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ac current in nonequilibrium mesoscopic systems
Computer Physics Communications 142 (2001) 436–441 www.elsevier.com/locate/cpc
ac current in nonequilibrium mesoscopic systems Jun-ichiro Ohe, Kousuke Yakubo ∗ Department of Applied Physics, Hokkaido University, Sapporo 060-8628, Japan
Abstract A transfer-matrix method to study ac transport in harmonically driven mesoscopic electron systems has been developed. This method makes it possible to calculate, without consuming a large amount of computing time and memory space, nonlinear ac currents in nonequilibrium systems by means of transmission and reflection amplitudes of electrons through sideband states produced by the oscillating potential. We also discuss a way to take a thermal average of the ac current at a finite temperature under a finite dc-bias voltage. Simple applications of this method exhibit the efficiency of this numerical technique. 2001 Elsevier Science B.V. All rights reserved. PACS: 72.10.-d; 73.23.-b; 73.23.Ad
1. Introduction Quantum transport in driven mesoscopic systems is an active research field due to advances in microfabrication technology [1]. Applying a time-varying potential with frequency ω to an electron through a resonant tunneling device, the electron can tunnel the system by absorbing/emitting the energy h¯ ω. This phenomenon, so-called photon-assisted tunneling (PAT), has been observed in a number of resonant systems, such as superconductorinsulator-superconductor junctions [2], superlatices [3], quantum dots [4,5], dual-gate field-effect transistors [6], and electron pumps [7]. In previous studies of PAT, dc transport of systems has been mainly examined. There also exist, however, ac currents with frequency ω, 2ω, 3ω, . . . in driven mesoscopic systems. Recent interests are shifting to ac transport in dynamical mesoscopic systems, though theoretical tools for these problems have not been enough developed yet. The Kubo formula [8] is widely used for calculating ac currents in quantum systems. This formula is not applicable for highly nonlinear transport such as resonant tunneling. Although the nonequilibrium Green’s function technique [9] can evaluate nonlinear transport of systems far from equilibrium, it is difficult to calculate quantitatively ac currents of complex systems. An efficient numerical method to calculate quantitatively the nonlinear ac current of a nonequilibrium quantum system with realistic parameter opens a new way to understanding quantum transport in dynamical mesoscopic systems. In this paper, we show a transfer-matrix method to calculate nonlinear ac currents induced in harmonically driven quantum systems. This method enables us to compute transmission and reflection amplitudes of electrons through dynamical mesoscopic systems without consuming a large amount of computing time and memory space [10], * Corresponding author.
E-mail address: [email protected] (K. Yakubo). 0010-4655/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 1 0 - 4 6 5 5 ( 0 1 ) 0 0 3 8 0 - 0
J.-i. Ohe, K. Yakubo / Computer Physics Communications 142 (2001) 436–441
437
and photo-induced ac currents from these amplitudes. We can obtain not only the ac current with frequency ω [j (ω)] but also higher harmonics ac currents j (2ω), j (3ω), . . . . It is found that the ac current depends on a spatial position of measurement because the ac current results from the interference between different sideband states with different wavenumbers. We apply this method to two types of driven mesoscopic systems. As the first application, the ac current induced an oscillating rectangular potential system is investigated. It is clarified that a resonant ac current appears at E = h¯ ω in this system, where E is the electron incident energy. We show that the ac current j (ω) is induced by a purely quantum mechanical effect in the sense that limω→0 j (ω) = 0, in contrast to the case of a system driven by a uniform electric field in which both classical and quantum ac currents appear. We also apply this method to a double-barrier resonant tunneling system driven coherently at the well region. It has been shown that the ac current j (ω) in this system exhibits PAT peaks and the magnitude of the thermally averaged ac current is as large as the PAT signal of the dc current. This paper is organized as follows: Section 2 gives a brief description of the transfer-matrix formalism and a way of taking the thermal average of the ac current. Numerical results of photoinduced ac currents in harmonically driven mesoscopic systems are presented in Section 3. Section 4 gives the concluding remarks.
