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Current–voltage characteristics in periodically-driven miniband superlattices
Materials Science and Engineering B75 (2000) 126 – 129 www.elsevier.com/locate/mseb
Current–voltage characteristics in periodically-driven miniband superlattices J.C. Cao *, X.L. Lei State Key Laboratory of Functional Materials for Informatics, Shanghai Institute of Metallurgy, Chinese Academy of Sciences, 865 Changning Road, Shanghai 200050, PR China
Abstract We have theoretically investigated current–voltage characteristics of periodically-driven GaAs-based miniband superlattice at room temperature by using the time-dependent hydrodynamic balance equations. With the amplitude of the ac signal varying in the parameter space, the superlattice is found to exhibit various types of spatiotemporal behavior and current–voltage characteristics. In the case of a large-amplitude ac signal, the solution is synchronized with the ac frequency, and the time-averaged current–voltage curve exhibits microwave-radiation-induced dc current suppression. A good agreement is obtained between the calculated dc current and the recent experiment for a GaAs/AlAs miniband superlattice. © 2000 Elsevier Science S.A. All rights reserved. Keywords: Miniband superlattice; Hydrodynamic balance equation; Current – voltage characteristics
1. Introduction In recent years, the transport properties of semiconductor superlattices driven by dc and ac biases have attracted many experimental and theoretical studies in the literature [1– 9]. Under the influence of external periodic bias, doped semiconductor superlattices exhibit many interesting phenomena related to self-sustaining periodic current oscillation. It is reported experimentally that self-sustaining current oscillation was detected in silicon-doped miniband superlattices biased in the negative differential mobility (NDM) [5,6]. On the other hand, experiments [8] confirm that the formation of domains and the dc current are strongly suppressed when an intense high-frequency field is applied to a doped miniband superlattice. Recent experiments [3,4] on sequential resonant tunneling superlattices show that the presence of a microwave signal can produce an alternative mode of operation and lead to a quite complex spatiotemporal behaviour. Theoretically, the above-mentioned spatiotemporal phenomena and current – voltage characteristics can be * Corresponding author. Present address: Advanced Devices Group, Institute for Microstructural Sciences, National Research Council Canada, 1200 Montreal Road, Otawa, Ontario K1A 0R6. Fax: + 1-613-9900202.
simulated as the response of the electron gas in superlattice system to an external periodic bias. Recently, chaotic dynamics in sequential resonant tunneling superlattice is theoretically investigated within the discrete drift-diffusion equations [1,2], in which the mechanism for negative differential velocity in the field–velocity curve originates from sequential resonant tunneling between adjacent quantum wells. In a recent paper [7], the hydrodynamic balance equations [10,11], which are fully three-dimensional in nature and take microscopic impurity and phonon scattering into account, are applied to the analyses of spatiotemporal dynamics in a GaAs-based miniband superlattice subject to a dc bias voltage. In this paper we present a study of nonlinear dynamics of miniband superlattice by applying an additional ac signal to the dc voltage bias at which the superlattice exhibits self-sustaining current oscillations. In contrast with those studies for sequential resonant tunneling superlattices [1,2], the NDM in the present case is due to the negative effective mass of Bloch electrons in the miniband.
