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Interaction-induced electromagnetic properties of a single channel quantum wire with a tunnel junction
Microelectronic Engineering 47 (1999) 341-344
Interaction-induced electromagnetic wire with a tunnel junction.
properties of a single channel quantum
A. Fechnera, M. Sassettib and B. Kramera aI. Institut fiir Theoretische Physik, Universitit Jungiusstrafle 9, D-20355 Hamburg
Hamburg
bIstituto di Fisica di Ingegneria, INFM, Universitb di Genova Via Dodecaneso 33, I-16146 Genova
The time- and space-dependent non-linear current through a single channel quantum wire with a tunnel barrier subject to an ac-electric field is investigated. General properties of the current are derived. The non-Fermi liquid behavior of the harmonics and the corresponding emitted electromagnetic powers are studied.
The dynamical interplay between currents and electromagnetic fields on a microscopic scale belongs to one of the fundamental and still open problems of condensed matter physics. One of the main difficulties is to describe the ac-phenomena in a self-consistent way, especially in the presence of interaction and non-linear effects [l]. In this paper, we address this question for a particular system, namely a single channel quantum wire with a barrier, treating the interaction within the Luttinger liquid model (21. The presence of both, the barrier and the electron-electron interaction, causes a non-linear dc currentvoltage characteristic reflecting non-Fermi liquid behavior [3]. However, experimental evidence confirming this result has been obtained only by exploiting the mapping to the fractional quantum Hall edge states (41. Due to its non-linearity, the system should be also a paradigm for interaction-induced harmonic generation. Here, we provide quantitative results for these ac-effects. Deriving an exact expression which links the ac-current at an arbitrary point of the system to the current at the position of the barrier allows us to provide quantitative results for the electromagnetic radiation emitted by the time dependent spatial current distribution along the wire. We consider a wire containing an impurity at the site x = 0 and driven by an ac-longitudinal electric field. The total Hamiltonian is H = Ho + H,, + Hv, where Ho = ii &o w(Qb+(Q(k) contains the dispersion w(k) = ]rC]v(k) of the collective charge density excitations (CDE). The renormalized wave-number dependent velocity v(lc) = ‘u~(l + V(~)/~~~FZIF)~/~ (21~Fermi velocity) reflects the Fourier transform of the electronelectron interaction potential V(x). The barrier is represented by the phase field 8(x) and the time-dependent driving voltage V(x,t) is contained [2], Hb = ub 42@(o)], in the term HV = eJ_+,” dq+)V(q t ) . The electric field E(z, t) = -W(x,t)/ax is not 0167-9317/99/$ - see front matter PII: SO167-9317(99)00229-4
0
1999 Elsevier Science B.V. All rights reserved.
342
A. Fechner et al. ! Microelectronic
Engineering 47 (1999) 341-344
assumed to have any specific shape apart from J_‘,” E(z, t)dz = V(t) being the externally applied voltage. The particle current operator is defined via the continuity equation, j(z,t) = -e&(z,t). The task is to determine the space and time dependence of the current as a function of the ac-electric field. First, we evaluate the non-equilibrium averaged current I* at the position of the barrier. Using the imaginary-time formalism one can integrate over the collective modes at x # 0 which leads to an effective driving voltage [5] l&(t)
=
Tdx ] &‘E(x, t’)r(x, t - t’) , -W
(1)
-W
where r(x,t) is the Fourier transform of r(x,w) = ae(x, w)/ae(O, w) and ao(x,w) is the ac-conductivity of the pure wire. In general, the effective driving voltage (1) depends on the spatial shape of the electric field. In the dc-limit, it depends only on the integral of the electric field, the external voltage V. Second, by generalizing the average over the collective modes, one can link the current at an arbitrary point x to the current at the position of the barrier, Al(x, t) = 1 dt’r(x, i! - t’)AI(O, t’)
.