2. Transfer-matrix formalism We consider, for simplicity, an one-dimensional mesoscopic system with a static potential Vdc (x) and a harmonically oscillating potential Vac (x) cos ωt. An electron in the system can be described by the Schrödinger equation, ∂ψ h¯ 2 ∂ 2 ih¯ (1) = − ∗ 2 + Vdc (x) + Vac (x) cos ωt ψ, ∂t 2m ∂x where m∗ is the electron effective mass. Hereafter, we assume that Vdc (x) asymptotically approaches finite values at x → ±∞ and the origin of the energy is chosen to be Vdc (−∞) = 0. In the transfer-matrix method, the system is divided into N segments. The width of these segments x is so small that the potentials Vdc (x) and Vac (x) can be regarded as constants in each segment. The wave function ψ l in the lth segment, namely the solution of Eq. (1) with constant Vdc and Vac , is written as ψl =
∞ l ik l x l Ap e p + Bpl e−ikp x p=−∞ ∞
×
q=−∞
Jq
Vacl exp −i(E + phω ¯ + q h¯ ω)t/h¯ , hω ¯
(2)
l + phω)/h, and V l and V l are values of V (x) where Jq is the qth-order Bessel function, kpl = 2m∗ (E − Vdc ¯ ¯ dc ac dc and Vac (x) in the lth segment, respectively. Suffixes p and q denote sideband indices. It should be noted that Eq. (2) l = 0. This can be easily confirmed gives a conventional solution of a time-independent Schrödinger equation if Vac by considering Jq (0) = δq0 and the boundary condition at the left-hand side of the system. From matching conditions between adjacent segments ψ l (x l , t) = ψ l+1 (x l , t) and ∂x ψ l (x l , t) = ∂x ψ l+1 (x l , t), we obtain the 1 l relating coefficients Al+1 transfer matrix T and Bpl+1 to Alp and Bpl , and AN p p and Bp can be calculated from l N l T [10]. Using calculated coefficients Ap , the wave function at a position x in the right-hand side of the system (far from scatter) is given by ikp x −i(E+p h¯ ω) ψ(x) = AN e , (3) pe p
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√ where kp = 2m∗ [E − Vdc (∞) − phω]/ h¯ . We can calculate the current density from the definition. From Eq. (3) ¯ and j = (ieh¯ /2m∗ )(ψ∂x ψ ∗ − ψ ∗ ∂x ψ), we have j= jdc (p) + jac (pω), (4) p
p
where jdc (p) =
h¯ kp
N
2 A , m∗ p
and jac (pω) =
(5)
N∗ i(kq+p −kq )x ipωt N∗ i(kq−p −kq )x −ipωt . ikq AN e + AN e q+p Aq e q−p Aq e
(6)
q
It should be noted, from Eq. (6), that jac (pω) depends on x due to the difference between wavenumbers of different sideband components. Next, we discuss the way of taking the thermal average of the ac current at a finite temperature and under a finite dc-bias voltage. The thermal average of the photoinduced ac current jac (pω) should be carefully taken, because jac (pω) arises from plural sideband components with different energies. Chemical potentials in the lefthand side and right-hand side reservoirs of the system are assumed to be µL and µR , respectively. The dc-bias voltage V is, then, given by (µL − µR )/e. The Fermi-distribution function of the left (right) reservoir is written as fL(R) (E) = 1/{exp[(E − µL(R))/kB T ] + 1}. We consider first the current due to electrons incident from the leftN∗ i(kp+q −kq )x eipωt ≡ j (p, q) hand side reservoir and transmitting to the right-hand side one. A term ikq AN p+q Aq e in Eq. (6), for example, gives finite contribution to the ac current only if the incident energy state in the left-hand side reservoir is occupied and both states with energies E + (p + q)h¯ ω and E + q h¯ ω in the right-hand side reservoir are unoccupied. Therefore, the thermal average of jac (pω) contributed from electrons incident from the left-hand side is given by
∞
JL→R (pω) =
q
dE D(E)fL (E)
0
× j (p, q) 1 − fR (E + q h¯ ω)fR (E + (q + p)h¯ ω) + j (−p, q) 1 − fR (E + q h¯ ω)fR (E + (q − p)h¯ ω) ,
(7)
where D(E) is the density of states of the reservoir. In contrast to dc currents, the ac current at a point in the right-hand side of the system is generated also by electrons incident from the right-hand side reservoir, which is similarly calculated to Eq. (7). Therefore, the total ac current is given by J (pω) = JL→R (pω) + JR→R (pω).
