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Negative differential conductivity in a confined superlattice
~
Solid State Communications, Vol. 86, No. 4, PP. 231-233, 1993. Printed in Great Britain.
0038-1098/93 $6.00+.00 Pergamon Press Ltd
NEGATIVE DIFFERENTIAL CONDUCTIVITY IN A CONFINED SUPERLATTICE X.L. Lei, N.J.M. Horing and H.L. Cui
Department of Physics g~ Engineering Physics, Stevens Institute of Technology, Hoboken, New Jersey 07030 and K.K. Thornber
NEC Research Institute, ~, Independence Way, Princeton, New Jersey 08540 (Received 13 January 1993 by A.A. Maradudin)
(Accepted for publication 9 February 1993) The effects of confinement on steady state miniband transport are examined here, particularly with respect to negative differential conductivity. The superlattice is taken to be confined in one of the two directions normal to the growth direction. Using a balance equation approach adjusted to incorporate superlattice band structure and transverse confinement-induced subbands, we present the results of numerical calculations which take account of impurity, acoustic phonon and polar optic phonon scatterings. Nonlinear drift velocity and electron temperature are exhibited as functions of applied electric field for a series of confined superlattices having miniband width A = 900 K. Negative differential conductivity is clearly manifested and the case of confinement width d~ = 45 nm appears to be suitable for use in high frequency oscillators even at 300 K lattice temperature.
In this Letter we report the results of accurate numerical calculations detailing the effects of confinement on negative differential conductivity (NDC) in steady state superlattice miniband transport in the growth (z) direction. The superlattice (period d in the z direction) is understood to be confined in the x-direction, (having extension d,) and we account for four associated transverse subbands. Our results for negative differential conductivity for a confined superlattice suggest this system to be suitable for use in high frequency devices, as anticipated for a bulk superlattice in the original proposal of Esaki and Tsu 1. In this connection, new experiments on confined superlattices, analogous to recent work 2'3 on bulk superlattices demonstrating NDC in Esaki-Tsu conduction, will be important for further confirmation of confined behavior. Such superlattice confinement brings with it the prospect of smaller NDC-based devices in the development of microeletronics4's. Our formulation of the balance equation transport theory 6-s, extended to incorporate superlattice band structure 9-12 in the tight binding approximation along the growth direction and adjusted to take account of confinement13-as to a narrow well of width d, in the xdirection while electrons move freely in the y-direction, is rather lengthy and will be presented elsewhere 13. The energy gap in the electron energy band structure is taken to be large enough so that only the lowest miniband (width A) is important, but allowance is made for population of up to four x-transverse energy levels (subbands). Within this framework, balance equations are formulated to describe steady state vertical transport in terms of momentum and energy balance, with the constant uniform electric field E = E~. driving the electrons in the z-growth direction, subject to scatterings by ran-
domly distributed background impurities, and three dimensional acoustic phonons (with deformation potential and piezoelectric couplings) and polar optic phonons. These balance equations determine the electron temperature 7~ and the electron drift velocity va, as described in full detail in Ref.13. Here, we present numerical solutions of the balance equations for confined GaAs-based superlattices suitable for use in high frequency oscillators, explicitly exhibiting vd and T~ as functions of the applied electric field E, and also account for their dependence on superlattice period d, miniband width A, lattice temperature T, as well as x-confinement extension d,. In particular, we focus on miniband width A = 900 K for lattice temperatures T = 300 K and T = 45 K, sampling x-extensions d, = 10, 20, 30, and 45 nm. (A few simplifying assumptions are made in our work. A longitudinal form factor is estimated by assuming the single well wavefunction ¢(z) = d -1/2 for (0 < z < d) and 6(z) = 0 otherwise. Also, a Thomas-Fermi-type static screening is used for charged impurity and polar optic phonon scatterings. The screening form is not critical to the impurity scattering, since its role is subsumed in a total linear lowtemperature mobility #(0), which is subject to experimental determination. This static screening approximation may slightly underestimate the contribution of the polar optic phonons5, but it turns out to have little effect on the present results.) All the material parameters used in the calculations are taken to have typical values for GaAs, the same as those used in Ref.16. All of our calculated results shown in the figures exhibit negative differential conductivity, with the drift velocity vd reaching a peak value vp at a critical electric field strength Ec, and then declining into the NDC regime. 231
232
NEGATIVE DIFFERENTIAL CONDUCTIVITY 2d-Superlattice
(a) lOO
•". 3d
S0
~
6o
To display the effects of transverse cofffinement on longitudinal miniband transport for electrons subject to one-dimensional confinement in the superlattice, we inelude the corresponding bulk superlattice results in the figures under the "3d" designation. In order to clarify the variation of transport properties with variation of d~ and to facilitate comparison between 2d and 3d geometries, the carrier sheet density for each superlattice has been so chosen that the 3d carrier density is kept at
T =300 K
," "'. 45nm : ~ O n m
A=900 K
i
o
~- 40i d~__10am ' " ' . . . . 20 0
~ "'"......
