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Intrinsic bistability and magnetic-field effects in resonant tunneling systems
F.LSEYIER
Intrinsic
Physica A 207 (1994) 457-460
bistability
and magnetic-field effects in resonant tunneling systems
P. Orellanaa, E. Andab, F. Claroa “Facultad de Fisica, Pontificia Vniversidad Catdlica de Chile, VicuKa Mackenna 4860, Casilla 306, Santiago 22, Chile bIn.stituto de Fisica, Vniversidad Federal Fluminense, Outeiro de Sao Joao Batista sin, Niteroi, Rio de Janeiro, Brazil
Abstract The intrinsic b&ability of an asymmetric double barrier device in the presence of a magnetic field in the direction of the current flow is discussed. A simple tight binding
formalism with a nonlinear Hartree term that represents the electron-electron interaction is shown to produce the magnetic field-dependent hysteresis loops characteristic of these systems. The width of the resulting bistable region oscillates with magnetic field, as observed in experiment.
Resonant tunneling in double barrier heterostructures has been extensively studied in the last decade. The first observation of resonant tunneling was done by Chang, Esaki and Tsui in 1974 on heterostructures grown by molecular beam epitaxy. The Z-V characteristics of these systems peaks at voltages near the quasibound states of the potential, followed by a negative differential resistance region [l]. The double barrier heterostructure has also been reported to exhibit hysteresis in the I-Vcurves [2]. Recently Eaves et al. reported the observation of an enhancement of the intrinsic bistability region when a magnetic field is applied parallel to the current direction [3,4]. In this paper we discuss magnetotunneling through a double barrier heterostructure. At zero field the hysteresis effect is known to be associated with the accumulation of charge in the enclosed well as the current increases, an effect that is not symmetric in the direction of current change [5]. A magnetic field applied parallel to the growth axis quantizes the motion in the perpendicular plane changing drastically the density of states of the system. Using a simple model we study the effect of this redistribution of energy states and their dependence on the magnetic field. Our results are in good qualitative agreement with experiment. We consider the transport properties of a double barrier heterostructure joining 037%4371/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDZ 037%4371(93)E0564-U
458
P. Orellana et al. I Physica A 207 (1994) 457-460
two reservoirs. Its nonlinear response is studied by introducing a fully selfconsistent scheme to treat the potential acting on the carriers and the charge distribution along the sample. We model the system by the one-band hamiltonian
where the operator cl, creates an electron with spin u at site i, tij is the hopping matrix between sites i and j, li the site energy and lJ the electron-electron coupling constant. In the mean-field (Hartree) approximation, the eigenvalue equation for the hamiltonian (1) becomes
We have expanded the eigenfunction in a tight binding basis with coefficients ua and assumed nearest-neighbor coupling only (fij = t, lJTiU, = U). The nonlinear term represents the electron-electron interaction. The magnetic field quantizes the motion parallel to the interface and the site particle density is given at zero temperature by
Here k, is the component of the incoming wavevector in the direction of the current, E, is the Fermi level in the emitter contact, En = heBlm* is the energy of the nth Landau level, I, = (hleB)“* the magnetic length, and IZ runs over all Landau levels below E,. The applied bias V is assumed to fall linearly throughout the structure. At a given bias V the current density / can be obtained from
dk, kT,,(k,, V>>
(4)
where T,(k,, V) is the electron transmission probability obtained by solving eq. (2). Solving self-consistently eqs. (2)-(4) we thus obtain the current, potential profile and electron charge density as a function of the applied bias for each value of the magnetic field. Asymmetric double barriers have been found to be advantageous for observation of the intrinsic bistability in experiments [6]. This is expected since the instability is due to accumulation of charge between the barriers, which is enhanced by either a thicker or a higher second barrier. Here we present results for an undoped GaAs/AlGaAs double barrier structure at 0 K, with the emitter and collector barrier thickness of 3.5 nm and 6 nm, respectively, and a well thickness of 10 nm. The conduction band offset is set at 300meV. The buffer layers are modeled as being uniformly doped up to 3 nm from either barrier, and to give a neutralizing free carrier concentration of 2 X 1017cm-3 at the contracts.
