- Email: [email protected]
Using non-equilibrium acoustic phonons to probe disordered quantum wires
ELSEVIER
Physica B 219&220 (1996) 4346
Using non-equilibrium acoustic phonons to probe disordered quantum wires M.P. Blencowe The Blackett Laboratory, Imperial College, London SW7 2BZ. UK
Abstract
Kent and co-workers at Nottingham recently performed an experiment in which non-equilibrium acoustic phonons were used to investigate the structure of an electron gas confined to a split-gate GaAs/(AIGa)As quantum wire. We explore the possibility of using non-equilibrium phonons to probe the effects of wire disorder on the electron gas.
In a recent experiment [1], Kent and co-workers at Nottingham demonstrated for the first time the use of non-equilibrium phonons as a probe of the electronic structure of a quantum wire. A split-gate GaAs/(A1Ga)As wire was employed, with metal film deposited on the back surface of the GaAs substrate serving as non-equilibrium phonon source when heated above the ambient temperature. The interaction of the confined electron gas with the incident phonon beam resulted in a detectable change in the wire resistance. As the gate voltage was varied, peaks in the resistance change were observed and found to occur at approximately the same gate voltage values as for the conductance step edges. The reason for the observed behaviour is as follows: the electron density of states is strongly peaked at the subband edges and thus the electrons will be scattered most strongly by the phonons as the Fermi level crosses a subband edge [2]. The Nottingham experiment clearly reveals the discrete subband structure of the confined electron states. In the present work, we shall explore the possible use of non-equilibrium phonons to probe the finer effects of wire disorder on the electrons. In the split-gate device, the electrons are scattered by the fluctuating potential of the remote ionized donor impurities [3] and also by the fluctuating wire width [4]. We shall consider the former type of disorder, leaving a brief discussion of the latter type to the end of the paper. We assume the combined processes of electron-electron scattering and elastic disorder scattering bring the electrons in the different subbands to thermal equilibrium in a characteristic time which is less than the electron-phonon scattering time. The wire resistance is viewed as a thermometer for the electron temperature Te. From the dependence of T~ on the Fermi level and incident phonon momentum, information about the electron states is obtained. We also suppose the incident phonon pulse is of sufficient duration to heat the electron gas up to the point where the power absorbed from the phonon beam is balanced by the radiated power. This enables a straightforward calculation of the electron temperature. For given electron temperature Te, heater temperature Th and ambient temperature Tl, the net absorbed power per unit wire length is given approximately by P ~ ~
)< Z M,N
2 .:
dQ] Qy I-ll E~(Q)]2
~c dxeiO'XzM(X)ZN(X)
dEDM'N"~(E'Q)Fs(E'O)'
0921-4526/96/$15.00 @ 1996 Elsevier Science B.V. All rights reserved SSDI 0921-4526(95)00644-3
(1)
M.P. Blencowe/ Physica B 219&220 (1996) 43 46
44 where
F,(E, Q) := {(e ~C~'(O)'kr~- 1) -1 - ( e a''(Q)'kr~ - 1) 1+ O,(Q)[(et,~,,.(O),kr~,_ 1) l _ (e~¢~(Q)/kr,
-', '1}
x [(e (E E~),'kr~+ 1) -I _ (e(E+~,,,(O)-Ev)/kr~+ 1) - I ]
(2)
and m,
DM,N,s(E,Q) := ~
I Qy I kM(E)-'kN(E--~o,(Q)) -I
X Z~_
I[IM(E)-I + IN(E -[- h o , ( O ) ) -1] [kM(E) ± kN(e 4- ho~(Q)) ± Oy]2 + ¼[IM(E)-I + lN(E -[- ho~(Q)) 112
with
kM(E)=(2m*'~'/2 h2 ]
E _ m.w (M--
1))1"2
(3)
(4)
and
1;¢1 ~ Cw-3kM ' Z
AFMM,kM,1 {exp[--¼~2(kM -- kM, )2] + exp[_¼~2(kM + kM, )2]},
(5)
MI
where f+~
N'MM, := / a -oo
f+oo
dx I
dx' ZM(X)ZM,(X)Z~c(x')zM,(X') exp[--~-2(x -- x')2].