0
5
10
Vol. 86, No 4
15
(a)
20
25
2d-Superlattice 160
30
E (kV/cm)
T=45 K
140 om
120
(b)
2d-Superlattice
"~-
A=900 K
.f'
=
100
8o
1800
-~ 60 ::" 3d 4O
1400
3d . . . . . . . .
2O
": /
~
/
" ' " . . . ..........
dz=10nm 5 ....
1000
1~0 . . . .
15 . . . .
20
E (kV/cm) 60o
d=10 nm 200
5
10
15
20
25
(b)
2d-Superlattice
30
1600
E (kV/cm)
1400
(c)
3d .."::"
~
~
1200
2d-Superlattice
1000 800
• ET-B
1.0 0.8
7
6OO
o 3d -- 2d
Jele¢e°°*°°"
T=45 K
4OO
x=
A=900 K
d = 10 nm
2OO
0.6
0
2
4
S
10
12
14
16
18
20
E (kV/cm)
0.4 0.2 0.0
6
d=10 nm ,
0
,
i
1
. . . .
i
2d-Superlattice
(c) . . . .
2
i
3
,
,
,
i
,
4
E/Ee
• ET-B
1.0
F i g u r e 1 Drift velocity Vd (a), and electron temperature T~ (b), as functions of the electric feld, as well as the normalized drift velocity Vd/Vp (c) as a function of the normalized electric field E/Ec, for a series of confined quantum well superlattices with differing xextensions. Here T = 300 K, A = 900K, d = 10 nm, and linear mobility #(0) = 1.0 m2/Vs at 4.2 K. Curves correspond to d~ = 10,20,30, and 45nm, and respective 2d densities N2 = 1.5,3.0,4.5 and 6.75 x 1011/cm2. Also shown are results for a 3d superlattice having the same d, A, and #(0), and with a carrier sheet density Ns = 1.5 x 1011/cm 2 per period. "ET-B" denotes results calculated using the Esaki-Tsu-Boltzmann theory.
o 3d
0.8
#
0.6 • ........
0.4 0.2 0.0
d=10 nm 1
2
3
4
E/E~ F i g u r e 2 Same as Figure 1, except that the lattice temperature here is T = 45 K.