(2)
Here, Al(x, t) = I(x,t) - le(x,t), and I and IO are the currents through the wire with and without impurity, respectively. Relation (2) is independent of the spatial distribution of the electric field. We use it in order to determine the current as a function of x, which is a necessary ingredient for the evaluation of the generated electromagnetic radiation. In the following, we evaluate the space and time dependent current explicitly for zerorange interactions, represented by a constant velocity u = v(rC+ 0) G uF/g of the CDE, + $ I(X#) = eilzlw/u~b(w)
2 +O”
/
dx’E(x’) [eilz-4W/v _ eW+14)wl~]
.
(3)
For an interaction of non-zero range, the dispersion relation of the collective modes is non-linear. Then, w/v on the right hand side of (3) has to be replaced by q(w), the wave number of the excitation with frequency w. In the limit of a large barrier the current at the barrier can be expressed in terms of forward and backward tunneling rates proportional to the tunneling probability A2,
J0
r
lb(t) = eA2 mdTe_S(‘) sin[R( )] sin [it!
dt’V&(t’)]
.
(4
The real and imaginary parts of the one-electron propagator (p inverse temperature) S(r) = ;
log(1
+ 0; 72) + ; log [gsinh
(a)]
,
R(T)
= a
tan-‘(w,r),
(5)
A. Fechner et al. I Microelectronic
343
Engineering 47 (1999) 341-344
respectively, have been derived by assuming the usual cutoff for frequencies above wC. In the presence of a monochromatic field of frequency St one obtains I*(t) =
fj
(6)
In(fl)e-‘n*t.
n=-a3
The complex amplitude I,(n) = I,‘(Q) + iI; may be decomposed into a real and an imaginary part. While 1: describes the in-phase part of the current, I; represents the contribution phase-shifted by f7r/2. These terms can be evaluated using (4). They consist of superpositions of the static non-linear current evaluated at integer multiples of the frequency, mR, weighted by the squares of the Bessel functions, &(z). The argument of the latter, z z eV,E/liR, depends on the spatial range I and also the shape of the ac part of the electric field via Vex as defined in (1). It scales as 0-l for RZ < 2)or as CZ2 for SX >> w. Because the qualitative behavior of the results does not depend on 1, we consider the simple case of a zero-range electric field, corresponding to an ac-voltage which drops at the barrier V(z,t) = -sgn(z)(Va + VI cos(Qt))/2. The behavior of the induced dc-current (n = 0) has been discussed previously in terms of the linear photoconductance [S]. Here, we concentrate on the results for the amplitudes of the harmonics (n # 0). They depend crucially on the strength of the interaction. At intermediate interaction strengths (g M 0.7) harmonic generation is strongly enhanced. For realistic parameters of etched AlGaAs/GaAs quantum wires we find a second harmonic current I-2 of the order of few nA and almost independent of temperature as long as T < 1K. With increasing n, the amplitudes I,, decay very rapidly. The dependencies of the harmonic currents on V, and Vl show non-Fermi liquid behavior. For instance, at low frequency and for temperatures kBT/g < eVl or eV1 > k&?/g > eV0 the harmonic currents are dominated by In+which is of the form
I:+’e-4K vo; 9) a
(2)n
(!5!)2’g-n-1
or
o( (2)’
(Z!L)2’g-’
,
(7)
for VO>> VI or VO< VI, respectively (s E (1 + (-1)*/2)). The proportionality constants do not contain the frequency and/or the voltages. Thus, the power law behaviors of the harmonics with VI or VOdisplay the interaction. The radiation field is obtained by solving Maxwell’s equations with the charge currents (3) and (4) as source. In the far field region, at a distance r much larger than the length of the wire and than the electromagnetic radiation lenght, r >> 2wc/nR, (c light velocity), the Poynting vector has radial direction with 131 E S a rw2. The time average of the, emitted power dP, per solid angle, de, in the nth harmonic, is
dpn -=-dQ
’ 11n12 -’ EOC2n2 0 c
'sin2~
.