(8)
3. Applications to ac current in driven mesoscopic systems In this section, we apply the present transfer-matrix method to electron transport in two driven mesoscopic systems. We calculate, at first, ac current in an oscillating rectangular potential system. The time-varying potential is given by V1 |x| < L/2, Vac (x) = (9) 0 otherwise.
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Fig. 1. ac current density j (ω) of an oscillating rectangular potential system as a function of the incident electron energy. The frequency of the oscillating potential is set to be hω ¯ = 1.0 meV. The inset shows the frequency dependence of the ac current density j (ω) for the same system. The incident electron energy is fixed at 10 meV.
No static potential exists in the system. The energy of the time-varying potential h¯ ω is 1.0 meV and the amplitude V1 is varied with L under the condition of V1 L = 100 meV·Å. Fig. 1 shows the ac current density j (ω) as a function of the incident electron energy. It is found that the ac current density has a resonant peak at E = h¯ ω. The −1st sideband wavefunction A−1 eik−1 x e−i(E−h¯ ω)t /h¯ becomes a constant in space and time at E = h¯ ω. The ac current from the quantum interference between the 0th and the −1st sideband states is, then, produced by the dominant 0th sideband component itself. The oscillatory behavior of j (ω) for E > h¯ ω results from the following reason. As mentioned below Eq. (6), the ac current j (ω) at a fixed incident energy E oscillates as a function of x. For a weak time-varying potential, the period of this spatial oscillation is approximately 2π/(k1 − k−1 ) which becomes large with increasing the incident energy E. Therefore, j (ω) at a fixed x oscillates with E. The inset of Fig. 1 shows the frequency dependence of the ac current density of the same system. The incident electron energy is fixed at 10 meV. This shows that the ac current approaches zero in the limit of zero frequency. This means that the ac current density in this system is induced by a purely quantum effect. It should be noted that both classical and quantum ac currents appear when the system is driven by a uniform electric field. The second application is ac transport in a driven double-barrier resonant tunneling system. This is the simplest model of an irradiated quantum dot. The double-barrier potential is given by V0 for x0 − ξdc /2 |x| x0 + ξdc /2, (10) Vdc (x) = 0 otherwise. Here we chose parameters as V0 = 30 meV, ξdc = 10 nm, and x0 = 40 nm. A time-varying potential applied to the double-barrier system is given by V1 for ξac |x|, (11) Vac (x) = 0 otherwise, where V1 = 0.5 meV and ξac = 35 nm. The frequency of the time-varying potential is chosen to be h¯ ω = 1.0 meV. We examine the ac current density jac (ω) at energies around a resonant level ε0 = 11.2 meV of the system. Fig. 2 represents the result of jac (ω) in a gray scale plot as a function of an incident electron energy E and the distance x from the origin (the center of two barriers). At a fixed position, one can find distinct sideband peaks of jac (ω) at
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Fig. 2. ac current density j (ω) of a driven double-barrier system as a function of the incident electron energy and the observing position (distance from the center of the double barrier). Bright portions indicate large amplitudes of j (ω).
Fig. 3. The thermal average of the ac current under a fixed dc-bias voltage (V = 0.1 mV) and at a finite temperature (T = 0.6 K). The system is the same with that for Fig. 2. The abscissa represents the chemical potential of the left-hand side electron reservoir.
E = ε0 ± h¯ ω. In contrast to the dc current, the amplitude of jac (ω) spatially oscillates as shown in Fig. 2, which arises from the beat of the interference between different sideband states. The thermal average of jac (ω) under a fixed dc-bias voltage (V = 0.1 mV) is shown in Fig. 3. The solid line shows the ac current of frequency ω (h¯ ω = 1.0 meV) as a function of the chemical potential µL of the left-hand side reservoir. The chemical potential µR of the right-hand side reservoir is always less than µL by 0.1 meV. Here, the temperature is chosen to be 0.6 K and x = 1.1 µm. From this result, the first sideband peak (at µL = ε0 − h¯ ω) survives even at finite temperatures, while the −1st sideband peak (at µL = ε0 + h¯ ω) disappears. The ac current mainly stems from the interference between the 0th and the ±1st sideband components. However, the −1st sideband state is always occupied in the right-hand side reservoir because of eV < h¯ ω. Therefore, the ac
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current is suppressed at µL = ε0 + h¯ ω. Moreover, we should mention that the magnitude of the ac current is as large as that of the dc current as shown in Fig. 3. This implies that photoinduced ac current and its PAT signals can be observed in actual experiments.
