 0.112 meV 0.116 meV
of considerable and competing importance as that of the impurity scattering effects in MQWS. It is anticipated that the results of the present theory combined with the disorder scattering theory [5, 6] would provide satisfactory explanation to the observed data on density-of-states [2]. Acknowledgements - - This work is supported by Formation de chercheurs et concertie (FCAR), Provincial Government of Quebec, Grant No 88-AR0270. I would like to thank Professor M. Singh for discussions and Professor C.M. Van Vliet for the hospitality at CRM, Universit6 de Montr6al.
5.
6. 7. 8. 9.
10. REFERENCES 1. 2.
3. 4.
M. Ya. Azbel, Phys. Rev. B33, 8844 (1986). J.P. Eisenstein, H.L. Stormer, V. Narayanmurti, A.Y. Cho, A.C. Grossard & C.W. Tu, Phys. Rev. Letts. 55, 875 (1985). T.P. Smith, B.B. Goldberg, P.J. Stiles & M. Heiblum, Phys. Rev. B32, 2696 (1985). E. Stahl, D. Weiss, G. Weimann, K. Von Klitzing & K. Ploog, J. Phys. C18, L783 (1985). T. Ando, A.B. Fowler & F. Stern, Rev. Mod. Phys. 54, 437 (1982) and references cited therein. P. Vasilopoulos, Phys. Rev. B33, 8587 (1986).
11. 12. 13. 14. 15. 16.
0.198 meV 0.341 meV 0.394 meV 0.435 meV 0.447 meV
T. Ando & Y. Uemura, J. Phys. Soc. Japan 36, 959 (1974). T. Ando, J. Phys. Soc. Japan 36, 1521 (1974); 37, 622 (1974); 37, 1233 (1974); 38, 989 (1975). R.R. Gerhardts, Z. Physik B2I, 275 (1975); B21, 285 (1975). M. Prasad and S. Fujita, Physica 91A, 1 (1978). M. Prasad, Phys. Status Solidi (b) 109, 11 (1982). M. Prasad, Phys. Letts. 70A, 127 (1979); J. Phys. C12, 5489 (1979); J. Phys. C13, 3239 (1980). G. Abstreiter, J.F. Koch, P. Groy & Y. Couder, Phys. Rev. B14, 2494 (1976); G. Abstreiter, J.P. Kotthaus, J.F. Koch & G. Dorda, Phys. Rev. BI4, 2480 (1976). B.J.F. Lin, D.C. Tsui & G. Weimann, Solid State Commun. 56, 287 (1985). M.P. Chaubey & O.P. Gupta, Solid State Commun. 59, 317 (1986). F. Stern & W.J. Howard, Phys. Rev. 163, 816 (1967). M.P. Chaubey & M. Singh, Phys. Rv. B34, 2385 (1986). D. Calecki, J.F. Palmier & A. Chomette, J. Phys. C17, 5017 (1984). M. Prasad & S. Fujita, Phys. Letts. 63A, 147 (1978). P.K. Basu & B.R. Nag, J. Phys. C14, 1519 (1981).
0038-1098/88 $3.00 + .00 Pergamon Press plc
DENSITY-OF-STATES OF A TWO-DIMENSIONAL ELECTRON GAS IN MULTIPLE QUANTUM WELL STRUCTURES DUE TO ELECTRON-PHONON INTERACTION M.P. Chaubey Department of Mathematics, Dawson College, 350 Selby Street, Montreal (Quebec) H3Z lW7, Canada
(Received 9 February 1988 by M.F. Collins) Density-of-states (DOS) of a quasi-two-dimensional electron gas subjected to a high magnetic field applied perpendicularly to the electronic layer as formed in multiple quantum well structures (MQWS) is calculated. The second moment (tr0) representing the broadening of the Gaussian density-of-states of the ground Landau level due to the electron-acoustic and electron-piezoelectric phonon interaction is theoretically and numerically computed for the MQWS in GaAs/AIAs. The contribution of the acoustic phonon scattering is found to be larger than that due to the electron-piezoelectric phonon interaction, however piezoelectric phonon scattering is significant in MQWS. Magnetic field (B) dependence of % due to the acoustic and piezoelectric scattering is given approximately a s B 1/2 and B 1/4 respectively. Obtained numerical values of a0 indicate that at low temperatures the combined effect of an acoustic and piezoelectric phonon scatterings is of considerable and competing importance as that of the impurity scattering effects in MQWS.
