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Non-linear generation of alternating current harmonics in quantum dot superlattice miniband transport
E ELSEVIER
lli~TIBIital~ SClIMB & ~
B
Materials Science and Engineering B35 (1995) 256-258
Non-linear generation of alternating current harmonics in quantum dot superlattice miniband transport X . L . L e P 'b, N . J . M .
H o r i n g b, H . L . C u i b, K . K .
Thornber c
aState Key Laboratory of Functional Materials for Informatics, Shanghai Institute of Metallurgy, Chinese Academy of Sciences, 865 Changning Road, Shanghai 200050, China bDepartment of Physics and Engineering Physics, Stevens Institute of Technology, Hoboken, NJ 07030, USA CNEC Research Institute Inc., 4 Independence Way, Princeton, NJ 08540, USA
Abstract
We have analyzed vertical miniband transport in GaAs-based superlattices subject to a d.c. field Eo which drifts the system into the regime of negative differential mobility jointly with a perturbing single-frequency electric field E,, sin(2ogt). Our balance equation formulation takes account of strong electron-electron interactions which tend to thermalize the system rapidly about the center-of-mass motion. In this, we consider E,o >~Eo, so that the perturbing a.c. field is strong, and the responding steady periodic miniband current (evolving after transients relax) exhibits harmonic content beyond the impressed single frequency, characteristic of non-linear time harmonic generation. The scattering mechanisms are bulk phonons and random impurities. The superlattice treated here is a one-dimensional line of periodically spaced quantum dots of period d = 15 nm, lateral diameter dr = 10 nm, quantum well width a = 8 nm, carrier line density N l d = 0.314 and miniband width xl = 220 K. The impressed period of oscillation is taken as 2n/co = 4 ps, and, using material parameters appropriate to GaAs, we determine the harmonic composition of the periodic miniband current as a function of the high frequency amplitude E,o. Keywords: Superlattices; Electron conduction; High-field effects; Gallium arsenide
1. Introduction
The E s a k i - T s u [1] prediction of negative differential conductance (NDC) in vertical superlattice transport is valid for a variety of geometrical configurations of the superlattice. We report here on aspects of this phenomenology in a superlattice composed of a line of periodically spaced quantum dots subject to a strong d.c. electric field driving N D C miniband transport along the axial direction, with electron motion sharply confined in both lateral and transverse directions. In particular, we analyze the current response of such a system which it is further subjected to a large parallel a.c. electric field E,o sin(cot) comparable with the strong d.c. bias field, which produces a highly non-linear current response with attendant harmonic generation. In this analysis, we consider a model superlattice system in which electrons are driven along the axial z direction through the (lowest) miniband formed by periodically spaced potential wells and barriers of finite height. In the lateral-transverse ( x - y ) plane, the quan-
tum dot electrons are confined in both lateral directions. The electron energy dispersion e~(kz) can be written as the sum of a transverse subband energy e,, related to the lateral degrees of freedom, and a tightbinding-type lowest miniband energy e(kz) related to the longitudinal motion: e~(kz) = e~ + e(kz), with e(kz)=(A/2)(1-coskzd) where d is the superlattice period along the z direction, - n/d < kz <<.n/d, and A is the miniband width. The transverse electron state and its energy are described by the transverse quantum number n.
2. Method
Within the framework of the balance equation transport theory [2-4], a transport state of the many-electron system is described by its center-of-mass (CM) m o m e n t u m Pd = Npd ( N being the total number of carriers) and the relative electron temperature Tc. For a semiconductor superlattice having symmetric lateral ge0921-5107/95/$09.50 © 1995
ElsevierScience S.A. All rights reserved SSDI 0921-5107(95)01325-3
X.L. Lei et al. / Materials Science and Engineering B35 (1995) 256 258 ometry, the application of a spatially uniform time-dependent electric field E ( t ) along the axial z direction results in the CM momentum P~ = (0, 0,pa) and the carrier drift velocity rd----(0, 0, vd) all parallel in the z direction, and the superlattice force and energy balance equations [5] take the form dv~/dt = eE/m* + A
(1)
dhe/dt = eEva - W
(2)
257