2. Hydrodynamic balance equations Considering an N-doped GaAs-based superlattice sandwiched between two heavily-doped N+-GaAs con-
0921-5107/00/$ - see front matter © 2000 Elsevier Science S.A. All rights reserved. PII: S 0 9 2 1 - 5 1 0 7 ( 0 0 ) 0 0 3 4 6 - 9
J.C. Cao, X.L. Lei / Materials Science and Engineering B75 (2000) 126–129
tact layers, the electrons in superlattice travel along the growth axis through the lowest miniband and in the lateral xy plane they move freely with a transverse energy ok = k 2 /2m with k =(kx, ky ) (m being the effective mass of the electron). The electron energy dispersion of the system can be written as the sum of the transverse energy and a tight-binding miniband energy related to the longitudinal motion, o(k , kz )=ok + D/2[1− cos(kzd)], in which d is the superlattice period, −p/d B kz 5p/d, and D is the miniband width. When the electric field E is applied along the symmetrical axis (the z-axis), the electron drift is also in this direction. The inverse-effective-mass tensor and the velocity dyadic tensor have only non-vanishing diagonal elements, the zz-components of which are denoted as 1/m *z and Bz, respectively. The z-component of the energy flow vector density is defined as Sz. We can express current density as j = − en6 and electric field as E = − (f/(x with f the electrostatic potential and write the hydrodynamic balance equations [10] in the following form, (n 1 (j = (t e (z
j −n =e
(1)
− y (( j/n) A (t
−y ((nBz ) j (j 1 (f − 2 × − en A (z e n (z m*z (z
((no) ((nSz ) (f =− −j − nW (t (z (z
(2) (3)
in which, e is the electron charge, n is the electron density, y is the average velocity, and o is the average energy of the electrons. The average acceleration A and energy-loss rate W, share the same expressions as those given in Ref. [11]. The electrostatic potential f is related to the charge density by the Poisson equation ( 2f e = (n−N) (z 2 os
(4)
with os the dielectric constant of the bulk material and N = N(z) the profile of the doping concentration. We can determine the unknown variables f(z, t), n(z, t), and Te(z, t), then all other useful quantities such as E(z, t), y(z, t), and j(z, t) by numerically solving Eqs. (1)–(4). The total current density J(t) is the sum of the conduction current density and the displacement current density. When a dc bias voltage Vdc and an ac bias voltage of frequency fac and amplitude Vac, V(t)= Vdc + Vac sin(2pfact) (5) are applied to the contacts, the boundary conditions are as follows, n(zL, t)=N(zL),
n(zR, t) =N(zR)
(6)
f(zR, t)− f(zL, t)=
kBT N(zR) + V(t) ln e N(zL)
Te(zL, t)= Te(zR, t)= T
127
(7) (8)
in which T is the lattice temperature, zL, and zR stand for the cathode contact and the anode contact, respectively. To numerically solve Eqs. (1)–(4), we apply the generalized Scharfetter –Gummel finite-difference technique to discretize the current continuity Eq. (2) and use the Crank–Nicolson implicit scheme to discretize all time derivatives. The maximum relative error is set to be 1 × 10 − 4 for all unknown variables f(z, t), n(z, t), and Te(z, t).
3. Self-sustaining current oscillations under dc bias In this section, we consider the case of a pure dc bias (Vac = 0). The parameters of the N+NN+ superlattice structure are as follows. The superlattice length is L= 0.55 mm with doping concentration N= 7×1016 cm − 3, the contact length is L/2 with doping concentration N+ = 2× 1018 cm − 3, miniband width D = 17 meV, period d=5.1 nm, and well width a=3.1 nm. The lattice temperature is T=300 K. The low-temperature (4.2 K) linear dc mobility is assumed to be m0 = 0.26 m2 V − 1 s − 1 for this superlattice system. All the material constants used in the calculations are typical values of bulk GaAs [11]. We obtain the spatial and temporal evolution of the potential, electron density, and electron temperature by numerically solving the hydrodynamic equations. Depending on the applied voltage Vdc, damped or undamped current self-oscillation is obtained. If the dc voltage bias is in the NDM regime, a small doping notch made in the superlattice structure can cause the growth of an electron accumulation layer and lead to current oscillation. The NDM in superlattice, which comes from the strong nonparabolicity of the miniband and related Bragg scattering, is different from the NDM in sequential resonant tunneling superlattice. Calculation indicates that the threshold voltage for undamped current oscillation, i.e. the self-sustaining current oscillation, is about Vth = 0.69 V for the abovementioned superlattice system. Once Vdc \Vth, after some transient period, traveling high-field domains are formed in the superlattice system. Fig. 1 shows the spatiotemporal electric field domain E(z, t) (3D plot) and the corresponding current density J(t) as functions of time (continuous line) at T= 300 K and Vdc =0.85 V. The electric field domain with a maximum of about 45 kV cm − 1 is periodically propagating from the cathode end, where it is formed, to the anode end, where it disappears, with a self-sustaining oscillating frequency f0 = 8.78 GHz (i.e. a period T0 #113.9 ps), resulting in an oscillating current.
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4. Synchronized oscillation and current suppression under dc and ac biases
Fig. 1. Spatiotemporal electric field domain E(z, t) (3D plot) and the corresponding time-dependent current density J(t) (continuous line) at T= 300 K and Vdc = 0.85 V.