P. Orellana et al. I Physica A 207 (1994) 457-460
459
In equilibrium and at B = 0 T, the Fermi level lies 19.2 meV above the asymptotic conduction band edge. The hopping constant was set at t = 2.16 eV and the Coulomb parameter U in eq. (2) was given the value 10 meV. Fig. 1 shows our I-V results for three values of the magnetic field, 0, 4 and 10 T. At all fields there are two peaks, a large one from the usual tunneling resonance in the space between the barriers, and a small one that reflects the presence of a quasibound state in the self-consistent potential well formed by the charge buildup just before the first barrier. In the figure solid lines are results when the bias is increased, and dashed lines are for decreasing bias. Note that the peaks are bistable, in accordance with the observations of Goldman et al. [2]. As is apparent from the figure, the main effect of the magnetic field is to broaden the bistable region, most noticeably in the large peak case, whose width grows from the negligible size at 0 T to about 7 meV at 10 T. This is due to an increase in the amount of charge buildup in between the barriers before it collapses and the charge flows out. The hump at 0.065 V for B = 4 T shows the contribution of three Landau levels, II = 0, 1 and 2. For the parameters chosen above B = 10 T the small peak in the current disappears. Fig. 2 shows the total accumulated charge (per unit area) between the barriers at the same values of the field as in Fig. 1. It is seen to follow a similar pattern as the current itself. The motion in the plane is quantized into discrete levels by the
Fig. 2. Fig. 1. I-V characteristic of our asymmetric and 10T. (See text for different lines.) Fig. 2. Voltage dependence of the integrated magnetic fields as in fig. 1.
double charge
barrier
at three
in the space
values between
of the magnetic the barriers
field: 0, 4
for the same
P. Orellana et al. I Physica A 207 (1994) 457-460
460
0055
” 0
1
““‘d”““““’ 2 3
4
5
5
7
8
9
10
B(T)
Fig. 3. Magnetic
field dependence
of the edges of the main resonance
hysteresis
region.
magnetic field, each with a degeneracy that grows linearly with B. The high degeneracy at a single energy and the ensuing structure in the density of states is the origin of the enhanced charge buildup. Fig. 3 shows the position of the edges of the bistable region. Oscillations in this figure correspond to tunneling channels moving out of the Fermi sea as the magnetic field is increased. Similar oscillations have been observed by Eaves et al. [7].
Acknowledgements
This work was supported in part by Fondecyt AndesIVitaelAntorchas, Grant 12021-10.
Grant 9210059 and Fundacion
References [l] [2] [3] [4]
L. Chang, L. Esaki and R. Tsu, Appl. Phys. Lett. 24 (1974) 593. V.J. Goldman, D.C. Tsui and J.E Cunningham, Phys. Rev. Lett. 58 (1987) 1256. A. Zaslavsky, D.C. Tsui, M. Santos and M. Shayegan, Phys. Rev. B 40 (1989) 9829. E.S. Alves, F.W. Sheard, G.A. Toombs, P.E. L. Eaves, M.L. Leadbeater, D.G. Hayes, Simmonds, M.S. Skolnick, M. Henini and O.H. Hughes, Solid State Electron. 32 (1989) 1101. [5] F.W. Sheard and G.A. Toombs, Appl. Phys. Lett. 52 (1988) 1408; W. Potz, J. Appl. Phys. 66 (1989) 245; T. Fiig and A.P. Jauho, Surf. Sci. 267 (1992) 392. [6] A. Zaslavsky, V.J. Goldman, D.C. Tsui and J.E. Cunningham Appl. Phys. Lett. 53 (1988) 1408. [7] M.L. Leadbeater and L. Eaves, Phys. Ser. T 35 (1991) 215.