Vol. 86, No. 4
NEGATIVE DIFFERENTIAL CONDUCTIVITY
the same value for all the 2d and 3d superlattice samples studied. Furthermore, for comparison purposes, the predictions based on Esaki-Tsu and Boltzmann theory (see Poef.13 for a detailed discussion) are also presented in the figures (under the designation " ET-B".) Figs.la-e show the calculated results at lattice temperature T = 300K for a series of superlattices having miniband width A = 900K, and linear mobility #(0) = 1.0m2/Vs at 4.2K. The confined superlattices have x-extensions d~ = 10,20,30 and 45nm and 2d density N2 = 1.5,3.0,4.5 and 6.75 × 10al/cm 2 respectively. The 3d-bulk superlattice has a carrier sheet density N, = 1.5 × 1011/cm 2. A significant feature is that this relatively wide miniband case exhibits not only high values of peak drift velocities but also steep negativedifferential-velocity in the functional dependence of vd vs E and the normalized counterpart vd/vp vs E/Ec. The electron temperature Te is also exhibited as a function of E. Obviously, such wide miniband width is desirable for a high frequency oscillator. Figs.2a-c show the corresponding results for a lower lattice temperature, T = 45 K. Again, a series of superlattice samples are considered: all have the same miniband width A = 900K and linear, low-field mobility /t(0) = 1.0m2/Vs at 4.2K, but different x-extensions
233
and 2d densities, with d, = 10, 20, 30 and 45nm, N2 = 1.5, 3.0, 4.5 and 6.75 x 10al/cm 2. The 3d-bulk superlattice has a carrier sheet density N, = 1.5 x 1011/cm 2. Both the electron drift velocity and the electron temperature are displayed as functions of the applied electric field for the various samples. Also shown is the the normalized peak drift velocity as a function of the normalized electric field. The results are very similar to the corresponding ones for T = 300 K. Meanwhile, as might have been expected, for this lower lattice temperature (T = 45 K) the absolute drift velocities are higher, and their peak values are attained at somewhat lower applied electric field strengths. In general, x-confinement of the superlattice results in peak drift velocities and NDC characteristics which are comparable with those of the unconfined 3d superlattice sample, but the critical fields are higher for the confined samples. However, the unconventional configuration of the confined superlattices makes it possible to exploit the nonresonant device geometry proposed by Cooper and Thornber 4, which has the potential of higher oscillation frequencies and wider band tunability. Acknowledgement-The authors thank the NEC Research Institute, Princeton, New Jersey for support of this work.
References
1 L. Esaki and R. Tsu, IBM J. Res. Dev. 14, 61 (1970). 2 A. Sibille, J.F. Palmier, H. Wang, and F. Mollot, Phys. Rev. Lett. 64, 52 (1990); A. Sibille, Proc.
6th Int. Conf. on Superlattices, Microstruetures and Microdevices, August 4-7, 1992, Xi'an, China;
3 4 5 6 7
M. Hadjazi, A. Sibille, P.J. Palmier, and Mollot, Electronics Lett. 27, 1101 (1992). H.T. Grahn, K. yon Klitzing, K. Ploog, and G.H. D5hler, Phys. Rev. B43, 12094 (1991). J.A. Cooper and K.K. Thornber, IEEE Electron Device Lett. EDL-6, 50 (1985). A.A. Ignatov, E.P. Dodin, and V.I. Shaskin, Mod. Phys. Lett, B 5, 1087 (1991). X.L. Lei and C.S. Ting, Phys. Rev. B32, 1112 (1985). X.L. Lei, J. Phys. C 18, L593 (1985).
8 X.L. Lei and N.J.M. Horing, Int. J. Mod. Phys. B 6, 805 (1992). 9 X.L. Lei, Phys. Star. Sol. (b), 170, 519 (1992). 10 X.L. Lei, N.J.M. Horing, and H.L. Cui, J. Phys. Condens. Matter 4, 9375 (1992). 11 X.L. Lei, N.J. Horing and H.L. Cui, Phys. Rev. Lett. 66, 3277 (1991). 12 X.L. Lei and I.C. da Cunha Lima, J. Appl. Phys. 71, 5517 (1992). 13 X.L. Lei, N.J. Horing, H.L. Cui, and K.K. Thornbet, submitted to Phys. Rev. B. 14 X.L. Lei, Proc. 21th Int. Conf. Semiconductor
Physics, August 10-15, 1992, Beijing, China. 15 X.L. Lei and X.F. Wang, Proc. 6th Int. Conf. Su-
perlattices, Microstructures gJ Microdevices, Aug. 1992, Xi'an, China. 16 X.L. Lei, J.L. Birman, and C.S. Ting, J. Appl. Phys. 58, 2270 (1985).