4. Conclusions A novel transfer-matrix method to investigate photon-assisted ac transport in dynamical mesoscopic systems has been developed and applied to photoinduced ac currents of coherently driven electron systems. This method enables us to calculate quantitatively ac currents of nonequilibrium quantum systems within a short computing time. The thermal average of the ac current at a finite temperature and under a finite dc-bias voltage is also discussed. We calculate the ac current in an oscillating rectangular potential system. It is found that the resonant peak of the ac current appears at the electron incident energy equal to hω. ¯ The ac current in this system is induced by a purely quantum effect. We also calculate the ac current in a driven double-barrier system which is simplest model of an irradiated quantum dot. Results show that there exist distinct sideband signals (ac PAT signals) in the incident electron energy dependence of the photoinduced ac current, as well as dc PAT signals. The magnitude of the ac current depends on the position in the system. This is because the ac current results from the interference between different sideband states with different wavenumbers. Finally, we demonstrated that these sideband peaks of the ac current survives even at a finite temperature (T = 0.6 K) and the magnitude of the ac current is as large as that of the dc current. This implies that the photoinduced ac current and its PAT signals predicted in this paper can be observed in actual experiments.
Acknowledgements We are grateful with T. Nakayama for valuable discussion. Numerical calculations were partially performed on the facilities of Supercomputer Center, Institute for Solid State Physics, University of Tokyo. One of the authors (J.O.) was supported by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists.
References [1] M. Grifoni, P. Hänggi, Phys. Rep. 304 (1998) 229. [2] A.H. Dayem, R.J. Martin, Phys. Rev. Lett. 8 (1962) 246. [3] P.S.S. Gimaraes, B.J. Keay, J.P. Kaminski, S.J. Allen, P.F. Hopkins, A.C. Gossard, L.T. Florez, J.P. Harbison, Phys. Rev. Lett. 70 (1993) 3792. [4] L.P. Kouwenhoven, S. Jauhar, J. Orenstein, P.L. McEuen, Y. Nagamune, J. Motohisa, H. Sakaki, Phys. Rev. Lett. 73 (1994) 3443. [5] T.H. Oosterkamp, L.P. Kouwenhoven, A.E.A. Koolen, N.C. van der Vaat, C.J.P.M. Harmans, Phys. Rev. Lett. 78 (1997) 1536. [6] S. Verghese, R.A. Wyss, T. Schapers, Q. Hu, A. Foster, M.J. Rooks, Phys. Rev. B 52 (1995) 14 834. [7] M. Covington, M.W. Keller, R.L. Kautz, J.M. Martinis, Phys. Rev. Lett. 84 (2000) 5192. [8] R. Kubo, J. Phys. Soc. Jpn. 12 (1957) 570. [9] Y. Goldin, Y. Avishai, Phys. Rev. B 55 (1997) 16 359. [10] K. Yakubo, Phys. Rev. E 57 (1998) 3602.
ac current in nonequilibrium mesoscopic systems Jun-ichiro Ohe, Kousuke Yakubo ∗ Department of Applied Physics, Hokkaido University, Sapporo 060-8628, Japan
Abstract A transfer-matrix method to study ac transport in harmonically driven mesoscopic electron systems has been developed. This method makes it possible to calculate, without consuming a large amount of computing time and memory space, nonlinear ac currents in nonequilibrium systems by means of transmission and reflection amplitudes of electrons through sideband states produced by the oscillating potential. We also discuss a way to take a thermal average of the ac current at a finite temperature under a finite dc-bias voltage. Simple applications of this method exhibit the efficiency of this numerical technique. 2001 Elsevier Science B.V. All rights reserved. PACS: 72.10.-d; 73.23.-b; 73.23.Ad
1. Introduction Quantum transport in driven mesoscopic systems is an active research field due to advances in microfabrication technology [1]. Applying a time-varying potential with frequency ω to an electron through a resonant tunneling device, the electron can tunnel the system by absorbing/emitting the energy h¯ ω. This phenomenon, so-called photon-assisted tunneling (PAT), has been observed in a number of resonant systems, such as superconductorinsulator-superconductor junctions [2], superlatices [3], quantum dots [4,5], dual-gate field-effect transistors [6], and electron pumps [7]. In previous studies of PAT, dc transport of systems has been mainly examined. There also exist, however, ac currents with frequency ω, 2ω, 3ω, . . . in driven mesoscopic systems. Recent interests are shifting to ac transport in dynamical mesoscopic systems, though theoretical tools for these problems have not been enough developed yet. The Kubo formula [8] is widely used for calculating ac currents in quantum systems. This formula is not applicable for highly nonlinear transport such as resonant tunneling. Although the nonequilibrium Green’s function technique [9] can evaluate nonlinear transport of systems far from equilibrium, it is difficult to calculate quantitatively ac currents of complex systems. An efficient numerical method to calculate quantitatively the nonlinear ac current of a nonequilibrium quantum system with realistic parameter opens a new way to understanding quantum transport in dynamical mesoscopic systems. In this paper, we show a transfer-matrix method to calculate nonlinear ac currents induced in harmonically driven quantum systems. This method enables us to compute transmission and reflection amplitudes of electrons through dynamical mesoscopic systems without consuming a large amount of computing time and memory space [10], * Corresponding author.