Now, we deal with the miniband superlattice system driven by both dc and ac biases. Under the influence of a periodically-driven bias, the miniband superlattice is considered to be a nonlinear dynamic system with Vdc, Vac and fac as controlling parameters. For relatively large amplitude of the ac signal we could expect that the ac signal possibly plays a major role. By performing a large number of calculations, we have determined how the solutions vary with the controlling parameters and confirmed the above statement in that for a fixed Vdc and a fixed fac, larger Vac usually leads to a periodic solution synchronized with the ac frequency fac. In Fig. 2 we present the current–voltage phase plots of the superlattice system corresponding to the application of the fixed dc bias Vdc = 0.85 V, fixed ac frequency fac = 16 GHz, and ac amplitudes Vac changing from 0.4 to 1.2 V with a step of 0.2 V, respectively. It is found that, for each amplitude Vac the current density is a periodical function whose actual frequency fs always coincides with the ac frequency fac. The phasespace orbits appear as simple closed loops for each amplitude Vac. To directly compare our calculated results with the experiment, we define the time-averaged current density Jav by integrating J(t) over one oscillating period Ts (= 1/fs), Jav = 1/Ts
&
t1 + T s
J(t) dt
t1
Fig. 2. Current – voltage phase plots at the fixed dc voltage Vdc =0.85 V, the fixed ac frequency fac = 16 GHz, and five different ac amplitudes Vac =0.4, 0.6, 0.8, 1.0, and 1.2 V, respectively.
in which t1 is a time after the transient response dies out. In Fig. 3 we show current–voltage curves at the fixed ac amplitude Vac = 0.95 V and four different ac frequencies fac = 45, 78, 180, and 320 GHz, respectively. It is seen that at a fixed ac amplitude, the ac signal of lower frequency exhibits a stronger effect in suppressing the dc current of the superlattice system. The time-average dc current is obtained by I Jav × S (here, S= 2.2 mm2, is the effective superlattice cross area). For a fixed ac frequency fac = 78 GHz and different ac amplitudes Vac = 0, 0.74, 0.94, and 1.27 V, we have calculated and shown in Fig. 4 the dc current (lines) I versus the dc biases ranging from 0 to 0.9 V for the above-mentioned superlattice system at T =300 K. For comparison, the experimental data (points) of Schomburg et al. [8] for four power levels P/P0 =0, 0.02, 0.04, and 0.08 of microwave signal are also shown in Fig. 4. The power level is approximately related to the ac amplitude by the relation Vac = V max ac P/P0
Fig. 3. Time-averaged dc current densities are shown as functions of the dc biases at Vac =0.94 V and four different fac = 45, 78, 180, and 320 GHz, respectively.
with V max ac # 4.9 V. The solid line in Fig. 4 represents the calculated dc current versus dc bias without a microwave signal (Vac = 0), which obviously shows the
J.C. Cao, X.L. Lei / Materials Science and Engineering B75 (2000) 126–129
129
equations for miniband semiconductor superlattices we have studied current–voltage characteristics of doped GaAs-based miniband superlattice driven by a periodic voltage in the form of Vdc + Vac sin(2pfact). It is found that when the applied dc voltage is greater than a threshold value there exists periodically-oscillating current related to traveling high-electric-field domains in the superlattice. In contrast, when the superlattice system is driven by periodical voltage, the spatiotemporal solutions exhibit a very rich structure with changing control parameters. At larger amplitude of ac signal, the frequencylocked 1:1 solution is maintained. The calculated time-averaged dc currents show a good agreement with the available experimental data. Fig. 4. Time-averaged dc current (lines) are shown as functions of the dc biases at fac =78 GHz and four different Vac = 0, 0.74, 0.94, and 1.27 V, which, respectively correspond to four different power levels P/P0 =0, 0.02, 0.04, and 0.08 adopted by the experiments.
field-domain-related negative differential conductance (NDC). In the presence of microwave signals (broken lines), the dc currents are strongly reduced in comparison with that of Vac =0 and NDCs completely disappear when Vac =0.74, 0.94, and 1.27 V. Quantitative agreements are obtained between our calculated results (lines) and the experimental data (points) [8]. Fig. 4 also shows that, for the fixed ac frequency fac and the fixed dc bias Vdc, the dc current decreases with increasing ac amplitude Vac. The reason for ac-field-induced dc suppression is easily understood from the recently-developed balance-equation formulation for high-frequency field transport [12 – 14]. The effect of a high-frequency ac field on dc transport shows up through the Bessel function factor with an argument directly proportional to Eac/v 2 (see, for example Eqs. 31, 32, 34 and 35 of Refs. [12–14]), where Eac is the amplitude and v= 2pfac is the angular frequency of the ac field.