Solid State Communications, Vol. 86, No. 4, PP. 231-233, 1993. Printed in Great Britain.
0038-1098/93 $6.00+.00 Pergamon Press Ltd
NEGATIVE DIFFERENTIAL CONDUCTIVITY IN A CONFINED SUPERLATTICE X.L. Lei, N.J.M. Horing and H.L. Cui
Department of Physics g~ Engineering Physics, Stevens Institute of Technology, Hoboken, New Jersey 07030 and K.K. Thornber
NEC Research Institute, ~, Independence Way, Princeton, New Jersey 08540 (Received 13 January 1993 by A.A. Maradudin)
(Accepted for publication 9 February 1993) The effects of confinement on steady state miniband transport are examined here, particularly with respect to negative differential conductivity. The superlattice is taken to be confined in one of the two directions normal to the growth direction. Using a balance equation approach adjusted to incorporate superlattice band structure and transverse confinement-induced subbands, we present the results of numerical calculations which take account of impurity, acoustic phonon and polar optic phonon scatterings. Nonlinear drift velocity and electron temperature are exhibited as functions of applied electric field for a series of confined superlattices having miniband width A = 900 K. Negative differential conductivity is clearly manifested and the case of confinement width d~ = 45 nm appears to be suitable for use in high frequency oscillators even at 300 K lattice temperature.
In this Letter we report the results of accurate numerical calculations detailing the effects of confinement on negative differential conductivity (NDC) in steady state superlattice miniband transport in the growth (z) direction. The superlattice (period d in the z direction) is understood to be confined in the x-direction, (having extension d,) and we account for four associated transverse subbands. Our results for negative differential conductivity for a confined superlattice suggest this system to be suitable for use in high frequency devices, as anticipated for a bulk superlattice in the original proposal of Esaki and Tsu 1. In this connection, new experiments on confined superlattices, analogous to recent work 2'3 on bulk superlattices demonstrating NDC in Esaki-Tsu conduction, will be important for further confirmation of confined behavior. Such superlattice confinement brings with it the prospect of smaller NDC-based devices in the development of microeletronics4's. Our formulation of the balance equation transport theory 6-s, extended to incorporate superlattice band structure 9-12 in the tight binding approximation along the growth direction and adjusted to take account of confinement13-as to a narrow well of width d, in the xdirection while electrons move freely in the y-direction, is rather lengthy and will be presented elsewhere 13. The energy gap in the electron energy band structure is taken to be large enough so that only the lowest miniband (width A) is important, but allowance is made for population of up to four x-transverse energy levels (subbands). Within this framework, balance equations are formulated to describe steady state vertical transport in terms of momentum and energy balance, with the constant uniform electric field E = E~. driving the electrons in the z-growth direction, subject to scatterings by ran-
domly distributed background impurities, and three dimensional acoustic phonons (with deformation potential and piezoelectric couplings) and polar optic phonons. These balance equations determine the electron temperature 7~ and the electron drift velocity va, as described in full detail in Ref.13. Here, we present numerical solutions of the balance equations for confined GaAs-based superlattices suitable for use in high frequency oscillators, explicitly exhibiting vd and T~ as functions of the applied electric field E, and also account for their dependence on superlattice period d, miniband width A, lattice temperature T, as well as x-confinement extension d,. In particular, we focus on miniband width A = 900 K for lattice temperatures T = 300 K and T = 45 K, sampling x-extensions d, = 10, 20, 30, and 45 nm. (A few simplifying assumptions are made in our work. A longitudinal form factor is estimated by assuming the single well wavefunction ¢(z) = d -1/2 for (0 < z < d) and 6(z) = 0 otherwise. Also, a Thomas-Fermi-type static screening is used for charged impurity and polar optic phonon scatterings. The screening form is not critical to the impurity scattering, since its role is subsumed in a total linear lowtemperature mobility #(0), which is subject to experimental determination. This static screening approximation may slightly underestimate the contribution of the polar optic phonons5, but it turns out to have little effect on the present results.) All the material parameters used in the calculations are taken to have typical values for GaAs, the same as those used in Ref.16. All of our calculated results shown in the figures exhibit negative differential conductivity, with the drift velocity vd reaching a peak value vp at a critical electric field strength Ec, and then declining into the NDC regime. 231
232
NEGATIVE DIFFERENTIAL CONDUCTIVITY 2d-Superlattice
(a) lOO
•". 3d
S0
~
6o
To display the effects of transverse cofffinement on longitudinal miniband transport for electrons subject to one-dimensional confinement in the superlattice, we inelude the corresponding bulk superlattice results in the figures under the "3d" designation. In order to clarify the variation of transport properties with variation of d~ and to facilitate comparison between 2d and 3d geometries, the carrier sheet density for each superlattice has been so chosen that the 3d carrier density is kept at
T =300 K
," "'. 45nm : ~ O n m
A=900 K
i
o
~- 40i d~__10am ' " ' . . . . 20 0
~ "'"......