(8)
This is similar to the result for a classical antenna, except for the factor (v/c)~ < 1. It depletes the emitted power considerably. In the near field, the emitted power can be evaluated numerically. Figure 1 shows the time-averaged Poynting vector of the ,nth harmonic as a function of the distance 2 from the barrier along the wire for different value of the distance R perpendicular to the wire.
344
A. Fechner et al. I Microelectronic
Engineering 47 (1999) 341-344
0 z j I t
-2 -
-2 -4
J/
-4 -3
-2
-1
0
xnR/2m
1
2
3
-3
-2
-1
0
1
2
3
xnS1/2m
Figure 1: Components of time-averaged Poynting vector, (&) (solid line) and (Sv) (dashed line) at cp = 0 and c/v = lo3 in units Se = chn2n211,,12 - 103/47r2v2 for R = 0.27rv/nR (left) and in units 10-5Sc for R = 27rv/nR (right). From the results shown in the left figure, one can estimate the emitted power radiated perpendicular to the wire in an area of the width Ax x O.l7rv/(nn) centered at the barrier. One finds that the emitted power per unit angle is at least (c/v)~ bigger than the one obtained in the far field region. Therefore, we expect that detecting the harmonics will be only possible close to the tunnel barrier, in the near field region. This work is supported by the DFG via SFB 508 of Universit%t Hamburg, by the EU via TMR project FMRX CT980180, and by INFM via PRA97(QTMD).
REFERENCES 1. 2. 3. 4.
0. Keller, Phys. Rep. 268 (1996) 85. F. D. M. Haldane, J. Phys. Cl4 (1981) 2585. C. L. Kane, M. P. A. Fisher, Phys. Rev. B 46 (1992) 15233. F. P. Mill&an, C. P. Umbach, R. A. Webb, Sol. St. Commun. 97 (1996) 309. A. M. Chang, L. N. Pfeiffer, K. W. West, Phys. Rev. Lett. 77 (1996) 2538. 5. M. Sassetti, B. Kramer, Phys. Rev. B54 (1996) R5203. 6. M. Sassetti, U. Weiss, B. Kramer, Sol. St. Commun. 97 (1996) 605.
Interaction-induced electromagnetic wire with a tunnel junction.
properties of a single channel quantum
A. Fechnera, M. Sassettib and B. Kramera aI. Institut fiir Theoretische Physik, Universitit Jungiusstrafle 9, D-20355 Hamburg
Hamburg
bIstituto di Fisica di Ingegneria, INFM, Universitb di Genova Via Dodecaneso 33, I-16146 Genova
The time- and space-dependent non-linear current through a single channel quantum wire with a tunnel barrier subject to an ac-electric field is investigated. General properties of the current are derived. The non-Fermi liquid behavior of the harmonics and the corresponding emitted electromagnetic powers are studied.
The dynamical interplay between currents and electromagnetic fields on a microscopic scale belongs to one of the fundamental and still open problems of condensed matter physics. One of the main difficulties is to describe the ac-phenomena in a self-consistent way, especially in the presence of interaction and non-linear effects [l]. In this paper, we address this question for a particular system, namely a single channel quantum wire with a barrier, treating the interaction within the Luttinger liquid model (21. The presence of both, the barrier and the electron-electron interaction, causes a non-linear dc currentvoltage characteristic reflecting non-Fermi liquid behavior [3]. However, experimental evidence confirming this result has been obtained only by exploiting the mapping to the fractional quantum Hall edge states (41. Due to its non-linearity, the system should be also a paradigm for interaction-induced harmonic generation. Here, we provide quantitative results for these ac-effects. Deriving an exact expression which links the ac-current at an arbitrary point of the system to the current at the position of the barrier allows us to provide quantitative results for the electromagnetic radiation emitted by the time dependent spatial current distribution along the wire. We consider a wire containing an impurity at the site x = 0 and driven by an ac-longitudinal electric field. The total Hamiltonian is H = Ho + H,, + Hv, where Ho = ii &o w(Qb+(Q(k) contains the dispersion w(k) = ]rC]v(k) of the collective charge density excitations (CDE). The renormalized wave-number dependent velocity v(lc) = ‘u~(l + V(~)/~~~FZIF)~/~ (21~Fermi velocity) reflects the Fourier transform of the electronelectron interaction potential V(x). The barrier is represented by the phase field 8(x) and the time-dependent driving voltage V(x,t) is contained [2], Hb = ub 42@(o)], in the term HV = eJ_+,” dq+)V(q t ) . The electric field E(z, t) = -W(x,t)/ax is not 0167-9317/99/$ - see front matter PII: SO167-9317(99)00229-4
