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Mobility and thermoelectric power of one dimensional electron gas in quantum well-wires
395
Superlattices and Microstructures, Vol. 6, No. 4, 1989
MOBILITY AND THSRMOELECTRIC POWER OF ONE DIMENSIONAL ELECTRON GAS IN QUANTUM WELL-WIRES C. K. Sarkar ÷, S. Kundu* and P. K. Basu* +Jadavpur University, Calcutta, India *Institute of Radio Physics and Electronics, Calcutta University, 92 Acharya Prafulla Chandra Road, Calcutta-700009, India (Received 8 August 1988) The mobility and diffusion thermopower of a one dimensional electron gas in a Quantum Well wire have been calculated by employing the Boltzmann equation, taking into account the ideal density of states function, all relevant scattering mechanisms and screening of scattering potentials. The variations of the mobility and thermopower with temperature for different IDEG densities are presented.
i. Introduction The mobility behaviour of the two dimensional electron-gas (2DEG) in heterojunction has been explained by a theoryl based on the Boltzmann equation assuming a constant density of states (DOG), the usual scattering mechanisms and proper screening. A theory of diffusion thermopower developed along the same line 2 is not however successful in explaining all the experimental data. It applies to the case of high quality (mobility) samples only and for low quality samples, deviations in the DOS function from its constant ~ature have to be assumed. In the present work, we extend our calculation for the mobility of a IDEG supported by a High Electron Mobility Transistor (HEMT) having a narrow channel. In the calqulation, we have considered all the scattering processes including surface roughness scattering, ideal DOS of carriers and proper screening. In addition, results for diffusion thermopower are also presented. It is expected that the experimental data on mobility and thermopower of the IDEG will soon be available and hence a comparison with the present calculated values will give an idea about the quality of the samples used. The expressions needed for our calculation are given in sec.2 and the results are discussed in sec.3. 2. Theory The IDEG is assumed to occupy the
0749-6036/89/080395+04 $02.00/0
lowest sub-~and, and the wave function is given by ° ~(x,y,z) = (b3/La) I/2 sin(~y/a) ze -bz/2 e ikx
(I)
where L and a are respectively, the length and width of the channel and b is the variational parameter. The scattering mechanisms considered are deformation potential acoustic (DP~, piezoelectric (PZ), impurity ( I ~ ) * both remote impurity (RI) and background impuiity (BI), and surface roughness (SR) D. The corresponding unscreened ,relaxation times are given as follows:
~D~
= 9
E12kB Tm*b/(aal13CLk)
(21
~
= [ (K2e2kBTm*b)/(32~3%ak) ] [2g(k)+g(ko) ] (3)
~
= [(~m*NBi)/(2~3k3)](e2/2~6S )2 [~+(ka)21Ko2(ka)-Kl2(ka)}] +
[(~m*NRI)/(2"I'*3k3)](e2/2~6S)2 [K12(ka) - Ko2(ka)] (4) ~S~ = (35/32~)[e2N1Hd/~s]2(m*/~3a 3) exp(-4k2d2/k)
(5)
In addition, the effect of polar optic phonon (PO) scattering has been included at higher temperature. The relaxation time is
© 1989AcademicPressLimited
396
Superlattices and Microstructures. VoL 6, No. 4, 1989
= e2 °b
The thermoelectric power of the IDEG is derived from the Boltzmann transport equation and is given by 7
[2g(qx)+g(qxo)](NQ+l)+(X+Xo )-I
S = ~
[2g(q~l+g(q~
where < > denotes average over the electron distribution function.