(6)
a -Do
In Eq. (1), m. is the effective electron mass and p the mass density of GaAs. The first sum is over the three phonon polarizations and Qy is the component of phonon momentum parallel to the wire. The function ~.s(O) denotes the usual (deformation + piezoelectric) electron-phonon interaction term. In Eq. (2), the stepfunction O,(O) is equal to one (and zero otherwise) for all Q such that the group velocity vector v~(Q) ( = Voo,(Q)) with origin at the heater surface points towards the wire. The indices M and N label the subbands and ZM denotes the Mth harmonic oscillator energy eigenfunction for frequency h/(m.w2), where w is the effective wire width. The approximate expression for the mean free path lM of an electron with initial wavenumber kM is given in Eq. (5). The dimensionless constant C characterizes the strength of the fluctuations, while the correlation length ~ gives the length scale over which the fluctuations vary significantly. From Eq. (3), we see that the main effect of the disorder is to relax momentum conservation parallel to the wire for the process of absorption and emission of phonons by the electrons. The resulting broadening of the peaks in electron temperature versus EF gives a measure of the disorder strength C. Unless the following condition is satisfied: h
- - > kTe
(7)
"~'M
where ZMI = (hkM/m.)lM 1, the disorder broadening will be masked by thermal broadening. This was most likely the situation in the Nottingham experiment which operated at temperatures of a few Kelvin. With typical mean free paths of the order of a micron and a wire width of a few tens of nanometers, the electron temperature must be no larger than a few tenths of a Kelvin to be in the disorder broadening regime (7). In our calculations, we use the same values for the wire parameters as those of the Nottingham device, i.e. an effective wire width w ~ 23nm, length L = 10gm and heater film with dimensions z2~x= 100lam and A y = 10lam located a distance D = 3801am directly below the wire channel. To be in the disorder broadening regime, we set Ti = 0.1 K, Th = 0.5 K and choose disorder strength C -- 0.05. In the two figures, we present, as a first approximation, the results of a calculation of the net absorbed power per unit wire length for fixed electron temperature Te - 7"1 = 0.1 K. (According to our above assumptions, we should properly solve for the Te versus Ev dependence by setting to zero the net absorbed power (1).) Fig. 1 gives the absorbed power
M.P. Blencowe/Physica B 219&220 (1996) 43-46
45
I
<___
I
0
2.5
2 1.5
0
<_____
1 0.5
~o
.<
0 0.5
0.6
0.7
0.8
0.9
1
i.i
EF (10 -~1 J) Fig. 1. The net absorbed power per unit wire length for ~ = 200 nm. I
3.5 I
3
\
2.5 2 1.5 1
!
/
r
0.5 0
<
o.s
0.6
0.7
0.8
0.9
~
z.~
E r ( I 0 -2~ J)
Fig. 2. The net absorbed power per unit wire length for ~ = 23 nm. for correlation length ~ = 200nm, while for Fig. 2 we have ¢ - w ( = 23nm). The considered Fermi level range begins just above the M = 1 subband edge and ends just below the M -- 3 subband edge. Only the dominant absorption processes are taken into account. The dominant processes for the given EF range are: (a) absorption o f fast (s = 2) transverse phonons causing the electrons to undergo M = 1 ~ N = 1 forward scattering transitions (long dashed line), and (b) absorption o f fast transverse phonons causing M = 2 ~ N = 2 forward scattering transitions (short dashed line). The solid line denotes the net absorption. Intrasubband backscattering and intersubband transitions give a much smaller contribution because o f the small momentum of the dominant frequency phonons emitted from the metal film at Th = 0.5 K. The fast transverse phonons give the dominant contribution because they have a much larger range o f Qy momenta than do the slow transverse and longitudinal phonons - a consequence o f focusing and the given heater geometry. Comparing the EF dependence of the absorbed power in the two plots, we see that the main qualitative difference appears in the behaviour of the absorbed power for the M - 1 -+ N = 1 transition as the Fermi level crosses the M = 2 subband edge: there is an abrupt change for ~ = 23 nm, while for ~ = 200 nm the absorbed power is completely unaffected. When ~ >> w, the matrix Afgg' is approximately diagonal and we have only intrasubband disorder scattering. On the other hand, when ~ ~ O ( w ) and ~ ~
46
M.P. Blencowe/ Physica B 219&220 (1996) 43-46
the definition of the matrix A/'MM'. The resulting Er dependence of the absorbed power would be qualitatively the same however.
Acknowledgements The author is grateful to A. Kent, A. Shik, A. MacKirmon and A. Naylor for helpful and stimulating discussions.
References [1] [2] [3] [4]
A. Naylor et al., in preparation. A.Y. Shik and L.J. Challis, Phys. Rev. B 47 (1993) 2082. J.A. Nixon and J.H. Davies, Phys. Rev. B 41 (1989) 7929. A. Kumar et al., Appl. Phys. Lett. 54 (1989) 1270.