where the d.c. component v0 and the harmonic coefficients /3ml and v~2 are given by vo = (w/27t) fo re#° Vd(t) d t v,.l = (~o/Tz)
v~(t) sin(mogt) dt
(8)
n/rJJ
Here, A is the total acceleration induced by both the impurity and the phonon scattering mechanisms [5], A = A~ + Ap, W is the electron energy loss rate due to phonon interactions [5], and va = (2/N) ~ [de(k. )/dk= ]f[e. (k: - Pd), Tel
(3)
n.k z
is the CM velocity, or the average drift velocity of the carrier in the z direction. Furthermore, 1/m* = (2/N) ~ [d2e(k:)/dkZ]f[e.(k: --Pd), Te]
(4)
n,k z
is the ensemble-averaged inverse effective mass of the CM, and he = (2/N) ~ e ( k : ) f [ e . ( k . - p a ) , Te]
(5)
tt,k=
is the ensemble-averaged electron energy per carrier. In these equations f(e, Tell = {exp[(e- p)/Te] + 1}-1 denotes the Fermi distribution function at the electron temperature T¢, and /~ is the chemical potential determined by the condition that the total number of electrons equals N:
v,.2 = (a~/~)
I~
Vd(t) COS(mtot) dt
3. Results
We have carried out numerical calculations of the large signal high frequency response for several GaAsbased superlattices using these equations. In this, realistic impurity, acoustic phonon and polar optical phonon scatterings in GaAs-A1GaAs systems are taken into consideration on a microscopic basis. Fig. 1 shows the calculated harmonic coefficients vii ,/)12, ~)31,v32 of the drift velocity as functions of the driving a.c. field amplitude E,o in the range 0.5 kV cm l ~< E,o <~ 30 kV cm 1 for the special case of zero bias E0 = 0 kV cm -1. (In this particular case v21 = 0 cm s - 1 and v22 = 0 cm s - 1.) Here, the GaAs-A1GaAs superlattice period d = 15 nm, lateral diameter dr of the quantum dot is 10 nm, quantum well width a = 8 nm and the miniband ld-Superlattiee Nld=0.314
d=15nm
/1=220 K
4
N=2
~f[e.(k_.--pd), T~]=2 ~f[e.(kz), n,k:
Te]
n,k z
3
The expressions for the impurity- and phonon-induced frictional accelerations Ai and Ap and the energy transfer rate W from the electron system to the phonon system have been given previously in Ref. [5]. They incorporate the role of impurity and phonon scatterings treated microscopically. Our study treats the application of both a d.c. electric field E0 and an a.c. field of frequency 09 and amplitude E,o, such that E(t) = Eo + E,,, sin(~ot). We determine the non-linear steady state time-dependent response of the drift velocity vd(t) and the electron temperature Te(t) to this time-dependent electric field by Fourier series analysis of the above bahmce equations. For the high frequency steady state, the drift velocity and the electron temperature of the system are periodic functions, and the drift velocity can be written in the form Vd(/) : Vo --b ~ *'rt~
[I)ml sin(me)t) + Urn2cos(mogt)] I
T
(6)
(7)
,,_2
g
=77 K T ¢ = 4 ps
Vl2
" "f'.. / /
-1
v31 ""'.....
.."
V32
-2 .3
/ °. . . .
0.5
t
1.0
. . . .
i
1.5
. . . .
i
. . . .
2.0
i
2-5
. . . .
3.0
Ew ( k V / c m ) Fig. 1. The calculated harmoniccoefficientsv~, v~2,v3~and v32of the drift velocityin the steady time-dependentresponse of a quantum dot superlattice (with period d = 15 nm, miniband width A = 220 K and carrier line density given by Nld= 0.314) is exhibitedas a function of the high frequency amplitude E,o. The To, bias Eo is zero, lattice temperature T = 77 K and the a.c. period is 4 ps. The range of E,, shown is 0.5-3.0 kV cm-~.
258
X.L. Lei et al. / Materials Science and Engineering B35 (1995) 256-258
width A = 220 K. The a.c. period is T~o= 2n/o9 = 4 ps, lattice temperature T = 77 K and the carrier line density is given by N~d= 0.314. Harmonic generation activity is clearly evident in this figure. Although results for E~o < 0.5 kV cm -1 are not shown here, we have verified that vl~ and v~2 reach maxima and minima respectively at E,~ ~ 0.2 kV c m - ~, and then they both approach zero as E~, goes to zero.
Acknowledgement One of the authors (N.J.M.H.) wishes to acknOwledge travel support of the US Office of Naval Research.
References [1] L. Esaki and R. Tsu, IBM J. Res. Dev., 14 (1970) 61. [2] X.L. Lei and C.S. Ting, Phys. Rev. B, 30 (1984) 4809; 32 (1985) 1112. [3] X.L. Lei and N.J.M. Horing, in C.S. Ting (ed.), Physics of Hot Electron Transport, World Scientific, Singapore, 1992, p. 1. [4] N.J.M. Horing, H.L. Cui and X.L. Lei, in C.S. Ting (ed.), Physics of Hot Electron Transport, World Scientific, Singapore, 1992, p. 133. [5] X.L. Lei, N.J.M. Horing and H.L. Cui, Phys. Rev. Lett., 66 (1991) 3277.
lli~TIBIital~ SClIMB & ~
B
Materials Science and Engineering B35 (1995) 256-258
Non-linear generation of alternating current harmonics in quantum dot superlattice miniband transport X . L . L e P 'b, N . J . M .