Intrinsic
Physica A 207 (1994) 457-460
bistability
and magnetic-field effects in resonant tunneling systems
P. Orellanaa, E. Andab, F. Claroa “Facultad de Fisica, Pontificia Vniversidad Catdlica de Chile, VicuKa Mackenna 4860, Casilla 306, Santiago 22, Chile bIn.stituto de Fisica, Vniversidad Federal Fluminense, Outeiro de Sao Joao Batista sin, Niteroi, Rio de Janeiro, Brazil
Abstract The intrinsic b&ability of an asymmetric double barrier device in the presence of a magnetic field in the direction of the current flow is discussed. A simple tight binding
formalism with a nonlinear Hartree term that represents the electron-electron interaction is shown to produce the magnetic field-dependent hysteresis loops characteristic of these systems. The width of the resulting bistable region oscillates with magnetic field, as observed in experiment.
Resonant tunneling in double barrier heterostructures has been extensively studied in the last decade. The first observation of resonant tunneling was done by Chang, Esaki and Tsui in 1974 on heterostructures grown by molecular beam epitaxy. The Z-V characteristics of these systems peaks at voltages near the quasibound states of the potential, followed by a negative differential resistance region [l]. The double barrier heterostructure has also been reported to exhibit hysteresis in the I-Vcurves [2]. Recently Eaves et al. reported the observation of an enhancement of the intrinsic bistability region when a magnetic field is applied parallel to the current direction [3,4]. In this paper we discuss magnetotunneling through a double barrier heterostructure. At zero field the hysteresis effect is known to be associated with the accumulation of charge in the enclosed well as the current increases, an effect that is not symmetric in the direction of current change [5]. A magnetic field applied parallel to the growth axis quantizes the motion in the perpendicular plane changing drastically the density of states of the system. Using a simple model we study the effect of this redistribution of energy states and their dependence on the magnetic field. Our results are in good qualitative agreement with experiment. We consider the transport properties of a double barrier heterostructure joining 037%4371/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDZ 037%4371(93)E0564-U
458
P. Orellana et al. I Physica A 207 (1994) 457-460
two reservoirs. Its nonlinear response is studied by introducing a fully selfconsistent scheme to treat the potential acting on the carriers and the charge distribution along the sample. We model the system by the one-band hamiltonian
where the operator cl, creates an electron with spin u at site i, tij is the hopping matrix between sites i and j, li the site energy and lJ the electron-electron coupling constant. In the mean-field (Hartree) approximation, the eigenvalue equation for the hamiltonian (1) becomes
We have expanded the eigenfunction in a tight binding basis with coefficients ua and assumed nearest-neighbor coupling only (fij = t, lJTiU, = U). The nonlinear term represents the electron-electron interaction. The magnetic field quantizes the motion parallel to the interface and the site particle density is given at zero temperature by
Here k, is the component of the incoming wavevector in the direction of the current, E, is the Fermi level in the emitter contact, En = heBlm* is the energy of the nth Landau level, I, = (hleB)“* the magnetic length, and IZ runs over all Landau levels below E,. The applied bias V is assumed to fall linearly throughout the structure. At a given bias V the current density / can be obtained from
dk, kT,,(k,, V>>
(4)
where T,(k,, V) is the electron transmission probability obtained by solving eq. (2). Solving self-consistently eqs. (2)-(4) we thus obtain the current, potential profile and electron charge density as a function of the applied bias for each value of the magnetic field. Asymmetric double barriers have been found to be advantageous for observation of the intrinsic bistability in experiments [6]. This is expected since the instability is due to accumulation of charge between the barriers, which is enhanced by either a thicker or a higher second barrier. Here we present results for an undoped GaAs/AlGaAs double barrier structure at 0 K, with the emitter and collector barrier thickness of 3.5 nm and 6 nm, respectively, and a well thickness of 10 nm. The conduction band offset is set at 300meV. The buffer layers are modeled as being uniformly doped up to 3 nm from either barrier, and to give a neutralizing free carrier concentration of 2 X 1017cm-3 at the contracts.