0
5
10
Vol. 86, No 4
15
(a)
20
25
2d-Superlattice 160
30
E (kV/cm)
T=45 K
140 om
120
(b)
2d-Superlattice
"~-
A=900 K
.f'
=
100
8o
1800
-~ 60 ::" 3d 4O
1400
3d . . . . . . . .
2O
": /
~
/
" ' " . . . ..........
dz=10nm 5 ....
1000
1~0 . . . .
15 . . . .
20
E (kV/cm) 60o
d=10 nm 200
5
10
15
20
25
(b)
2d-Superlattice
30
1600
E (kV/cm)
1400
(c)
3d .."::"
~
~
1200
2d-Superlattice
1000 800
• ET-B
1.0 0.8
7
6OO
o 3d -- 2d
Jele¢e°°*°°"
T=45 K
4OO
x=
A=900 K
d = 10 nm
2OO
0.6
0
2
4
S
10
12
14
16
18
20
E (kV/cm)
0.4 0.2 0.0
6
d=10 nm ,
0
,
i
1
. . . .
i
2d-Superlattice
(c) . . . .
2
i
3
,
,
,
i
,
4
E/Ee
• ET-B
1.0
F i g u r e 1 Drift velocity Vd (a), and electron temperature T~ (b), as functions of the electric feld, as well as the normalized drift velocity Vd/Vp (c) as a function of the normalized electric field E/Ec, for a series of confined quantum well superlattices with differing xextensions. Here T = 300 K, A = 900K, d = 10 nm, and linear mobility #(0) = 1.0 m2/Vs at 4.2 K. Curves correspond to d~ = 10,20,30, and 45nm, and respective 2d densities N2 = 1.5,3.0,4.5 and 6.75 x 1011/cm2. Also shown are results for a 3d superlattice having the same d, A, and #(0), and with a carrier sheet density Ns = 1.5 x 1011/cm 2 per period. "ET-B" denotes results calculated using the Esaki-Tsu-Boltzmann theory.
o 3d
0.8
#
0.6 • ........
0.4 0.2 0.0
d=10 nm 1
2
3
4
E/E~ F i g u r e 2 Same as Figure 1, except that the lattice temperature here is T = 45 K.