5. Summary Based on the time-dependent hydrodynamic balance
.
Acknowledgements The Shanghai Foundation for Young Scientists (QiMing Stars) and the National Natural Science Foundation of China are gratefully acknowledged for support of this work.
References [1] O.M. Bulashenko, L.L. Bonilla, Phys. Rev. B52 (1995) 7849. [2] O.M. Bulashenko, M.J. Garcı´a, L.L. Bonilla, Phys. Rev. B53 (1996) 10008. [3] Y.H. Zhang, J. Kastrup, R. Klann, K.H. Ploog, H.T. Grahn, Phys. Rev. Lett. 77 (1996) 3001. [4] K.J. Luo, H.T. Grahn, K.H. Ploog, L.L. Bonilla, Phys. Rev. Lett. 81 (1998) 1290. [5] H. Le Person, C. Mint, L. Boni, J.F. Palmier, F. Mollot, Appl. Phys. Lett. 60 (1992) 2397. [6] K. Hofbeck, et al., Phys. Lett. 218A (1996) 349. [7] J.C. Cao, X.L. Lei, Phys. Rev. B59 (1999) 2199. [8] E. Schomburg, et al., Appl. Phys. Lett. 68 (1996) 1096. [9] A.A. Ignatov, E. Schomburg, J. Grenzer, K.F. Renk, E.P. Dodin, Z. Phys. B 98 (1995) 187. [10] X.L. Lei, Phys. Stat. Solid. (b) 192 (1995) K1. [11] X.L. Lei, N.J.M. Horing, H.L. Cui, Phys. Rev. Lett. 66 (1991) 3277. [12] X.L. Lei, J. Appl. Phys. 84 (1998) 1396. [13] X.L. Lei, J. Phys. Condens. Matter 10 (1998) 3201. [14] X.L. Lei, H.L. Cui, Eur. Phys. J. B 4 (1998) 513.
Current–voltage characteristics in periodically-driven miniband superlattices J.C. Cao *, X.L. Lei State Key Laboratory of Functional Materials for Informatics, Shanghai Institute of Metallurgy, Chinese Academy of Sciences, 865 Changning Road, Shanghai 200050, PR China
Abstract We have theoretically investigated current–voltage characteristics of periodically-driven GaAs-based miniband superlattice at room temperature by using the time-dependent hydrodynamic balance equations. With the amplitude of the ac signal varying in the parameter space, the superlattice is found to exhibit various types of spatiotemporal behavior and current–voltage characteristics. In the case of a large-amplitude ac signal, the solution is synchronized with the ac frequency, and the time-averaged current–voltage curve exhibits microwave-radiation-induced dc current suppression. A good agreement is obtained between the calculated dc current and the recent experiment for a GaAs/AlAs miniband superlattice. © 2000 Elsevier Science S.A. All rights reserved. Keywords: Miniband superlattice; Hydrodynamic balance equation; Current – voltage characteristics
1. Introduction In recent years, the transport properties of semiconductor superlattices driven by dc and ac biases have attracted many experimental and theoretical studies in the literature [1– 9]. Under the influence of external periodic bias, doped semiconductor superlattices exhibit many interesting phenomena related to self-sustaining periodic current oscillation. It is reported experimentally that self-sustaining current oscillation was detected in silicon-doped miniband superlattices biased in the negative differential mobility (NDM) [5,6]. On the other hand, experiments [8] confirm that the formation of domains and the dc current are strongly suppressed when an intense high-frequency field is applied to a doped miniband superlattice. Recent experiments [3,4] on sequential resonant tunneling superlattices show that the presence of a microwave signal can produce an alternative mode of operation and lead to a quite complex spatiotemporal behaviour. Theoretically, the above-mentioned spatiotemporal phenomena and current – voltage characteristics can be * Corresponding author. Present address: Advanced Devices Group, Institute for Microstructural Sciences, National Research Council Canada, 1200 Montreal Road, Otawa, Ontario K1A 0R6. Fax: + 1-613-9900202.