)]NQ}
(6)
0
where,
k
o
= (k 2 + ~ 2 / a 2 ) 1 / 2
g ( x ) = ( 4 b 2 + 9 b x + 6 x 2 ) / ~ x [ b + 2 x ] 3} x = E/kBT xo = ~o/kBT 2
qx
x' = 0 '
2 q x ,X I
o
42/a2 +
qx,., : c2{2X ×o-2X(* xo)}
1/2
C2 = (2m*kBT/~2) I/2
Here, E 1 is the deformation potential constant, CL is the elastic constant K is the dimensionless piezoelectric constant, NBI and NR! are respectively the background and remote impurity densities, KQ and K 1 are the modified Bessel functlons of the second kind of order 0 and 1 respectively, H is the mean asperity height, d is the correlation length, 6£o and 6SI represent respectively ~ e free space-permittivity and the dielectric constants of the semiconductor for zero and infinite frequencies and ~ o is the phonon energy. The above expressions are modified due to screening and the screened dielectric constant @t non-zero temperatures i~ given by o
7dE%(q,EFI[4kBT
<(q) =
cosh2(u)]-l(7)
where
(i0)
3. Results and Discussion The following values of parameters are used for calculating the mobility and thermopower for the IDEG formed in a narrow channel GaAs-AIGaAs heterostructure : a = i0 nm, El=7eV , K=0.064, CL=139.7 GNm -2, m*=0.067 mo, £=0=12.9, H=0.43 nm, d=l.5 nm, NBI = I ( ~ m-3, 6SI = 10.8. Remote impurity scattering has been neglected. The variations of the mobility with temperature are shown in Figs.l and 2 for a IDEG density of IxlO 8 m -I. The unscreened values of the mobility are shown in Fig.l whereas the screened values are plotted in Fig.2. It appears from Fig.l, that the values of mobility are quite high compared to the values for the 2DEG. As expected the background impurity scattering is insignificant at higher temperatures for the chosen value of impurity density. Also the mobilities limited by piezoelectric and surface roughness scattering
Unscreened N = xlOam-I
103
PO
. .-'"
T>
104
IMP --->
lu}
..." I0 z
u = [(E-EF)/kBT]
and
_EF]
/
Io
-t" SR
T
%
0
%(q'EF)
[ < ~ > - i
q+2k E 4e2m * : %G + ~'--'~"-- F(q)q i n ~ .(8) 2
F(q) = (--~R~ il-2Kl(qR)Jl(qR);
\
S o
I03
\\t
q~,~ p \
t z-*
z
(9)
~ R ~ being the background dielectric co5Ntant. There is a logarithmic singularity at q = 2kF, k F being the wavevector at the Fermi energy for zerotemperature. R is the radius of the hypothetical circular wire that can replace the actual QWW and yet have the same screening effect.
I0
T 0
~ I 102 80 I60 240 Temperature (°K)
Fig. i. Variation of mobility with temperature for different scattering processes. T indicates overall mobility excluding PO scattering contribution
Super/attices and Microstructures, VoL 6, No. 4, 1989
397
Screened I0
.
:.
T
-
' ~
T>
I05
:. -"~-,.44~ ]" •
:
,\\'
SR
u~
\po
~"
\~\
T>
\
IOem -'
% I03
\
=
':.~-,
~,,
\
I04
~-.-___
I x I
0
i
108 I
0
K
I
4
I
8
12
q/kF T~ " ~ I02
L
0
£ig.2.
Fig.3.
~D P
I I \1~ 80 160 Temperature (°K)
I
Variation of normalised screened dielectric constant with normaliSed wave vector
I03
240
Variation of mobility with temperature including the effect of screening. T : same as in Fig.l.