0
1999 Elsevier Science B.V. All rights reserved.
342
A. Fechner et al. ! Microelectronic
Engineering 47 (1999) 341-344
assumed to have any specific shape apart from J_‘,” E(z, t)dz = V(t) being the externally applied voltage. The particle current operator is defined via the continuity equation, j(z,t) = -e&(z,t). The task is to determine the space and time dependence of the current as a function of the ac-electric field. First, we evaluate the non-equilibrium averaged current I* at the position of the barrier. Using the imaginary-time formalism one can integrate over the collective modes at x # 0 which leads to an effective driving voltage [5] l&(t)
=
Tdx ] &‘E(x, t’)r(x, t - t’) , -W
(1)
-W
where r(x,t) is the Fourier transform of r(x,w) = ae(x, w)/ae(O, w) and ao(x,w) is the ac-conductivity of the pure wire. In general, the effective driving voltage (1) depends on the spatial shape of the electric field. In the dc-limit, it depends only on the integral of the electric field, the external voltage V. Second, by generalizing the average over the collective modes, one can link the current at an arbitrary point x to the current at the position of the barrier, Al(x, t) = 1 dt’r(x, i! - t’)AI(O, t’)
.
(2)
Here, Al(x, t) = I(x,t) - le(x,t), and I and IO are the currents through the wire with and without impurity, respectively. Relation (2) is independent of the spatial distribution of the electric field. We use it in order to determine the current as a function of x, which is a necessary ingredient for the evaluation of the generated electromagnetic radiation. In the following, we evaluate the space and time dependent current explicitly for zerorange interactions, represented by a constant velocity u = v(rC+ 0) G uF/g of the CDE, + $ I(X#) = eilzlw/u~b(w)
2 +O”
/
dx’E(x’) [eilz-4W/v _ eW+14)wl~]
.
(3)
For an interaction of non-zero range, the dispersion relation of the collective modes is non-linear. Then, w/v on the right hand side of (3) has to be replaced by q(w), the wave number of the excitation with frequency w. In the limit of a large barrier the current at the barrier can be expressed in terms of forward and backward tunneling rates proportional to the tunneling probability A2,
J0
r
lb(t) = eA2 mdTe_S(‘) sin[R( )] sin [it!
dt’V&(t’)]
.
(4
The real and imaginary parts of the one-electron propagator (p inverse temperature) S(r) = ;
log(1
+ 0; 72) + ; log [gsinh
(a)]
,
R(T)
= a
tan-‘(w,r),
(5)
A. Fechner et al. I Microelectronic
343
Engineering 47 (1999) 341-344
respectively, have been derived by assuming the usual cutoff for frequencies above wC. In the presence of a monochromatic field of frequency St one obtains I*(t) =
fj
(6)
In(fl)e-‘n*t.