H o r i n g b, H . L . C u i b, K . K .
Thornber c
aState Key Laboratory of Functional Materials for Informatics, Shanghai Institute of Metallurgy, Chinese Academy of Sciences, 865 Changning Road, Shanghai 200050, China bDepartment of Physics and Engineering Physics, Stevens Institute of Technology, Hoboken, NJ 07030, USA CNEC Research Institute Inc., 4 Independence Way, Princeton, NJ 08540, USA
Abstract
We have analyzed vertical miniband transport in GaAs-based superlattices subject to a d.c. field Eo which drifts the system into the regime of negative differential mobility jointly with a perturbing single-frequency electric field E,, sin(2ogt). Our balance equation formulation takes account of strong electron-electron interactions which tend to thermalize the system rapidly about the center-of-mass motion. In this, we consider E,o >~Eo, so that the perturbing a.c. field is strong, and the responding steady periodic miniband current (evolving after transients relax) exhibits harmonic content beyond the impressed single frequency, characteristic of non-linear time harmonic generation. The scattering mechanisms are bulk phonons and random impurities. The superlattice treated here is a one-dimensional line of periodically spaced quantum dots of period d = 15 nm, lateral diameter dr = 10 nm, quantum well width a = 8 nm, carrier line density N l d = 0.314 and miniband width xl = 220 K. The impressed period of oscillation is taken as 2n/co = 4 ps, and, using material parameters appropriate to GaAs, we determine the harmonic composition of the periodic miniband current as a function of the high frequency amplitude E,o. Keywords: Superlattices; Electron conduction; High-field effects; Gallium arsenide
1. Introduction
The E s a k i - T s u [1] prediction of negative differential conductance (NDC) in vertical superlattice transport is valid for a variety of geometrical configurations of the superlattice. We report here on aspects of this phenomenology in a superlattice composed of a line of periodically spaced quantum dots subject to a strong d.c. electric field driving N D C miniband transport along the axial direction, with electron motion sharply confined in both lateral and transverse directions. In particular, we analyze the current response of such a system which it is further subjected to a large parallel a.c. electric field E,o sin(cot) comparable with the strong d.c. bias field, which produces a highly non-linear current response with attendant harmonic generation. In this analysis, we consider a model superlattice system in which electrons are driven along the axial z direction through the (lowest) miniband formed by periodically spaced potential wells and barriers of finite height. In the lateral-transverse ( x - y ) plane, the quan-
tum dot electrons are confined in both lateral directions. The electron energy dispersion e~(kz) can be written as the sum of a transverse subband energy e,, related to the lateral degrees of freedom, and a tightbinding-type lowest miniband energy e(kz) related to the longitudinal motion: e~(kz) = e~ + e(kz), with e(kz)=(A/2)(1-coskzd) where d is the superlattice period along the z direction, - n/d < kz <<.n/d, and A is the miniband width. The transverse electron state and its energy are described by the transverse quantum number n.
2. Method
Within the framework of the balance equation transport theory [2-4], a transport state of the many-electron system is described by its center-of-mass (CM) m o m e n t u m Pd = Npd ( N being the total number of carriers) and the relative electron temperature Tc. For a semiconductor superlattice having symmetric lateral ge0921-5107/95/$09.50 © 1995
ElsevierScience S.A. All rights reserved SSDI 0921-5107(95)01325-3
X.L. Lei et al. / Materials Science and Engineering B35 (1995) 256 258 ometry, the application of a spatially uniform time-dependent electric field E ( t ) along the axial z direction results in the CM momentum P~ = (0, 0,pa) and the carrier drift velocity rd----(0, 0, vd) all parallel in the z direction, and the superlattice force and energy balance equations [5] take the form dv~/dt = eE/m* + A
(1)
dhe/dt = eEva - W
(2)
257
where the d.c. component v0 and the harmonic coefficients /3ml and v~2 are given by vo = (w/27t) fo re#° Vd(t) d t v,.l = (~o/Tz)
v~(t) sin(mogt) dt