Physica B 219&220 (1996) 4346
Using non-equilibrium acoustic phonons to probe disordered quantum wires M.P. Blencowe The Blackett Laboratory, Imperial College, London SW7 2BZ. UK
Abstract
Kent and co-workers at Nottingham recently performed an experiment in which non-equilibrium acoustic phonons were used to investigate the structure of an electron gas confined to a split-gate GaAs/(AIGa)As quantum wire. We explore the possibility of using non-equilibrium phonons to probe the effects of wire disorder on the electron gas.
In a recent experiment [1], Kent and co-workers at Nottingham demonstrated for the first time the use of non-equilibrium phonons as a probe of the electronic structure of a quantum wire. A split-gate GaAs/(A1Ga)As wire was employed, with metal film deposited on the back surface of the GaAs substrate serving as non-equilibrium phonon source when heated above the ambient temperature. The interaction of the confined electron gas with the incident phonon beam resulted in a detectable change in the wire resistance. As the gate voltage was varied, peaks in the resistance change were observed and found to occur at approximately the same gate voltage values as for the conductance step edges. The reason for the observed behaviour is as follows: the electron density of states is strongly peaked at the subband edges and thus the electrons will be scattered most strongly by the phonons as the Fermi level crosses a subband edge [2]. The Nottingham experiment clearly reveals the discrete subband structure of the confined electron states. In the present work, we shall explore the possible use of non-equilibrium phonons to probe the finer effects of wire disorder on the electrons. In the split-gate device, the electrons are scattered by the fluctuating potential of the remote ionized donor impurities [3] and also by the fluctuating wire width [4]. We shall consider the former type of disorder, leaving a brief discussion of the latter type to the end of the paper. We assume the combined processes of electron-electron scattering and elastic disorder scattering bring the electrons in the different subbands to thermal equilibrium in a characteristic time which is less than the electron-phonon scattering time. The wire resistance is viewed as a thermometer for the electron temperature Te. From the dependence of T~ on the Fermi level and incident phonon momentum, information about the electron states is obtained. We also suppose the incident phonon pulse is of sufficient duration to heat the electron gas up to the point where the power absorbed from the phonon beam is balanced by the radiated power. This enables a straightforward calculation of the electron temperature. For given electron temperature Te, heater temperature Th and ambient temperature Tl, the net absorbed power per unit wire length is given approximately by P ~ ~
)< Z M,N
2 .:
dQ] Qy I-ll E~(Q)]2
~c dxeiO'XzM(X)ZN(X)
dEDM'N"~(E'Q)Fs(E'O)'
0921-4526/96/$15.00 @ 1996 Elsevier Science B.V. All rights reserved SSDI 0921-4526(95)00644-3
(1)
M.P. Blencowe/ Physica B 219&220 (1996) 43 46
44 where
F,(E, Q) := {(e ~C~'(O)'kr~- 1) -1 - ( e a''(Q)'kr~ - 1) 1+ O,(Q)[(et,~,,.(O),kr~,_ 1) l _ (e~¢~(Q)/kr,
-', '1}
x [(e (E E~),'kr~+ 1) -I _ (e(E+~,,,(O)-Ev)/kr~+ 1) - I ]
(2)
and m,
DM,N,s(E,Q) := ~
I Qy I kM(E)-'kN(E--~o,(Q)) -I
X Z~_
I[IM(E)-I + IN(E -[- h o , ( O ) ) -1] [kM(E) ± kN(e 4- ho~(Q)) ± Oy]2 + ¼[IM(E)-I + lN(E -[- ho~(Q)) 112
with
kM(E)=(2m*'~'/2 h2 ]
E _ m.w (M--
1))1"2
(3)
(4)
and
1;¢1 ~ Cw-3kM ' Z
AFMM,kM,1 {exp[--¼~2(kM -- kM, )2] + exp[_¼~2(kM + kM, )2]},
(5)
MI
where f+~
N'MM, := / a -oo
f+oo
dx I
dx' ZM(X)ZM,(X)Z~c(x')zM,(X') exp[--~-2(x -- x')2].