simulated as the response of the electron gas in superlattice system to an external periodic bias. Recently, chaotic dynamics in sequential resonant tunneling superlattice is theoretically investigated within the discrete drift-diffusion equations [1,2], in which the mechanism for negative differential velocity in the field–velocity curve originates from sequential resonant tunneling between adjacent quantum wells. In a recent paper [7], the hydrodynamic balance equations [10,11], which are fully three-dimensional in nature and take microscopic impurity and phonon scattering into account, are applied to the analyses of spatiotemporal dynamics in a GaAs-based miniband superlattice subject to a dc bias voltage. In this paper we present a study of nonlinear dynamics of miniband superlattice by applying an additional ac signal to the dc voltage bias at which the superlattice exhibits self-sustaining current oscillations. In contrast with those studies for sequential resonant tunneling superlattices [1,2], the NDM in the present case is due to the negative effective mass of Bloch electrons in the miniband.
2. Hydrodynamic balance equations Considering an N-doped GaAs-based superlattice sandwiched between two heavily-doped N+-GaAs con-
0921-5107/00/$ - see front matter © 2000 Elsevier Science S.A. All rights reserved. PII: S 0 9 2 1 - 5 1 0 7 ( 0 0 ) 0 0 3 4 6 - 9
J.C. Cao, X.L. Lei / Materials Science and Engineering B75 (2000) 126–129
tact layers, the electrons in superlattice travel along the growth axis through the lowest miniband and in the lateral xy plane they move freely with a transverse energy ok = k 2 /2m with k =(kx, ky ) (m being the effective mass of the electron). The electron energy dispersion of the system can be written as the sum of the transverse energy and a tight-binding miniband energy related to the longitudinal motion, o(k , kz )=ok + D/2[1− cos(kzd)], in which d is the superlattice period, −p/d B kz 5p/d, and D is the miniband width. When the electric field E is applied along the symmetrical axis (the z-axis), the electron drift is also in this direction. The inverse-effective-mass tensor and the velocity dyadic tensor have only non-vanishing diagonal elements, the zz-components of which are denoted as 1/m *z and Bz, respectively. The z-component of the energy flow vector density is defined as Sz. We can express current density as j = − en6 and electric field as E = − (f/(x with f the electrostatic potential and write the hydrodynamic balance equations [10] in the following form, (n 1 (j = (t e (z
j −n =e
(1)
− y (( j/n) A (t
−y ((nBz ) j (j 1 (f − 2 × − en A (z e n (z m*z (z
((no) ((nSz ) (f =− −j − nW (t (z (z
(2) (3)
in which, e is the electron charge, n is the electron density, y is the average velocity, and o is the average energy of the electrons. The average acceleration A and energy-loss rate W, share the same expressions as those given in Ref. [11]. The electrostatic potential f is related to the charge density by the Poisson equation ( 2f e = (n−N) (z 2 os
(4)
with os the dielectric constant of the bulk material and N = N(z) the profile of the doping concentration. We can determine the unknown variables f(z, t), n(z, t), and Te(z, t), then all other useful quantities such as E(z, t), y(z, t), and j(z, t) by numerically solving Eqs. (1)–(4). The total current density J(t) is the sum of the conduction current density and the displacement current density. When a dc bias voltage Vdc and an ac bias voltage of frequency fac and amplitude Vac, V(t)= Vdc + Vac sin(2pfact) (5) are applied to the contacts, the boundary conditions are as follows, n(zL, t)=N(zL),
n(zR, t) =N(zR)
(6)
f(zR, t)− f(zL, t)=
kBT N(zR) + V(t) ln e N(zL)
Te(zL, t)= Te(zR, t)= T
127
(7) (8)
in which T is the lattice temperature, zL, and zR stand for the cathode contact and the anode contact, respectively. To numerically solve Eqs. (1)–(4), we apply the generalized Scharfetter –Gummel finite-difference technique to discretize the current continuity Eq. (2) and use the Crank–Nicolson implicit scheme to discretize all time derivatives. The maximum relative error is set to be 1 × 10 − 4 for all unknown variables f(z, t), n(z, t), and Te(z, t).