E-mail address: [email protected] (K. Yakubo). 0010-4655/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 1 0 - 4 6 5 5 ( 0 1 ) 0 0 3 8 0 - 0
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437
and photo-induced ac currents from these amplitudes. We can obtain not only the ac current with frequency ω [j (ω)] but also higher harmonics ac currents j (2ω), j (3ω), . . . . It is found that the ac current depends on a spatial position of measurement because the ac current results from the interference between different sideband states with different wavenumbers. We apply this method to two types of driven mesoscopic systems. As the first application, the ac current induced an oscillating rectangular potential system is investigated. It is clarified that a resonant ac current appears at E = h¯ ω in this system, where E is the electron incident energy. We show that the ac current j (ω) is induced by a purely quantum mechanical effect in the sense that limω→0 j (ω) = 0, in contrast to the case of a system driven by a uniform electric field in which both classical and quantum ac currents appear. We also apply this method to a double-barrier resonant tunneling system driven coherently at the well region. It has been shown that the ac current j (ω) in this system exhibits PAT peaks and the magnitude of the thermally averaged ac current is as large as the PAT signal of the dc current. This paper is organized as follows: Section 2 gives a brief description of the transfer-matrix formalism and a way of taking the thermal average of the ac current. Numerical results of photoinduced ac currents in harmonically driven mesoscopic systems are presented in Section 3. Section 4 gives the concluding remarks.
2. Transfer-matrix formalism We consider, for simplicity, an one-dimensional mesoscopic system with a static potential Vdc (x) and a harmonically oscillating potential Vac (x) cos ωt. An electron in the system can be described by the Schrödinger equation, ∂ψ h¯ 2 ∂ 2 ih¯ (1) = − ∗ 2 + Vdc (x) + Vac (x) cos ωt ψ, ∂t 2m ∂x where m∗ is the electron effective mass. Hereafter, we assume that Vdc (x) asymptotically approaches finite values at x → ±∞ and the origin of the energy is chosen to be Vdc (−∞) = 0. In the transfer-matrix method, the system is divided into N segments. The width of these segments x is so small that the potentials Vdc (x) and Vac (x) can be regarded as constants in each segment. The wave function ψ l in the lth segment, namely the solution of Eq. (1) with constant Vdc and Vac , is written as ψl =
∞ l ik l x l Ap e p + Bpl e−ikp x p=−∞ ∞
×
q=−∞
Jq
Vacl exp −i(E + phω ¯ + q h¯ ω)t/h¯ , hω ¯
(2)
l + phω)/h, and V l and V l are values of V (x) where Jq is the qth-order Bessel function, kpl = 2m∗ (E − Vdc ¯ ¯ dc ac dc and Vac (x) in the lth segment, respectively. Suffixes p and q denote sideband indices. It should be noted that Eq. (2) l = 0. This can be easily confirmed gives a conventional solution of a time-independent Schrödinger equation if Vac by considering Jq (0) = δq0 and the boundary condition at the left-hand side of the system. From matching conditions between adjacent segments ψ l (x l , t) = ψ l+1 (x l , t) and ∂x ψ l (x l , t) = ∂x ψ l+1 (x l , t), we obtain the 1 l relating coefficients Al+1 transfer matrix T and Bpl+1 to Alp and Bpl , and AN p p and Bp can be calculated from l N l T [10]. Using calculated coefficients Ap , the wave function at a position x in the right-hand side of the system (far from scatter) is given by ikp x −i(E+p h¯ ω) ψ(x) = AN e , (3) pe p
