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Theory of resonant magneto-polaron in GaAs-GaAlAs heterostructure
Solid State Communications, Vo1.56,No.l, pp.43-46,1985. Printedin Great Britain.
0038-1098/85 $3.00+ .OO PergamonPressLtd.
THEORY OF RESONANT MAGNETO-POLARON IN GaAs-GaAlAs HETEROSTRUCTURE Wlodek Zawadzki* Physikalisches Institut, Montanuniversit?it Leoben,A-8700 Leoben Austria (Received
29 June 1985 by F.Bassani)
Energies of resonant polarons related to interaction of inversion electrons in GaAs with optic phonons in the presence of a magnetic field are calculated and compared with recent cyclotron resonance measurements on a GaAs-GaAlAs heterostructure. The nonparabolicity of the conduction band is accounted for in the triangular potential approximation. The screening of polar interaction is neglected. Using an intensity of the confining electric field as a fitting parameter a very good agreement between the theory and the experiment is obtained.
explained by band's nonparabolicity alone, which provided evidence for the resonant magneto-polar effects. However, no attempt has been made totreat theoretically resonant polarons in this important two-dimensional structure and to reach a qualitative description of the data. This is the purpose of the present communication.
INTRODUCTION It has been recognised some time ago that a polar interaction between two-dimensional electrons and optic phonons should lead to observable effects in the presence of a magnetic field (II, similar as in the three-dimensional case 121. After the first experimental observation of resonant polarons for inversion electrons in InSb by Horst et al. 131 this problem has attrac ted considerable attention of the theorists 14-71. The only attempt to describe two-dimensional magneto-polarons in a real system (metal-oxide-semiconductor structure involving InSb) has been made by Lassnig and Zawadzki 14 I. However, an investigation of polarons in InSb has one advantage but several disadvantages. The advantage is that one can reach experimentally the resonant condition fihw%wL at relatively low magnetic fiel&, so that it is possible to investigate polaron energies both below and above the resonance. The disadvantages are: InSb is very weakly polar, its conduction band is strongly nonparabolic, the MOS electrons have low mobilities and, last not least, electrons populate simultaneously a few electric subbands. All this makes a quantitative description of experimental data in InSb difficult and ambiguous. Resonant magneto-polarons have been recently observed in cyclotron resonance experiments with a GaAs-GaAlAs heterostructure 181. These results are free of the above disadvantages. It has been demonstrated that the observed increase of the effective mass can not be
THEORY We assume that the potential in GaAs near the interface with GaAlAs has a triangular form: V(z)=eFz for z>O and V(z)=m for z
* Permanent address: Institute of Physics, Polish Academy of Sciences 02668 Warsaw, Poland.
43
54
THEORY OF RESONANT MAGNETO-POLARON IN GaAs-GaAIAs HETEROSTRUCTURE
Vol. 56, No. I
lar potential within this model has been described in [gland the case of parallel electric and magnetic fields has been treated in 1101. The problem of finding the energies is reduced to a solution of a transcendental equation. However, if the nonparabolicity is weak, which is the case in GaAs, the equation for energies may be reduced to the following form (cf.18 1)
y/2+425k(T) Eg
l/2
5
(i+l) (1) 4
2m0
where
EL = -
47reFh (Eg+22
&cl
2
+
[(j Eg 2+E$w~(~+$)]
l/2 (2)
2
GaAWs - GaAs
Ed represents the energy of motion transverse to magnetic field, wc=eH/ m*c with m* denoting the effective mass a? the ban8 edge. Once the root so of Eq(1) is found for given F and H, one obtains the energies from the formula E in=Zo(Eg+2El) +
EL
* 2
Fig.1. Triangular potential well, first electric subband and cyclotron resonance transitions below the resonance with optic phonons in a GaAs-GaAlAs heterostructure (schematically).