5x 106m-'
, 200 /
are ~bout one order of magnitude higher than deformation potential scattering. The temperature dependence of overall mobility excluding polar optic phonon scattering is therefore primarily dictated by deformation potential acoustic phonon scattering. Above about 150 K, however, the LO phonon scattering is found to dominate. Inclusion of screening alters the nature of variation of impurity and surface roughness scattering limited mobility i.e., the values first decrease and then increase with temperature. As expected the effect of screening is to reduce scattering rates and hence to increase the values of mobility for all the processes. As shown in Fig.3, the higher is the electron concentration, the stronger is the effect of screening and hence the higher is the value of mobility. Fig.4 depicts the variation of the screened values of the thermopower with temperature for three different values of the IDEG density. As expected from Hq.(lO), the thermoelectric power is a strong function of fermi energy EF and the values decrease with increaslng concentration. It has been found that impurity scattering dominates at lowest temperature while deformation potential acoustic phonon scattering becomes effective at higher
t-~ I00
5x 0 % 1 -
z
-
//
x I0'
/
//
L
0 Fig.4.
I
]
I
80 160 Temperature (°K)
i
240
Variation of Thermoelectric power (T.E.P.) with temperature.
temperature. The piezoelectric and surface roughness scatterings are weaker. Since deformation potential acoustic phonon scattering is dominant at higher temperature, Eq.(2) suggests that/ is temperature independent and when this factor is large compared to HF, S is constant. This is indeed the case for N = 5xlO 6. In the absence of any experiment the present values cannot be compared with experimental data. The nature of variation of S is similar to that for the 2DEG, the values are however lligher for the IDEG.
Acknowledgement : The authors are grateful to the Council of Scientific and Industrial i{esearch and the Indian
398
Superlattices and Microstructures, Vol. 6, No. 4, 1989
National Science Academy for their financial support in this research work.
3. S. Das Sarma and W.Y. Lai, Physical Review B, 32, 1401 (1985). 4. J. Lee and M.O. Vassell, Journal of Physics C, 17, 2525 (1984). 5. P.K. Basu, C.K. Sarkar and S. Kundu, Superlattices and Microstructures, 2, 247 (1987). 6. G. Fishman~ Physical Review B, 34~ 2394 (1986). 7. B.R. Nag, Electron Transport in Compound Semiconductors (Springer, Berlin) 217 (1980).
References
:-
i. W. Walukiewicz, W.E. Ruda, J. Logowski and H. Gatos, Physical Review B, 80, 4571 (1984). 2. S. Kundu, C.K. Sarkar and P.K. Basu, Journal of Applied Physics, 61, 5080 (1987).
Superlattices and Microstructures, Vol. 6, No. 4, 1989
MOBILITY AND THSRMOELECTRIC POWER OF ONE DIMENSIONAL ELECTRON GAS IN QUANTUM WELL-WIRES C. K. Sarkar ÷, S. Kundu* and P. K. Basu* +Jadavpur University, Calcutta, India *Institute of Radio Physics and Electronics, Calcutta University, 92 Acharya Prafulla Chandra Road, Calcutta-700009, India (Received 8 August 1988) The mobility and diffusion thermopower of a one dimensional electron gas in a Quantum Well wire have been calculated by employing the Boltzmann equation, taking into account the ideal density of states function, all relevant scattering mechanisms and screening of scattering potentials. The variations of the mobility and thermopower with temperature for different IDEG densities are presented.