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√ where kp = 2m∗ [E − Vdc (∞) − phω]/ h¯ . We can calculate the current density from the definition. From Eq. (3) ¯ and j = (ieh¯ /2m∗ )(ψ∂x ψ ∗ − ψ ∗ ∂x ψ), we have j= jdc (p) + jac (pω), (4) p
p
where jdc (p) =
h¯ kp
N
2 A , m∗ p
and jac (pω) =
(5)
N∗ i(kq+p −kq )x ipωt N∗ i(kq−p −kq )x −ipωt . ikq AN e + AN e q+p Aq e q−p Aq e
(6)
q
It should be noted, from Eq. (6), that jac (pω) depends on x due to the difference between wavenumbers of different sideband components. Next, we discuss the way of taking the thermal average of the ac current at a finite temperature and under a finite dc-bias voltage. The thermal average of the photoinduced ac current jac (pω) should be carefully taken, because jac (pω) arises from plural sideband components with different energies. Chemical potentials in the lefthand side and right-hand side reservoirs of the system are assumed to be µL and µR , respectively. The dc-bias voltage V is, then, given by (µL − µR )/e. The Fermi-distribution function of the left (right) reservoir is written as fL(R) (E) = 1/{exp[(E − µL(R))/kB T ] + 1}. We consider first the current due to electrons incident from the leftN∗ i(kp+q −kq )x eipωt ≡ j (p, q) hand side reservoir and transmitting to the right-hand side one. A term ikq AN p+q Aq e in Eq. (6), for example, gives finite contribution to the ac current only if the incident energy state in the left-hand side reservoir is occupied and both states with energies E + (p + q)h¯ ω and E + q h¯ ω in the right-hand side reservoir are unoccupied. Therefore, the thermal average of jac (pω) contributed from electrons incident from the left-hand side is given by
∞
JL→R (pω) =
q
dE D(E)fL (E)
0
× j (p, q) 1 − fR (E + q h¯ ω)fR (E + (q + p)h¯ ω) + j (−p, q) 1 − fR (E + q h¯ ω)fR (E + (q − p)h¯ ω) ,
(7)
where D(E) is the density of states of the reservoir. In contrast to dc currents, the ac current at a point in the right-hand side of the system is generated also by electrons incident from the right-hand side reservoir, which is similarly calculated to Eq. (7). Therefore, the total ac current is given by J (pω) = JL→R (pω) + JR→R (pω).
(8)
3. Applications to ac current in driven mesoscopic systems In this section, we apply the present transfer-matrix method to electron transport in two driven mesoscopic systems. We calculate, at first, ac current in an oscillating rectangular potential system. The time-varying potential is given by V1 |x| < L/2, Vac (x) = (9) 0 otherwise.
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Fig. 1. ac current density j (ω) of an oscillating rectangular potential system as a function of the incident electron energy. The frequency of the oscillating potential is set to be hω ¯ = 1.0 meV. The inset shows the frequency dependence of the ac current density j (ω) for the same system. The incident electron energy is fixed at 10 meV.