3. Self-sustaining current oscillations under dc bias In this section, we consider the case of a pure dc bias (Vac = 0). The parameters of the N+NN+ superlattice structure are as follows. The superlattice length is L= 0.55 mm with doping concentration N= 7×1016 cm − 3, the contact length is L/2 with doping concentration N+ = 2× 1018 cm − 3, miniband width D = 17 meV, period d=5.1 nm, and well width a=3.1 nm. The lattice temperature is T=300 K. The low-temperature (4.2 K) linear dc mobility is assumed to be m0 = 0.26 m2 V − 1 s − 1 for this superlattice system. All the material constants used in the calculations are typical values of bulk GaAs [11]. We obtain the spatial and temporal evolution of the potential, electron density, and electron temperature by numerically solving the hydrodynamic equations. Depending on the applied voltage Vdc, damped or undamped current self-oscillation is obtained. If the dc voltage bias is in the NDM regime, a small doping notch made in the superlattice structure can cause the growth of an electron accumulation layer and lead to current oscillation. The NDM in superlattice, which comes from the strong nonparabolicity of the miniband and related Bragg scattering, is different from the NDM in sequential resonant tunneling superlattice. Calculation indicates that the threshold voltage for undamped current oscillation, i.e. the self-sustaining current oscillation, is about Vth = 0.69 V for the abovementioned superlattice system. Once Vdc \Vth, after some transient period, traveling high-field domains are formed in the superlattice system. Fig. 1 shows the spatiotemporal electric field domain E(z, t) (3D plot) and the corresponding current density J(t) as functions of time (continuous line) at T= 300 K and Vdc =0.85 V. The electric field domain with a maximum of about 45 kV cm − 1 is periodically propagating from the cathode end, where it is formed, to the anode end, where it disappears, with a self-sustaining oscillating frequency f0 = 8.78 GHz (i.e. a period T0 #113.9 ps), resulting in an oscillating current.
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4. Synchronized oscillation and current suppression under dc and ac biases
Fig. 1. Spatiotemporal electric field domain E(z, t) (3D plot) and the corresponding time-dependent current density J(t) (continuous line) at T= 300 K and Vdc = 0.85 V.
Now, we deal with the miniband superlattice system driven by both dc and ac biases. Under the influence of a periodically-driven bias, the miniband superlattice is considered to be a nonlinear dynamic system with Vdc, Vac and fac as controlling parameters. For relatively large amplitude of the ac signal we could expect that the ac signal possibly plays a major role. By performing a large number of calculations, we have determined how the solutions vary with the controlling parameters and confirmed the above statement in that for a fixed Vdc and a fixed fac, larger Vac usually leads to a periodic solution synchronized with the ac frequency fac. In Fig. 2 we present the current–voltage phase plots of the superlattice system corresponding to the application of the fixed dc bias Vdc = 0.85 V, fixed ac frequency fac = 16 GHz, and ac amplitudes Vac changing from 0.4 to 1.2 V with a step of 0.2 V, respectively. It is found that, for each amplitude Vac the current density is a periodical function whose actual frequency fs always coincides with the ac frequency fac. The phasespace orbits appear as simple closed loops for each amplitude Vac. To directly compare our calculated results with the experiment, we define the time-averaged current density Jav by integrating J(t) over one oscillating period Ts (= 1/fs), Jav = 1/Ts
&
t1 + T s
J(t) dt
t1
Fig. 2. Current – voltage phase plots at the fixed dc voltage Vdc =0.85 V, the fixed ac frequency fac = 16 GHz, and five different ac amplitudes Vac =0.4, 0.6, 0.8, 1.0, and 1.2 V, respectively.
in which t1 is a time after the transient response dies out. In Fig. 3 we show current–voltage curves at the fixed ac amplitude Vac = 0.95 V and four different ac frequencies fac = 45, 78, 180, and 320 GHz, respectively. It is seen that at a fixed ac amplitude, the ac signal of lower frequency exhibits a stronger effect in suppressing the dc current of the superlattice system. The time-average dc current is obtained by I Jav × S (here, S= 2.2 mm2, is the effective superlattice cross area). For a fixed ac frequency fac = 78 GHz and different ac amplitudes Vac = 0, 0.74, 0.94, and 1.27 V, we have calculated and shown in Fig. 4 the dc current (lines) I versus the dc biases ranging from 0 to 0.9 V for the above-mentioned superlattice system at T =300 K. For comparison, the experimental data (points) of Schomburg et al. [8] for four power levels P/P0 =0, 0.02, 0.04, and 0.08 of microwave signal are also shown in Fig. 4. The power level is approximately related to the ac amplitude by the relation Vac = V max ac P/P0
Fig. 3. Time-averaged dc current densities are shown as functions of the dc biases at Vac =0.94 V and four different fac = 45, 78, 180, and 320 GHz, respectively.