i. Introduction The mobility behaviour of the two dimensional electron-gas (2DEG) in heterojunction has been explained by a theoryl based on the Boltzmann equation assuming a constant density of states (DOG), the usual scattering mechanisms and proper screening. A theory of diffusion thermopower developed along the same line 2 is not however successful in explaining all the experimental data. It applies to the case of high quality (mobility) samples only and for low quality samples, deviations in the DOS function from its constant ~ature have to be assumed. In the present work, we extend our calculation for the mobility of a IDEG supported by a High Electron Mobility Transistor (HEMT) having a narrow channel. In the calqulation, we have considered all the scattering processes including surface roughness scattering, ideal DOS of carriers and proper screening. In addition, results for diffusion thermopower are also presented. It is expected that the experimental data on mobility and thermopower of the IDEG will soon be available and hence a comparison with the present calculated values will give an idea about the quality of the samples used. The expressions needed for our calculation are given in sec.2 and the results are discussed in sec.3. 2. Theory The IDEG is assumed to occupy the
0749-6036/89/080395+04 $02.00/0
lowest sub-~and, and the wave function is given by ° ~(x,y,z) = (b3/La) I/2 sin(~y/a) ze -bz/2 e ikx
(I)
where L and a are respectively, the length and width of the channel and b is the variational parameter. The scattering mechanisms considered are deformation potential acoustic (DP~, piezoelectric (PZ), impurity ( I ~ ) * both remote impurity (RI) and background impuiity (BI), and surface roughness (SR) D. The corresponding unscreened ,relaxation times are given as follows:
~D~
= 9
E12kB Tm*b/(aal13CLk)
(21
~
= [ (K2e2kBTm*b)/(32~3%ak) ] [2g(k)+g(ko) ] (3)
~
= [(~m*NBi)/(2~3k3)](e2/2~6S )2 [~+(ka)21Ko2(ka)-Kl2(ka)}] +
[(~m*NRI)/(2"I'*3k3)](e2/2~6S)2 [K12(ka) - Ko2(ka)] (4) ~S~ = (35/32~)[e2N1Hd/~s]2(m*/~3a 3) exp(-4k2d2/k)
(5)
In addition, the effect of polar optic phonon (PO) scattering has been included at higher temperature. The relaxation time is
© 1989AcademicPressLimited
396
Superlattices and Microstructures. VoL 6, No. 4, 1989
= e2 °b
The thermoelectric power of the IDEG is derived from the Boltzmann transport equation and is given by 7
[2g(qx)+g(qxo)](NQ+l)+(X+Xo )-I
S = ~
[2g(q~l+g(q~
where < > denotes average over the electron distribution function.
)]NQ}
(6)
0
where,
k
o
= (k 2 + ~ 2 / a 2 ) 1 / 2
g ( x ) = ( 4 b 2 + 9 b x + 6 x 2 ) / ~ x [ b + 2 x ] 3} x = E/kBT xo = ~o/kBT 2
qx
x' = 0 '
2 q x ,X I
o
42/a2 +
qx,., : c2{2X ×o-2X(* xo)}
1/2
C2 = (2m*kBT/~2) I/2
Here, E 1 is the deformation potential constant, CL is the elastic constant K is the dimensionless piezoelectric constant, NBI and NR! are respectively the background and remote impurity densities, KQ and K 1 are the modified Bessel functlons of the second kind of order 0 and 1 respectively, H is the mean asperity height, d is the correlation length, 6£o and 6SI represent respectively ~ e free space-permittivity and the dielectric constants of the semiconductor for zero and infinite frequencies and ~ o is the phonon energy. The above expressions are modified due to screening and the screened dielectric constant @t non-zero temperatures i~ given by o
7dE%(q,EFI[4kBT
<(q) =
cosh2(u)]-l(7)
where
(i0)
3. Results and Discussion The following values of parameters are used for calculating the mobility and thermopower for the IDEG formed in a narrow channel GaAs-AIGaAs heterostructure : a = i0 nm, El=7eV , K=0.064, CL=139.7 GNm -2, m*=0.067 mo, £=0=12.9, H=0.43 nm, d=l.5 nm, NBI = I ( ~ m-3, 6SI = 10.8. Remote impurity scattering has been neglected. The variations of the mobility with temperature are shown in Figs.l and 2 for a IDEG density of IxlO 8 m -I. The unscreened values of the mobility are shown in Fig.l whereas the screened values are plotted in Fig.2. It appears from Fig.l, that the values of mobility are quite high compared to the values for the 2DEG. As expected the background impurity scattering is insignificant at higher temperatures for the chosen value of impurity density. Also the mobilities limited by piezoelectric and surface roughness scattering
Unscreened N = xlOam-I
103
PO
. .-'"
T>
104
IMP --->
lu}
..." I0 z
u = [(E-EF)/kBT]
and
_EF]
/
Io
-t" SR
T
%
0
%(q'EF)
[
q+2k E 4e2m * : %G + ~'--'~"-- F(q)q i n ~ .(8) 2
F(q) = (--~R~ il-2Kl(qR)Jl(qR);
\
S o
I03
\\t
q~,~ p \
t z-*
z
(9)
~ R ~ being the background dielectric co5Ntant. There is a logarithmic singularity at q = 2kF, k F being the wavevector at the Fermi energy for zerotemperature. R is the radius of the hypothetical circular wire that can replace the actual QWW and yet have the same screening effect.