No static potential exists in the system. The energy of the time-varying potential h¯ ω is 1.0 meV and the amplitude V1 is varied with L under the condition of V1 L = 100 meV·Å. Fig. 1 shows the ac current density j (ω) as a function of the incident electron energy. It is found that the ac current density has a resonant peak at E = h¯ ω. The −1st sideband wavefunction A−1 eik−1 x e−i(E−h¯ ω)t /h¯ becomes a constant in space and time at E = h¯ ω. The ac current from the quantum interference between the 0th and the −1st sideband states is, then, produced by the dominant 0th sideband component itself. The oscillatory behavior of j (ω) for E > h¯ ω results from the following reason. As mentioned below Eq. (6), the ac current j (ω) at a fixed incident energy E oscillates as a function of x. For a weak time-varying potential, the period of this spatial oscillation is approximately 2π/(k1 − k−1 ) which becomes large with increasing the incident energy E. Therefore, j (ω) at a fixed x oscillates with E. The inset of Fig. 1 shows the frequency dependence of the ac current density of the same system. The incident electron energy is fixed at 10 meV. This shows that the ac current approaches zero in the limit of zero frequency. This means that the ac current density in this system is induced by a purely quantum effect. It should be noted that both classical and quantum ac currents appear when the system is driven by a uniform electric field. The second application is ac transport in a driven double-barrier resonant tunneling system. This is the simplest model of an irradiated quantum dot. The double-barrier potential is given by V0 for x0 − ξdc /2 |x| x0 + ξdc /2, (10) Vdc (x) = 0 otherwise. Here we chose parameters as V0 = 30 meV, ξdc = 10 nm, and x0 = 40 nm. A time-varying potential applied to the double-barrier system is given by V1 for ξac |x|, (11) Vac (x) = 0 otherwise, where V1 = 0.5 meV and ξac = 35 nm. The frequency of the time-varying potential is chosen to be h¯ ω = 1.0 meV. We examine the ac current density jac (ω) at energies around a resonant level ε0 = 11.2 meV of the system. Fig. 2 represents the result of jac (ω) in a gray scale plot as a function of an incident electron energy E and the distance x from the origin (the center of two barriers). At a fixed position, one can find distinct sideband peaks of jac (ω) at
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Fig. 2. ac current density j (ω) of a driven double-barrier system as a function of the incident electron energy and the observing position (distance from the center of the double barrier). Bright portions indicate large amplitudes of j (ω).
Fig. 3. The thermal average of the ac current under a fixed dc-bias voltage (V = 0.1 mV) and at a finite temperature (T = 0.6 K). The system is the same with that for Fig. 2. The abscissa represents the chemical potential of the left-hand side electron reservoir.
E = ε0 ± h¯ ω. In contrast to the dc current, the amplitude of jac (ω) spatially oscillates as shown in Fig. 2, which arises from the beat of the interference between different sideband states. The thermal average of jac (ω) under a fixed dc-bias voltage (V = 0.1 mV) is shown in Fig. 3. The solid line shows the ac current of frequency ω (h¯ ω = 1.0 meV) as a function of the chemical potential µL of the left-hand side reservoir. The chemical potential µR of the right-hand side reservoir is always less than µL by 0.1 meV. Here, the temperature is chosen to be 0.6 K and x = 1.1 µm. From this result, the first sideband peak (at µL = ε0 − h¯ ω) survives even at finite temperatures, while the −1st sideband peak (at µL = ε0 + h¯ ω) disappears. The ac current mainly stems from the interference between the 0th and the ±1st sideband components. However, the −1st sideband state is always occupied in the right-hand side reservoir because of eV < h¯ ω. Therefore, the ac
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current is suppressed at µL = ε0 + h¯ ω. Moreover, we should mention that the magnitude of the ac current is as large as that of the dc current as shown in Fig. 3. This implies that photoinduced ac current and its PAT signals can be observed in actual experiments.
4. Conclusions A novel transfer-matrix method to investigate photon-assisted ac transport in dynamical mesoscopic systems has been developed and applied to photoinduced ac currents of coherently driven electron systems. This method enables us to calculate quantitatively ac currents of nonequilibrium quantum systems within a short computing time. The thermal average of the ac current at a finite temperature and under a finite dc-bias voltage is also discussed. We calculate the ac current in an oscillating rectangular potential system. It is found that the resonant peak of the ac current appears at the electron incident energy equal to hω. ¯ The ac current in this system is induced by a purely quantum effect. We also calculate the ac current in a driven double-barrier system which is simplest model of an irradiated quantum dot. Results show that there exist distinct sideband signals (ac PAT signals) in the incident electron energy dependence of the photoinduced ac current, as well as dc PAT signals. The magnitude of the ac current depends on the position in the system. This is because the ac current results from the interference between different sideband states with different wavenumbers. Finally, we demonstrated that these sideband peaks of the ac current survives even at a finite temperature (T = 0.6 K) and the magnitude of the ac current is as large as that of the dc current. This implies that the photoinduced ac current and its PAT signals predicted in this paper can be observed in actual experiments.
Acknowledgements We are grateful with T. Nakayama for valuable discussion. Numerical calculations were partially performed on the facilities of Supercomputer Center, Institute for Solid State Physics, University of Tokyo. One of the authors (J.O.) was supported by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists.
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