Vol. 86, No. 4
NEGATIVE DIFFERENTIAL CONDUCTIVITY
the same value for all the 2d and 3d superlattice samples studied. Furthermore, for comparison purposes, the predictions based on Esaki-Tsu and Boltzmann theory (see Poef.13 for a detailed discussion) are also presented in the figures (under the designation " ET-B".) Figs.la-e show the calculated results at lattice temperature T = 300K for a series of superlattices having miniband width A = 900K, and linear mobility #(0) = 1.0m2/Vs at 4.2K. The confined superlattices have x-extensions d~ = 10,20,30 and 45nm and 2d density N2 = 1.5,3.0,4.5 and 6.75 × 10al/cm 2 respectively. The 3d-bulk superlattice has a carrier sheet density N, = 1.5 × 1011/cm 2. A significant feature is that this relatively wide miniband case exhibits not only high values of peak drift velocities but also steep negativedifferential-velocity in the functional dependence of vd vs E and the normalized counterpart vd/vp vs E/Ec. The electron temperature Te is also exhibited as a function of E. Obviously, such wide miniband width is desirable for a high frequency oscillator. Figs.2a-c show the corresponding results for a lower lattice temperature, T = 45 K. Again, a series of superlattice samples are considered: all have the same miniband width A = 900K and linear, low-field mobility /t(0) = 1.0m2/Vs at 4.2K, but different x-extensions
233
and 2d densities, with d, = 10, 20, 30 and 45nm, N2 = 1.5, 3.0, 4.5 and 6.75 x 10al/cm 2. The 3d-bulk superlattice has a carrier sheet density N, = 1.5 x 1011/cm 2. Both the electron drift velocity and the electron temperature are displayed as functions of the applied electric field for the various samples. Also shown is the the normalized peak drift velocity as a function of the normalized electric field. The results are very similar to the corresponding ones for T = 300 K. Meanwhile, as might have been expected, for this lower lattice temperature (T = 45 K) the absolute drift velocities are higher, and their peak values are attained at somewhat lower applied electric field strengths. In general, x-confinement of the superlattice results in peak drift velocities and NDC characteristics which are comparable with those of the unconfined 3d superlattice sample, but the critical fields are higher for the confined samples. However, the unconventional configuration of the confined superlattices makes it possible to exploit the nonresonant device geometry proposed by Cooper and Thornber 4, which has the potential of higher oscillation frequencies and wider band tunability. Acknowledgement-The authors thank the NEC Research Institute, Princeton, New Jersey for support of this work.
References
1 L. Esaki and R. Tsu, IBM J. Res. Dev. 14, 61 (1970). 2 A. Sibille, J.F. Palmier, H. Wang, and F. Mollot, Phys. Rev. Lett. 64, 52 (1990); A. Sibille, Proc.
6th Int. Conf. on Superlattices, Microstruetures and Microdevices, August 4-7, 1992, Xi'an, China;
3 4 5 6 7
M. Hadjazi, A. Sibille, P.J. Palmier, and Mollot, Electronics Lett. 27, 1101 (1992). H.T. Grahn, K. yon Klitzing, K. Ploog, and G.H. D5hler, Phys. Rev. B43, 12094 (1991). J.A. Cooper and K.K. Thornber, IEEE Electron Device Lett. EDL-6, 50 (1985). A.A. Ignatov, E.P. Dodin, and V.I. Shaskin, Mod. Phys. Lett, B 5, 1087 (1991). X.L. Lei and C.S. Ting, Phys. Rev. B32, 1112 (1985). X.L. Lei, J. Phys. C 18, L593 (1985).
8 X.L. Lei and N.J.M. Horing, Int. J. Mod. Phys. B 6, 805 (1992). 9 X.L. Lei, Phys. Star. Sol. (b), 170, 519 (1992). 10 X.L. Lei, N.J.M. Horing, and H.L. Cui, J. Phys. Condens. Matter 4, 9375 (1992). 11 X.L. Lei, N.J. Horing and H.L. Cui, Phys. Rev. Lett. 66, 3277 (1991). 12 X.L. Lei and I.C. da Cunha Lima, J. Appl. Phys. 71, 5517 (1992). 13 X.L. Lei, N.J. Horing, H.L. Cui, and K.K. Thornbet, submitted to Phys. Rev. B. 14 X.L. Lei, Proc. 21th Int. Conf. Semiconductor
Physics, August 10-15, 1992, Beijing, China. 15 X.L. Lei and X.F. Wang, Proc. 6th Int. Conf. Su-
perlattices, Microstructures gJ Microdevices, Aug. 1992, Xi'an, China. 16 X.L. Lei, J.L. Birman, and C.S. Ting, J. Appl. Phys. 58, 2270 (1985).