I0
T 0
~ I 102 80 I60 240 Temperature (°K)
Fig. i. Variation of mobility with temperature for different scattering processes. T indicates overall mobility excluding PO scattering contribution
Super/attices and Microstructures, VoL 6, No. 4, 1989
397
Screened I0
.
:.
T
-
' ~
T>
I05
:. -"~-,.44~ ]" •
:
,\\'
SR
u~
\po
~"
\~\
T>
\
IOem -'
% I03
\
=
':.~-,
~,,
\
I04
~-.-___
I x I
0
i
108 I
0
K
I
4
I
8
12
q/kF T~ " ~ I02
L
0
£ig.2.
Fig.3.
~D P
I I \1~ 80 160 Temperature (°K)
I
Variation of normalised screened dielectric constant with normaliSed wave vector
I03
240
Variation of mobility with temperature including the effect of screening. T : same as in Fig.l.
5x 106m-'
, 200 /
are ~bout one order of magnitude higher than deformation potential scattering. The temperature dependence of overall mobility excluding polar optic phonon scattering is therefore primarily dictated by deformation potential acoustic phonon scattering. Above about 150 K, however, the LO phonon scattering is found to dominate. Inclusion of screening alters the nature of variation of impurity and surface roughness scattering limited mobility i.e., the values first decrease and then increase with temperature. As expected the effect of screening is to reduce scattering rates and hence to increase the values of mobility for all the processes. As shown in Fig.3, the higher is the electron concentration, the stronger is the effect of screening and hence the higher is the value of mobility. Fig.4 depicts the variation of the screened values of the thermopower with temperature for three different values of the IDEG density. As expected from Hq.(lO), the thermoelectric power is a strong function of fermi energy EF and the values decrease with increaslng concentration. It has been found that impurity scattering dominates at lowest temperature while deformation potential acoustic phonon scattering becomes effective at higher
t-~ I00
5x 0 % 1 -
z
-
//
x I0'
/
//
L
0 Fig.4.
I
]
I
80 160 Temperature (°K)
i
240
Variation of Thermoelectric power (T.E.P.) with temperature.
temperature. The piezoelectric and surface roughness scatterings are weaker. Since deformation potential acoustic phonon scattering is dominant at higher temperature, Eq.(2) suggests that
Acknowledgement : The authors are grateful to the Council of Scientific and Industrial i{esearch and the Indian
398
Superlattices and Microstructures, Vol. 6, No. 4, 1989
National Science Academy for their financial support in this research work.
3. S. Das Sarma and W.Y. Lai, Physical Review B, 32, 1401 (1985). 4. J. Lee and M.O. Vassell, Journal of Physics C, 17, 2525 (1984). 5. P.K. Basu, C.K. Sarkar and S. Kundu, Superlattices and Microstructures, 2, 247 (1987). 6. G. Fishman~ Physical Review B, 34~ 2394 (1986). 7. B.R. Nag, Electron Transport in Compound Semiconductors (Springer, Berlin) 217 (1980).
References
:-
i. W. Walukiewicz, W.E. Ruda, J. Logowski and H. Gatos, Physical Review B, 80, 4571 (1984). 2. S. Kundu, C.K. Sarkar and P.K. Basu, Journal of Applied Physics, 61, 5080 (1987).