(3)
The left-hand side of Eq(1) presents an expansion in powers of zr+1'2.0ne deals typically with values ~~0.01 so that z7/2 and higher-order terms are certainly negligible, The standard formula for the energies (parabolic band) is obtained by neglecting sI with respect neglecting the second term in to E LHS 8; Eq(1) and expanding the root in Choosing a higher electric Eq(2). field F makes the potential well narrower, the energy of electric subband increase (cf. Fig.1) and Eqs.(l)-(3) give larger negative deviations of EO,-EoO from heH/m;c, since the electron is on average further away from the band edge and the nonparabolic effects are stronger. Considering the electron-phonon polar interaction we do not include the screening since the dynamic polarisation effects near the resonance fihwzfiwcho, are not well understood until present. The polaron pinning of the upper Landau level due to virtual phonon emission is described by the Wigner-Brillouin perturbation theory 2 AE,=I] I] ’ (4) q AE,-E,-E,-&w~ A nonresonant contribution of higher Landau levels has been neglected and we suppressed the electric subband index. The Wigner-Brillouin perturbation theory is valid as long as the imaginary part of the self energy is small, as long as real phonon-emission irzcesses do not occur 1111. This is satisfied for hwcChwL. The electron wavefunctions are Y on=exp
Y - Y,
(ik,x)$n (~1
f. (z 1
(5)
where yo=kxL2, L=(fic/eH)"2 and $n are
harmonic oscillator functions. For the first electric subband we take f0
=
($)1’3zexp(-
$z)
For the triangular potential the variational calculation gives b=2(3eFm*/2fi2)1'3. The Frijhlich interaction is (7) where 27re2BhwL
A=[
(1K - ; )]I/2 m 0
V
In the above form the interaction does not depend on electron mass, which is important since in our case the mass is not constant. For the phonon emission there is 2
I
-i%z]fo>]2=(,+$)-3
(9)
The matrix elements for x- and y-dependent functions (5) are standard. The denominator of Eq(4) does not depend on q and one obtains AE,= Ck
(C2+d'2
(10)
where C = (El-Eo -hwL)/2 and Z is the matrixelement squared summed over phonon states. The summation is carried out in cylindrical coordinates. Integrations over $Iand q, can be done analytically.The final result is
+;y2x I;=BiemX/x
dx (l+yJx)S
(11)
Vol. 56, No. I
45
THEORYOF RESONANTMAGNETO-POLARON IN GaAs-GaAlAsHETEROSTRUCTURE
in which D= -1 2J2
e2hW _-_&(1_1) L ken
(12) KO
and y =J2/bL. Parameter y characterises anisotropy of the electron wavefunction. For a very thin inversion layer (ideal 2D gas) y=O at all magnetic fields. In our case O
5
IO
15
H(T)
25
Fig.2. Deviation of the cyclotron resonance transition energy from heH/m*c for inversion electrons in a GaAs-GaAlxs heterostructureversus magnetic field. Points-recalculated experimental data of Horst et al. 181 , dashed line theoretical (includinq band's nonparabolicity) , solid line-- theoretical (including band's nonparabolicity and resonant polaron effect).
nant situation by Price 1121, the screening of the polar interaction becomes significant for electron densities above 1011cm-2, so that for the heterostructure in question (ns=4x1011cm-2) the screening effects should not be very pronounced. The only possibility to separate clearly both contributions is to measure the energy of the upper polaron branch at magnetic fields above the resonance with optic phonons. Thus the problem requires further experimental and theoretical investigation.
46
THEORY OF RESONANT MAGNETO-POLARON IN GaAs-GaAlAs HETEROSTRUCTURE
Acknowledgements - I acknowledge helpful discussions with Dr.U.Merkt and Dr.R.Lassnig. It is my pleasure to thank Prof.G.Bauer for generous hospitality during my stay in Austria. Note - Just before sending this work to the publisher a recent paper of Sigg et al. 1131, concerned with the similar
Vol. 56, No.
subject, came to our attention. The authors include in the theory a static screening of the polar interaction. On the other hand the influence of nonparabolicity does not seem to be well accounted for. In order to describe their experimental data two parameters have had to be adjusted: the extent of electron wavefunction and the strength of the screening.
REFERENCES 1.
2. 3. 4.
5. 6. 7.
S.Das Sarma and A.Madhukar, Phys. 2823 (1980). Rev. E, E.J.Johnson and D-M-Larsen, Phys. Rev.Lett. 16, 655 (1966). M.Horst, U.Merkt and J.P.Kotthaus, Phys.Rev.Lett. 50, 754 (1983). R.Lassniq and W.Zawadzki, Surface Science k, 388 (1984) EXDRSSiOn (16) for C. in this paper should be multiFlied by 2~. S.Das Sarma, Phys.Rev.Lett. 52, 859 (1984). D.M.Larsen, Phys.Rev. E, 4595(1984). F.M.Peeters and J.T.Devreese, Phys. Rev. e, 3689 (1985).
8. M.Horst, U.Merkt, W.Zawadzki, J.C. Maan and K.Ploog, Solid State Commun. 53, 403 (1985). 9. W.ZawadzE, J.Phys.C: Solid State Phys. 16, 229 (1983). IO. W.Zawazki, in "Two-Dimensional Systems, Heterostructures and Superlattices" (Ed.G.Bauer et al.), p.2, Springer-Verlag (1984). II. G.D.Mahan, "Many-Particle Physics", Plenum Press (19851, p.489. 12. P.J.Price, J.Vac.Sci.Technol.19, 599 (1981). 13. H.Sigg, P.Wyder and J.A.A.J.Perenboom, Phys.Rev.m,5253 (1985).