(8)
n/rJJ
Here, A is the total acceleration induced by both the impurity and the phonon scattering mechanisms [5], A = A~ + Ap, W is the electron energy loss rate due to phonon interactions [5], and va = (2/N) ~ [de(k. )/dk= ]f[e. (k: - Pd), Tel
(3)
n.k z
is the CM velocity, or the average drift velocity of the carrier in the z direction. Furthermore, 1/m* = (2/N) ~ [d2e(k:)/dkZ]f[e.(k: --Pd), Te]
(4)
n,k z
is the ensemble-averaged inverse effective mass of the CM, and he = (2/N) ~ e ( k : ) f [ e . ( k . - p a ) , Te]
(5)
tt,k=
is the ensemble-averaged electron energy per carrier. In these equations f(e, Tell = {exp[(e- p)/Te] + 1}-1 denotes the Fermi distribution function at the electron temperature T¢, and /~ is the chemical potential determined by the condition that the total number of electrons equals N:
v,.2 = (a~/~)
I~
Vd(t) COS(mtot) dt
3. Results
We have carried out numerical calculations of the large signal high frequency response for several GaAsbased superlattices using these equations. In this, realistic impurity, acoustic phonon and polar optical phonon scatterings in GaAs-A1GaAs systems are taken into consideration on a microscopic basis. Fig. 1 shows the calculated harmonic coefficients vii ,/)12, ~)31,v32 of the drift velocity as functions of the driving a.c. field amplitude E,o in the range 0.5 kV cm l ~< E,o <~ 30 kV cm 1 for the special case of zero bias E0 = 0 kV cm -1. (In this particular case v21 = 0 cm s - 1 and v22 = 0 cm s - 1.) Here, the GaAs-A1GaAs superlattice period d = 15 nm, lateral diameter dr of the quantum dot is 10 nm, quantum well width a = 8 nm and the miniband ld-Superlattiee Nld=0.314
d=15nm
/1=220 K
4
N=2
~f[e.(k_.--pd), T~]=2 ~f[e.(kz), n,k:
Te]
n,k z
3
The expressions for the impurity- and phonon-induced frictional accelerations Ai and Ap and the energy transfer rate W from the electron system to the phonon system have been given previously in Ref. [5]. They incorporate the role of impurity and phonon scatterings treated microscopically. Our study treats the application of both a d.c. electric field E0 and an a.c. field of frequency 09 and amplitude E,o, such that E(t) = Eo + E,,, sin(~ot). We determine the non-linear steady state time-dependent response of the drift velocity vd(t) and the electron temperature Te(t) to this time-dependent electric field by Fourier series analysis of the above bahmce equations. For the high frequency steady state, the drift velocity and the electron temperature of the system are periodic functions, and the drift velocity can be written in the form Vd(/) : Vo --b ~ *'rt~
[I)ml sin(me)t) + Urn2cos(mogt)] I
T
(6)
(7)
,,_2
g
=77 K T ¢ = 4 ps
Vl2
" "f'.. / /
-1
v31 ""'.....
.."
V32
-2 .3
/ °. . . .
0.5
t
1.0
. . . .
i
1.5
. . . .
i
. . . .
2.0
i
2-5
. . . .
3.0
Ew ( k V / c m ) Fig. 1. The calculated harmoniccoefficientsv~, v~2,v3~and v32of the drift velocityin the steady time-dependentresponse of a quantum dot superlattice (with period d = 15 nm, miniband width A = 220 K and carrier line density given by Nld= 0.314) is exhibitedas a function of the high frequency amplitude E,o. The To, bias Eo is zero, lattice temperature T = 77 K and the a.c. period is 4 ps. The range of E,, shown is 0.5-3.0 kV cm-~.
258
X.L. Lei et al. / Materials Science and Engineering B35 (1995) 256-258
width A = 220 K. The a.c. period is T~o= 2n/o9 = 4 ps, lattice temperature T = 77 K and the carrier line density is given by N~d= 0.314. Harmonic generation activity is clearly evident in this figure. Although results for E~o < 0.5 kV cm -1 are not shown here, we have verified that vl~ and v~2 reach maxima and minima respectively at E,~ ~ 0.2 kV c m - ~, and then they both approach zero as E~, goes to zero.
Acknowledgement One of the authors (N.J.M.H.) wishes to acknOwledge travel support of the US Office of Naval Research.
References [1] L. Esaki and R. Tsu, IBM J. Res. Dev., 14 (1970) 61. [2] X.L. Lei and C.S. Ting, Phys. Rev. B, 30 (1984) 4809; 32 (1985) 1112. [3] X.L. Lei and N.J.M. Horing, in C.S. Ting (ed.), Physics of Hot Electron Transport, World Scientific, Singapore, 1992, p. 1. [4] N.J.M. Horing, H.L. Cui and X.L. Lei, in C.S. Ting (ed.), Physics of Hot Electron Transport, World Scientific, Singapore, 1992, p. 133. [5] X.L. Lei, N.J.M. Horing and H.L. Cui, Phys. Rev. Lett., 66 (1991) 3277.