(6)
a -Do
In Eq. (1), m. is the effective electron mass and p the mass density of GaAs. The first sum is over the three phonon polarizations and Qy is the component of phonon momentum parallel to the wire. The function ~.s(O) denotes the usual (deformation + piezoelectric) electron-phonon interaction term. In Eq. (2), the stepfunction O,(O) is equal to one (and zero otherwise) for all Q such that the group velocity vector v~(Q) ( = Voo,(Q)) with origin at the heater surface points towards the wire. The indices M and N label the subbands and ZM denotes the Mth harmonic oscillator energy eigenfunction for frequency h/(m.w2), where w is the effective wire width. The approximate expression for the mean free path lM of an electron with initial wavenumber kM is given in Eq. (5). The dimensionless constant C characterizes the strength of the fluctuations, while the correlation length ~ gives the length scale over which the fluctuations vary significantly. From Eq. (3), we see that the main effect of the disorder is to relax momentum conservation parallel to the wire for the process of absorption and emission of phonons by the electrons. The resulting broadening of the peaks in electron temperature versus EF gives a measure of the disorder strength C. Unless the following condition is satisfied: h
- - > kTe
(7)
"~'M
where ZMI = (hkM/m.)lM 1, the disorder broadening will be masked by thermal broadening. This was most likely the situation in the Nottingham experiment which operated at temperatures of a few Kelvin. With typical mean free paths of the order of a micron and a wire width of a few tens of nanometers, the electron temperature must be no larger than a few tenths of a Kelvin to be in the disorder broadening regime (7). In our calculations, we use the same values for the wire parameters as those of the Nottingham device, i.e. an effective wire width w ~ 23nm, length L = 10gm and heater film with dimensions z2~x= 100lam and A y = 10lam located a distance D = 3801am directly below the wire channel. To be in the disorder broadening regime, we set Ti = 0.1 K, Th = 0.5 K and choose disorder strength C -- 0.05. In the two figures, we present, as a first approximation, the results of a calculation of the net absorbed power per unit wire length for fixed electron temperature Te - 7"1 = 0.1 K. (According to our above assumptions, we should properly solve for the Te versus Ev dependence by setting to zero the net absorbed power (1).) Fig. 1 gives the absorbed power
M.P. Blencowe/Physica B 219&220 (1996) 43-46
45
I
<___
I
0
2.5
2 1.5
0
<_____
1 0.5
~o
.<
0 0.5
0.6
0.7
0.8
0.9
1
i.i
EF (10 -~1 J) Fig. 1. The net absorbed power per unit wire length for ~ = 200 nm. I
3.5 I
3
\
2.5 2 1.5 1
!
/
r
0.5 0
<
o.s
0.6
0.7
0.8
0.9
~
z.~
E r ( I 0 -2~ J)
Fig. 2. The net absorbed power per unit wire length for ~ = 23 nm. for correlation length ~ = 200nm, while for Fig. 2 we have ¢ - w ( = 23nm). The considered Fermi level range begins just above the M = 1 subband edge and ends just below the M -- 3 subband edge. Only the dominant absorption processes are taken into account. The dominant processes for the given EF range are: (a) absorption o f fast (s = 2) transverse phonons causing the electrons to undergo M = 1 ~ N = 1 forward scattering transitions (long dashed line), and (b) absorption o f fast transverse phonons causing M = 2 ~ N = 2 forward scattering transitions (short dashed line). The solid line denotes the net absorption. Intrasubband backscattering and intersubband transitions give a much smaller contribution because o f the small momentum of the dominant frequency phonons emitted from the metal film at Th = 0.5 K. The fast transverse phonons give the dominant contribution because they have a much larger range o f Qy momenta than do the slow transverse and longitudinal phonons - a consequence o f focusing and the given heater geometry. Comparing the EF dependence of the absorbed power in the two plots, we see that the main qualitative difference appears in the behaviour of the absorbed power for the M - 1 -+ N = 1 transition as the Fermi level crosses the M = 2 subband edge: there is an abrupt change for ~ = 23 nm, while for ~ = 200 nm the absorbed power is completely unaffected. When ~ >> w, the matrix Afgg' is approximately diagonal and we have only intrasubband disorder scattering. On the other hand, when ~ ~ O ( w ) and ~ ~
46
M.P. Blencowe/ Physica B 219&220 (1996) 43-46
the definition of the matrix A/'MM'. The resulting Er dependence of the absorbed power would be qualitatively the same however.
Acknowledgements The author is grateful to A. Kent, A. Shik, A. MacKirmon and A. Naylor for helpful and stimulating discussions.
References [1] [2] [3] [4]
A. Naylor et al., in preparation. A.Y. Shik and L.J. Challis, Phys. Rev. B 47 (1993) 2082. J.A. Nixon and J.H. Davies, Phys. Rev. B 41 (1989) 7929. A. Kumar et al., Appl. Phys. Lett. 54 (1989) 1270.