0038-1098/85 $3.00+ .OO PergamonPressLtd.
THEORY OF RESONANT MAGNETO-POLARON IN GaAs-GaAlAs HETEROSTRUCTURE Wlodek Zawadzki* Physikalisches Institut, Montanuniversit?it Leoben,A-8700 Leoben Austria (Received
29 June 1985 by F.Bassani)
Energies of resonant polarons related to interaction of inversion electrons in GaAs with optic phonons in the presence of a magnetic field are calculated and compared with recent cyclotron resonance measurements on a GaAs-GaAlAs heterostructure. The nonparabolicity of the conduction band is accounted for in the triangular potential approximation. The screening of polar interaction is neglected. Using an intensity of the confining electric field as a fitting parameter a very good agreement between the theory and the experiment is obtained.
explained by band's nonparabolicity alone, which provided evidence for the resonant magneto-polar effects. However, no attempt has been made totreat theoretically resonant polarons in this important two-dimensional structure and to reach a qualitative description of the data. This is the purpose of the present communication.
INTRODUCTION It has been recognised some time ago that a polar interaction between two-dimensional electrons and optic phonons should lead to observable effects in the presence of a magnetic field (II, similar as in the three-dimensional case 121. After the first experimental observation of resonant polarons for inversion electrons in InSb by Horst et al. 131 this problem has attrac ted considerable attention of the theorists 14-71. The only attempt to describe two-dimensional magneto-polarons in a real system (metal-oxide-semiconductor structure involving InSb) has been made by Lassnig and Zawadzki 14 I. However, an investigation of polarons in InSb has one advantage but several disadvantages. The advantage is that one can reach experimentally the resonant condition fihw%wL at relatively low magnetic fiel&, so that it is possible to investigate polaron energies both below and above the resonance. The disadvantages are: InSb is very weakly polar, its conduction band is strongly nonparabolic, the MOS electrons have low mobilities and, last not least, electrons populate simultaneously a few electric subbands. All this makes a quantitative description of experimental data in InSb difficult and ambiguous. Resonant magneto-polarons have been recently observed in cyclotron resonance experiments with a GaAs-GaAlAs heterostructure 181. These results are free of the above disadvantages. It has been demonstrated that the observed increase of the effective mass can not be
THEORY We assume that the potential in GaAs near the interface with GaAlAs has a triangular form: V(z)=eFz for z>O and V(z)=m for z
* Permanent address: Institute of Physics, Polish Academy of Sciences 02668 Warsaw, Poland.
43
54
THEORY OF RESONANT MAGNETO-POLARON IN GaAs-GaAIAs HETEROSTRUCTURE
Vol. 56, No. I
lar potential within this model has been described in [gland the case of parallel electric and magnetic fields has been treated in 1101. The problem of finding the energies is reduced to a solution of a transcendental equation. However, if the nonparabolicity is weak, which is the case in GaAs, the equation for energies may be reduced to the following form (cf.18 1)
y/2+425k(T) Eg
l/2
5
(i+l) (1) 4
2m0
where
EL = -
47reFh (Eg+22
&cl
2
+
[(j Eg 2+E$w~(~+$)]
l/2 (2)
2
GaAWs - GaAs
Ed represents the energy of motion transverse to magnetic field, wc=eH/ m*c with m* denoting the effective mass a? the ban8 edge. Once the root so of Eq(1) is found for given F and H, one obtains the energies from the formula E in=Zo(Eg+2El) +
EL
* 2
Fig.1. Triangular potential well, first electric subband and cyclotron resonance transitions below the resonance with optic phonons in a GaAs-GaAlAs heterostructure (schematically).