P. Orellana et al. I Physica A 207 (1994) 457-460
459
In equilibrium and at B = 0 T, the Fermi level lies 19.2 meV above the asymptotic conduction band edge. The hopping constant was set at t = 2.16 eV and the Coulomb parameter U in eq. (2) was given the value 10 meV. Fig. 1 shows our I-V results for three values of the magnetic field, 0, 4 and 10 T. At all fields there are two peaks, a large one from the usual tunneling resonance in the space between the barriers, and a small one that reflects the presence of a quasibound state in the self-consistent potential well formed by the charge buildup just before the first barrier. In the figure solid lines are results when the bias is increased, and dashed lines are for decreasing bias. Note that the peaks are bistable, in accordance with the observations of Goldman et al. [2]. As is apparent from the figure, the main effect of the magnetic field is to broaden the bistable region, most noticeably in the large peak case, whose width grows from the negligible size at 0 T to about 7 meV at 10 T. This is due to an increase in the amount of charge buildup in between the barriers before it collapses and the charge flows out. The hump at 0.065 V for B = 4 T shows the contribution of three Landau levels, II = 0, 1 and 2. For the parameters chosen above B = 10 T the small peak in the current disappears. Fig. 2 shows the total accumulated charge (per unit area) between the barriers at the same values of the field as in Fig. 1. It is seen to follow a similar pattern as the current itself. The motion in the plane is quantized into discrete levels by the
Fig. 2. Fig. 1. I-V characteristic of our asymmetric and 10T. (See text for different lines.) Fig. 2. Voltage dependence of the integrated magnetic fields as in fig. 1.
double charge
barrier
at three
in the space
values between
of the magnetic the barriers
field: 0, 4
for the same
P. Orellana et al. I Physica A 207 (1994) 457-460
460
0055
” 0
1
““‘d”““““’ 2 3
4
5
5
7
8
9
10
B(T)
Fig. 3. Magnetic
field dependence
of the edges of the main resonance
hysteresis
region.
magnetic field, each with a degeneracy that grows linearly with B. The high degeneracy at a single energy and the ensuing structure in the density of states is the origin of the enhanced charge buildup. Fig. 3 shows the position of the edges of the bistable region. Oscillations in this figure correspond to tunneling channels moving out of the Fermi sea as the magnetic field is increased. Similar oscillations have been observed by Eaves et al. [7].
Acknowledgements
This work was supported in part by Fondecyt AndesIVitaelAntorchas, Grant 12021-10.
Grant 9210059 and Fundacion
References [l] [2] [3] [4]
L. Chang, L. Esaki and R. Tsu, Appl. Phys. Lett. 24 (1974) 593. V.J. Goldman, D.C. Tsui and J.E Cunningham, Phys. Rev. Lett. 58 (1987) 1256. A. Zaslavsky, D.C. Tsui, M. Santos and M. Shayegan, Phys. Rev. B 40 (1989) 9829. E.S. Alves, F.W. Sheard, G.A. Toombs, P.E. L. Eaves, M.L. Leadbeater, D.G. Hayes, Simmonds, M.S. Skolnick, M. Henini and O.H. Hughes, Solid State Electron. 32 (1989) 1101. [5] F.W. Sheard and G.A. Toombs, Appl. Phys. Lett. 52 (1988) 1408; W. Potz, J. Appl. Phys. 66 (1989) 245; T. Fiig and A.P. Jauho, Surf. Sci. 267 (1992) 392. [6] A. Zaslavsky, V.J. Goldman, D.C. Tsui and J.E. Cunningham Appl. Phys. Lett. 53 (1988) 1408. [7] M.L. Leadbeater and L. Eaves, Phys. Ser. T 35 (1991) 215.