n=-a3
The complex amplitude I,(n) = I,‘(Q) + iI; may be decomposed into a real and an imaginary part. While 1: describes the in-phase part of the current, I; represents the contribution phase-shifted by f7r/2. These terms can be evaluated using (4). They consist of superpositions of the static non-linear current evaluated at integer multiples of the frequency, mR, weighted by the squares of the Bessel functions, &(z). The argument of the latter, z z eV,E/liR, depends on the spatial range I and also the shape of the ac part of the electric field via Vex as defined in (1). It scales as 0-l for RZ < 2)or as CZ2 for SX >> w. Because the qualitative behavior of the results does not depend on 1, we consider the simple case of a zero-range electric field, corresponding to an ac-voltage which drops at the barrier V(z,t) = -sgn(z)(Va + VI cos(Qt))/2. The behavior of the induced dc-current (n = 0) has been discussed previously in terms of the linear photoconductance [S]. Here, we concentrate on the results for the amplitudes of the harmonics (n # 0). They depend crucially on the strength of the interaction. At intermediate interaction strengths (g M 0.7) harmonic generation is strongly enhanced. For realistic parameters of etched AlGaAs/GaAs quantum wires we find a second harmonic current I-2 of the order of few nA and almost independent of temperature as long as T < 1K. With increasing n, the amplitudes I,, decay very rapidly. The dependencies of the harmonic currents on V, and Vl show non-Fermi liquid behavior. For instance, at low frequency and for temperatures kBT/g < eVl or eV1 > k&?/g > eV0 the harmonic currents are dominated by In+which is of the form
I:+’e-4K vo; 9) a
(2)n
(!5!)2’g-n-1
or
o( (2)’
(Z!L)2’g-’
,
(7)
for VO>> VI or VO< VI, respectively (s E (1 + (-1)*/2)). The proportionality constants do not contain the frequency and/or the voltages. Thus, the power law behaviors of the harmonics with VI or VOdisplay the interaction. The radiation field is obtained by solving Maxwell’s equations with the charge currents (3) and (4) as source. In the far field region, at a distance r much larger than the length of the wire and than the electromagnetic radiation lenght, r >> 2wc/nR, (c light velocity), the Poynting vector has radial direction with 131 E S a rw2. The time average of the, emitted power dP, per solid angle, de, in the nth harmonic, is
dpn -=-dQ
’ 11n12 -’ EOC2n2 0 c
'sin2~
.
(8)
This is similar to the result for a classical antenna, except for the factor (v/c)~ < 1. It depletes the emitted power considerably. In the near field, the emitted power can be evaluated numerically. Figure 1 shows the time-averaged Poynting vector of the ,nth harmonic as a function of the distance 2 from the barrier along the wire for different value of the distance R perpendicular to the wire.
344
A. Fechner et al. I Microelectronic
Engineering 47 (1999) 341-344
0 z j I t
-2 -
-2 -4
J/
-4 -3
-2
-1
0
xnR/2m
1
2
3
-3
-2
-1
0
1
2
3
xnS1/2m
Figure 1: Components of time-averaged Poynting vector, (&) (solid line) and (Sv) (dashed line) at cp = 0 and c/v = lo3 in units Se = chn2n211,,12 - 103/47r2v2 for R = 0.27rv/nR (left) and in units 10-5Sc for R = 27rv/nR (right). From the results shown in the left figure, one can estimate the emitted power radiated perpendicular to the wire in an area of the width Ax x O.l7rv/(nn) centered at the barrier. One finds that the emitted power per unit angle is at least (c/v)~ bigger than the one obtained in the far field region. Therefore, we expect that detecting the harmonics will be only possible close to the tunnel barrier, in the near field region. This work is supported by the DFG via SFB 508 of Universit%t Hamburg, by the EU via TMR project FMRX CT980180, and by INFM via PRA97(QTMD).
REFERENCES 1. 2. 3. 4.
0. Keller, Phys. Rep. 268 (1996) 85. F. D. M. Haldane, J. Phys. Cl4 (1981) 2585. C. L. Kane, M. P. A. Fisher, Phys. Rev. B 46 (1992) 15233. F. P. Mill&an, C. P. Umbach, R. A. Webb, Sol. St. Commun. 97 (1996) 309. A. M. Chang, L. N. Pfeiffer, K. W. West, Phys. Rev. Lett. 77 (1996) 2538. 5. M. Sassetti, B. Kramer, Phys. Rev. B54 (1996) R5203. 6. M. Sassetti, U. Weiss, B. Kramer, Sol. St. Commun. 97 (1996) 605.