with V max ac # 4.9 V. The solid line in Fig. 4 represents the calculated dc current versus dc bias without a microwave signal (Vac = 0), which obviously shows the
J.C. Cao, X.L. Lei / Materials Science and Engineering B75 (2000) 126–129
129
equations for miniband semiconductor superlattices we have studied current–voltage characteristics of doped GaAs-based miniband superlattice driven by a periodic voltage in the form of Vdc + Vac sin(2pfact). It is found that when the applied dc voltage is greater than a threshold value there exists periodically-oscillating current related to traveling high-electric-field domains in the superlattice. In contrast, when the superlattice system is driven by periodical voltage, the spatiotemporal solutions exhibit a very rich structure with changing control parameters. At larger amplitude of ac signal, the frequencylocked 1:1 solution is maintained. The calculated time-averaged dc currents show a good agreement with the available experimental data. Fig. 4. Time-averaged dc current (lines) are shown as functions of the dc biases at fac =78 GHz and four different Vac = 0, 0.74, 0.94, and 1.27 V, which, respectively correspond to four different power levels P/P0 =0, 0.02, 0.04, and 0.08 adopted by the experiments.
field-domain-related negative differential conductance (NDC). In the presence of microwave signals (broken lines), the dc currents are strongly reduced in comparison with that of Vac =0 and NDCs completely disappear when Vac =0.74, 0.94, and 1.27 V. Quantitative agreements are obtained between our calculated results (lines) and the experimental data (points) [8]. Fig. 4 also shows that, for the fixed ac frequency fac and the fixed dc bias Vdc, the dc current decreases with increasing ac amplitude Vac. The reason for ac-field-induced dc suppression is easily understood from the recently-developed balance-equation formulation for high-frequency field transport [12 – 14]. The effect of a high-frequency ac field on dc transport shows up through the Bessel function factor with an argument directly proportional to Eac/v 2 (see, for example Eqs. 31, 32, 34 and 35 of Refs. [12–14]), where Eac is the amplitude and v= 2pfac is the angular frequency of the ac field.
5. Summary Based on the time-dependent hydrodynamic balance
.
Acknowledgements The Shanghai Foundation for Young Scientists (QiMing Stars) and the National Natural Science Foundation of China are gratefully acknowledged for support of this work.
References [1] O.M. Bulashenko, L.L. Bonilla, Phys. Rev. B52 (1995) 7849. [2] O.M. Bulashenko, M.J. Garcı´a, L.L. Bonilla, Phys. Rev. B53 (1996) 10008. [3] Y.H. Zhang, J. Kastrup, R. Klann, K.H. Ploog, H.T. Grahn, Phys. Rev. Lett. 77 (1996) 3001. [4] K.J. Luo, H.T. Grahn, K.H. Ploog, L.L. Bonilla, Phys. Rev. Lett. 81 (1998) 1290. [5] H. Le Person, C. Mint, L. Boni, J.F. Palmier, F. Mollot, Appl. Phys. Lett. 60 (1992) 2397. [6] K. Hofbeck, et al., Phys. Lett. 218A (1996) 349. [7] J.C. Cao, X.L. Lei, Phys. Rev. B59 (1999) 2199. [8] E. Schomburg, et al., Appl. Phys. Lett. 68 (1996) 1096. [9] A.A. Ignatov, E. Schomburg, J. Grenzer, K.F. Renk, E.P. Dodin, Z. Phys. B 98 (1995) 187. [10] X.L. Lei, Phys. Stat. Solid. (b) 192 (1995) K1. [11] X.L. Lei, N.J.M. Horing, H.L. Cui, Phys. Rev. Lett. 66 (1991) 3277. [12] X.L. Lei, J. Appl. Phys. 84 (1998) 1396. [13] X.L. Lei, J. Phys. Condens. Matter 10 (1998) 3201. [14] X.L. Lei, H.L. Cui, Eur. Phys. J. B 4 (1998) 513.