IN RECENT years, several theoretical and experimental investigations [1,2] have been made to observe the density-of-states of a two-dimensional electron gas subjected to a high magnetic field applied perpendicularly to the layers as realized in the single quantum wells, multiple quantum wells and heterolayers of GaAs/A1As sytems [3]. Energy spectrum of such a system is completely quantized and the usual optical and transport quantities diverge due to the presence of the delta function singularities in the density-of-states. This divergence can be removed either by considering the inelastic [4] scattering of the electrons with low energy acoustic phonons or by considering the multiple elastic scattering [5-7] of the electrons with disorder, acoustic [8] and piezoelectric phonons or surface roughness or any combination of the two. However, in the limit of high magnetic field and low phonon energy, the inelastic scattering by acoustical or piezoelectric phonons gives vanishing conductivity [4] in the absence of the mixing between the Landau levels. Thus, the broadening of the density-of-states due to the inelastic scattering mechanism is not expected to be significant at low temperatures and high magnetic fields. Multiple scattering by disorder (called the Self Consistent Born Approximation (SCBA)) as developed by Ando [5] and Gerhardts [6] broadens the delta function singularity into semi-elliptical and
Gaussian shapes respectively with width varying in both cases as x/~ where B is the applied magnetic field. This x/~ variation of the width due to the disorder scattering has been widely observed in the density-ofstates [2] and cyclotron resonance measurements [9] in agreement with the theory [5-7]. Tsui and co-workers [10] have recently observed the effect of acoustic and piezoelectric phonon scattering competing with the long range coulomb scattering in the heterojunctions at low temperatures. Chaubey and Gupta [1 l] subsequently investigated the broadening of the density-ofstates due to the acoustic phonon scattering using the form factor given by Stern Howard's [12] variational wave function. The broadening of the density-ofstates as calculated [l l] was found to vary as x/B and in agreement with the observed magnetic field dependence experimentally [2]. In the present Communication, an earlier analysis [11] of the effect of the electron phonon interaction on the density-of-states (DOS) of a two dimensional electron gas as formed in heterojunctions is extended to the multiple quantum well structures (MQWS). The second moment which represents the broadening of the density-of-states is explicitly calculated for the electron acoustic and electron piezoelectric phonon interaction in a multiple quantum well structure in GaAs/AIAs system. The magnitude of the broadening
625
DENSITY-OF-STATES OF A TWO-DIMENSIONAL ELECTRON GAS Vol. 67, No. 6
626
(a0) of the density-of-states of the ground Landau level due to the combined scattering from the acoustic and piezoelectric phonons is of the same order of magnitude as that obtained from the short range scattering theory [5], with the magnetic field dependence given as v/B in agreement with the experimental data [2]. Density-of-states Do(E ) of the ground Landau level is found to be given by the following expression [6, 11]
Do(E) =
l
1
2x12 ~ 0 . 0 exp
[
(E ~Eo)2.7 2og
_]'
and ep, the subband energy is found by solving the following transcendental equation
kpa+ 2 t a n - l ( ~ )
=
0.2
0.ac +
o.P~,
No(Q) =
0.U,Phin the limit of high magnetic field and Self Consistent Born Approximation is given by [8, 11, 13]
a (°2 = 4 j(2~)3 ( d3Q [v~°(Q)12[1 + 2N0(O)] e -½t2q~ N,Ph (3)
where the index (i) stands for the acoustic and piezoelectric phonon contributions to the width
Iv~(Q)I: =
fie 2 ~
2c, (q~ + q2),/2,
(4)
e2p2fi V~
IVP~(Q)I2 = 2~(q~ + q2),/2,
(5)
E, P, V~, cL and e~ are the deformation potential, piezoelectric constant, velocity of sound, elastic constant and static dielectric constant of GaAs respectively. The form factor for the multiple quantum well is given by [13, 14]
[ IFp~(q=)l z =
1 6 c o s 4 ( ~ -~)
-]2 7""W - -
+
~
--
+
'
/
[e~ ' ° ~ - 1]-',fl
-
ksT'
(9)
ks and T being the Boltzmann's constant and temperature respectively. Equation (3) is numerically evaluated by making use of the equations (4)-(7) and (8) for the ground Landau level (N = 0) and the ground subband (p = 1) for the acoustic and piezoelectric phonon interactions separately. The parameters are deformation potential E = 13.5eV, the static permittivity ~ = 12.9 e0, ~ = 5.37 x 103msec, m*/m = 0.0667 dimensionless piezoelectric constant [16] P = 0.064, c L = 2 × 1 0 1 ° J m 3. The quantum well parameters are a = b = 50 A, w = 0.1 eV. Variation of a0.phwith magnetic field is presented in Table 1 for acoustic as well as for piezoelectric phonon interaction. The magnetic field dependence of the width of the Gaussian density due to the deformation potential and piezoelectric interactions are approximately given by B ~/2 and B ~/4respectively. The value of a0.Phwhich lies between 0.2 meV and 0.5 meV with the increase of the magnetic field is of the same order of magnitude as obtained using the impurity scattering theory in the Self Consistent Born Approximation. It is further noteworthy that as compared to a single electron layer [11] the electron phonon interaction effects in multilayers are reduced by about a factor of 2. The contribution of the acous-
(2)
x [Z~)(½12q~)]EIFpp(q~)[2,
(8)
1
(1)
0.2
p~,p = 1,2 . . . . .