(3)
The left-hand side of Eq(1) presents an expansion in powers of zr+1'2.0ne deals typically with values ~~0.01 so that z7/2 and higher-order terms are certainly negligible, The standard formula for the energies (parabolic band) is obtained by neglecting sI with respect neglecting the second term in to E LHS 8; Eq(1) and expanding the root in Choosing a higher electric Eq(2). field F makes the potential well narrower, the energy of electric subband increase (cf. Fig.1) and Eqs.(l)-(3) give larger negative deviations of EO,-EoO from heH/m;c, since the electron is on average further away from the band edge and the nonparabolic effects are stronger. Considering the electron-phonon polar interaction we do not include the screening since the dynamic polarisation effects near the resonance fihwzfiwcho, are not well understood until present. The polaron pinning of the upper Landau level due to virtual phonon emission is described by the Wigner-Brillouin perturbation theory
Y - Y,
(ik,x)$n (~1
f. (z 1
(5)
where yo=kxL2, L=(fic/eH)"2 and $n are
harmonic oscillator functions. For the first electric subband we take f0
=
($)1’3zexp(-
$z)
For the triangular potential the variational calculation gives b=2(3eFm*/2fi2)1'3. The Frijhlich interaction is (7) where 27re2BhwL
A=[
(1K - ; )]I/2 m 0
V
In the above form the interaction does not depend on electron mass, which is important since in our case the mass is not constant. For the phonon emission there is 2
I
-i%z]fo>]2=(,+$)-3
(9)
The matrix elements for x- and y-dependent functions (5) are standard. The denominator of Eq(4) does not depend on q and one obtains AE,= Ck
(C2+d'2
(10)
where C = (El-Eo -hwL)/2 and Z is the matrixelement squared summed over phonon states. The summation is carried out in cylindrical coordinates. Integrations over $Iand q, can be done analytically.The final result is
+;y2x I;=BiemX/x
dx (l+yJx)S
(11)
Vol. 56, No. I
45
THEORYOF RESONANTMAGNETO-POLARON IN GaAs-GaAlAsHETEROSTRUCTURE
in which D= -1 2J2
e2hW _-_&(1_1) L ken
(12) KO
and y =J2/bL. Parameter y characterises anisotropy of the electron wavefunction. For a very thin inversion layer (ideal 2D gas) y=O at all magnetic fields. In our case O
5
IO
15
H(T)
25
Fig.2. Deviation of the cyclotron resonance transition energy from heH/m*c for inversion electrons in a GaAs-GaAlxs heterostructureversus magnetic field. Points-recalculated experimental data of Horst et al. 181 , dashed line theoretical (includinq band's nonparabolicity) , solid line-- theoretical (including band's nonparabolicity and resonant polaron effect).
nant situation by Price 1121, the screening of the polar interaction becomes significant for electron densities above 1011cm-2, so that for the heterostructure in question (ns=4x1011cm-2) the screening effects should not be very pronounced. The only possibility to separate clearly both contributions is to measure the energy of the upper polaron branch at magnetic fields above the resonance with optic phonons. Thus the problem requires further experimental and theoretical investigation.
46
THEORY OF RESONANT MAGNETO-POLARON IN GaAs-GaAlAs HETEROSTRUCTURE
Acknowledgements - I acknowledge helpful discussions with Dr.U.Merkt and Dr.R.Lassnig. It is my pleasure to thank Prof.G.Bauer for generous hospitality during my stay in Austria. Note - Just before sending this work to the publisher a recent paper of Sigg et al. 1131, concerned with the similar
Vol. 56, No.
subject, came to our attention. The authors include in the theory a static screening of the polar interaction. On the other hand the influence of nonparabolicity does not seem to be well accounted for. In order to describe their experimental data two parameters have had to be adjusted: the extent of electron wavefunction and the strength of the screening.
REFERENCES 1.
2. 3. 4.
5. 6. 7.
S.Das Sarma and A.Madhukar, Phys. 2823 (1980). Rev. E, E.J.Johnson and D-M-Larsen, Phys. Rev.Lett. 16, 655 (1966). M.Horst, U.Merkt and J.P.Kotthaus, Phys.Rev.Lett. 50, 754 (1983). R.Lassniq and W.Zawadzki, Surface Science k, 388 (1984) EXDRSSiOn (16) for C. in this paper should be multiFlied by 2~. S.Das Sarma, Phys.Rev.Lett. 52, 859 (1984). D.M.Larsen, Phys.Rev. E, 4595(1984). F.M.Peeters and J.T.Devreese, Phys. Rev. e, 3689 (1985).
8. M.Horst, U.Merkt, W.Zawadzki, J.C. Maan and K.Ploog, Solid State Commun. 53, 403 (1985). 9. W.ZawadzE, J.Phys.C: Solid State Phys. 16, 229 (1983). IO. W.Zawazki, in "Two-Dimensional Systems, Heterostructures and Superlattices" (Ed.G.Bauer et al.), p.2, Springer-Verlag (1984). II. G.D.Mahan, "Many-Particle Physics", Plenum Press (19851, p.489. 12. P.J.Price, J.Vac.Sci.Technol.19, 599 (1981). 13. H.Sigg, P.Wyder and J.A.A.J.Perenboom, Phys.Rev.m,5253 (1985).