L°N(X) in equation (3) is Laguerre polynomials. Q describes the wave vector of a phonon of frequency ~°o, Q = (q~l,qz), qtt being the two-dimensional vector in the plane of the layer.
where 0.o is the full width at half the maximum of the above Gaussian function, E0 = h~,/2 scaled by the ground subband energy and l = (cfi/eB) aj2. The width 0.0.Phdue to the electron phonon interaction is decomposed as follows [15], 2 0.0.Ph
=
. e-
2zpb
2Zp(X2p+ kp2)COS(~-~)+ q-.sin(~-~)(k2p-3Z2p-q2-_)] z, x I sin (~ b/2)
+
- - 7 - -+ - - 21,;-
a is the width of the quantum well, b is the interquanturn well separation and w stands for the height of the quantum well. Also d = a + b k, =
, \,/2 (~2eP);Xp
=
[2m*(w -- ep).-]'/2 L ~2 ] ,
(7)
-
q2)2 :
+
(6)
tic phonon scattering is larger than that due to the electron piezoelectric phonon interaction, however the piezoelectric phonon scattering is significant in GaAs/ AlAs MQWS. Obtained numerical values of a0 indicate that at low temperatures, the combined effect of the acoustic and piezoelectric scattering mechanisms is
Vol. 67, No. 6 DENSITY-OF-STATES OF A TWO-DIMENSIONAL ELECTRON GAS
627
Table 1. The variation of ao.ph with magnetic fieM (B) in GaAs/A1As multiple quantum well structures at T = 10 K Magnetic field B
Second moment of the density D0(E)
O'o,ph = [O'o,ac2 + O.O.pz2 ] 1/2
5 Tesla 10 Tesla 15 Tesla 20 Tesla 25 Tesla
Deformation potential
Piezoelectric interaction
(~o,.c)
(~o,Pz)
0.175 meV 0.325 meV 0.379 meV 0.420 meV 0.432 meV
0.092 meV 0.102 meV 0.108 meV 0.112 meV 0.116 meV
of considerable and competing importance as that of the impurity scattering effects in MQWS. It is anticipated that the results of the present theory combined with the disorder scattering theory [5, 6] would provide satisfactory explanation to the observed data on density-of-states [2]. Acknowledgements - - This work is supported by Formation de chercheurs et concertie (FCAR), Provincial Government of Quebec, Grant No 88-AR0270. I would like to thank Professor M. Singh for discussions and Professor C.M. Van Vliet for the hospitality at CRM, Universit6 de Montr6al.
5.
6. 7. 8. 9.
10. REFERENCES 1. 2.
3. 4.
M. Ya. Azbel, Phys. Rev. B33, 8844 (1986). J.P. Eisenstein, H.L. Stormer, V. Narayanmurti, A.Y. Cho, A.C. Grossard & C.W. Tu, Phys. Rev. Letts. 55, 875 (1985). T.P. Smith, B.B. Goldberg, P.J. Stiles & M. Heiblum, Phys. Rev. B32, 2696 (1985). E. Stahl, D. Weiss, G. Weimann, K. Von Klitzing & K. Ploog, J. Phys. C18, L783 (1985). T. Ando, A.B. Fowler & F. Stern, Rev. Mod. Phys. 54, 437 (1982) and references cited therein. P. Vasilopoulos, Phys. Rev. B33, 8587 (1986).
11. 12. 13. 14. 15. 16.
0.198 meV 0.341 meV 0.394 meV 0.435 meV 0.447 meV
T. Ando & Y. Uemura, J. Phys. Soc. Japan 36, 959 (1974). T. Ando, J. Phys. Soc. Japan 36, 1521 (1974); 37, 622 (1974); 37, 1233 (1974); 38, 989 (1975). R.R. Gerhardts, Z. Physik B2I, 275 (1975); B21, 285 (1975). M. Prasad and S. Fujita, Physica 91A, 1 (1978). M. Prasad, Phys. Status Solidi (b) 109, 11 (1982). M. Prasad, Phys. Letts. 70A, 127 (1979); J. Phys. C12, 5489 (1979); J. Phys. C13, 3239 (1980). G. Abstreiter, J.F. Koch, P. Groy & Y. Couder, Phys. Rev. B14, 2494 (1976); G. Abstreiter, J.P. Kotthaus, J.F. Koch & G. Dorda, Phys. Rev. BI4, 2480 (1976). B.J.F. Lin, D.C. Tsui & G. Weimann, Solid State Commun. 56, 287 (1985). M.P. Chaubey & O.P. Gupta, Solid State Commun. 59, 317 (1986). F. Stern & W.J. Howard, Phys. Rev. 163, 816 (1967). M.P. Chaubey & M. Singh, Phys. Rv. B34, 2385 (1986). D. Calecki, J.F. Palmier & A. Chomette, J. Phys. C17, 5017 (1984). M. Prasad & S. Fujita, Phys. Letts. 63A, 147 (1978). P.K. Basu & B.R. Nag, J. Phys. C14, 1519 (1981).