- Email: [email protected]
Quantum-classical crossover in carrier transport
.__ BB
cm
18 August
1997
d
PHYSICS
me
Physics Letters A 233 ( 1997)
ELSJZVIER
Quantum-classical
LETTERS
A
115-120
crossover in carrier transport Akihisa Tomita ’
Opto-Electronics Research Laboratories. NEC Corp., 34 Miyukigaoku, Tsukuba, Ibaraki 30.5, Japan Received 29 January
received 29 May 1997; accepted for publication 29 May 1997 Communicated by L.J. Sham
1997; revised manuscript
Abstract Relaxation effects on the carrier transport have been studied using a density matrix approach. The analysis shows that the transport characteristics are determined by the ratio of the mean free path IF to the active region length L. The Landauer formula and the classical conductivity are obtained in the limits 2&/L < 1 and 2&/L > 1. @ 1997 Elsevier Science
B.V. PACS: 72.10.-d; 73.40.-c; Keywords: Carrier transport;
85.42+m Scattering;
Quantum
transport
device
Recently, an increasing number of studies [ l] have been reported on the control of carrier transport by a microscopic potential, such as quantum interference switching devices, resonant tunneling diodes, and multiple quantum well barrier structures in quantum well lasers. These devices are based on the quantum coherence of the carrier wave. In actual devices, however, the carrier transport is affected by the carrier relaxation arising from carrier-phonon scattering and carrier-carrier scattering. The relaxation forces carriers to quasiequilibrium. It also broadens the energy spectrum and erases the phase memory. These effects will reduce quantum interference. If the relaxation is strong, the carriers act as classical particles, and the quantum devices will no longer work. The performance of the quantum devices are limited by the carrier relaxation, especially at high temperatures. This issue has been attracting researchers’ attention [ 2-51. Non-equilibrium fluctuations have been also discussed [ 6-81. This Letter will investigate the effect of the carrier relaxation on the transport within a density matrix formalism including the coupling with reservoirs. The theory is simple and can provide a general treatment of the interaction. It will show a quantum-classical crossover and a criterion for the appropriate description of the carrier motion. In other words, we will answer the question whether we can treat a wire of length L as a classical conductor. The intuitive answer would be yes, if the wire is longer than the mean free path. The density matrix analysis will show that our intuition is correct. The model of a quantum transport device is depicted in Fig. 1. The device consists of three parts: an “active” region between lead regions, which connect the active region and power supplies. The device function is realized by the carrier action in the active region; the carrier may tunnel the barriers, or may interfere with itself, or may emit a photon. Therefore, predicting the device performance will require detailed information ’ E-mail: [email protected]. 0375-9601/97/$17.00 @ 1997 Elsevier Science B.V. All rights reserved. PIISO375-9601(96)00430-l
A. Tomita/Physics
116
Letters A 233 (1997) 115-120
ACTIVE REGlON LEAD REGION
I 3 I
I I LEAD l REGION
: j
POWERlA_ SUPPLY
POWER SUPPLY
I I I I
I
I 5 :-
-___; I.
INTERmCE
I
i
x. c
l :
,NTERFr(CB
Fig. 1. Model of a quantum transport device. The device consists of an active region between two lead regions, which are connected power supplies. The potential near the end of the active region (xl < x < .I_, x+ < n < xr) is assumed to be flat.
to
in the active region, such as energy distribution of the carriers. The carrier dynamics should be considered quantum mechanically. A density matrix provides a proper description of the carriers in the active region. The lead regions act as reservoirs in the density matrix equation. The detailed definition of the reservoirs will be given later. On the other hand, the current is all we need to know in the lead regions. The carrier motion is described by a diffusion equation. The carrier flux in the active region provides a boundary condition for the diffusion equation; the active region can be treated as a source and a sink of carriers. Treating all the regions by quantum-mechanically would be intractable, though it is possible in principle. We will define a current density operator and its expectation values in the active region. The scattering state wave functions [9] provide a good starting point at which no carrier relaxation takes place. A density matrix p is obtained with the bases of the scattering states. Only the one-dimensional problem will be considered here for simplicity. The carriers are assumed to move in the x direction. The potential is assumed to be flat near the ends of the active region (XI < x < I_, x + < x < x,), where the wave number k is a good quantum number. The wave function for the particle incident from the right side of the active region of the wave number k (k < 0) is written as t(k) exp( ikx)
for xl < x < x-
exp(ik’x)
for x+ < x < xr
+ r(k) exp( -ik’x)
(1)
and that from the left side of the active region (k > 0) exp(ikx)
+ r(k) exp( -ikx)
for x1 < x < xfor x+ < x < xr
(2)
The wave number k for x1 < x < x_ (or k’ for x+ < x < x,) specifies the particle state in the whole active region (we here discard bound states). The scattering states ( 1) and (2) are orthonormal [ 91. Rigorously, one should calculate the scattering states self-consistently by taking into account the space charge potential as well as the heterostructure potential. We assume here that the scattering states have already been obtained, and will not concern ourselves with their calculation. The current density operator Jk,k/ is defined by the scattering state wave functions as
Suppose we want to measure the current density near the end of the active region (xl < x < x- or x+ < x < x,). The wave function (1) or (2) can be safely applied. We assume the measurement is done over a length larger than the de Broglie wavelength. Then the orthogonality of the function exp(ikx) reduces the current density operator to
A. Tomita/Physics
(1 -
efi
Jk,k’ = m*L
I Itf
Irl*wk,k’
Letters A 233 (1997) 115-120
(k,k’ > 0)
k&y
(k, k'
< 0)
-rt’k&-kf
(k > 0, k’ < 0)
r* tk&_k’
(k < 0, k’ > 0)
(4)
for XI < x < x_. The current density for x + < x < xr can be obtained numbers. The expectation value of the current density is obtained by Tr(pJ): Jl = ;x$[(l
- ir(k)l*)p
111
k,k - It(-
P-k,-k
-
r(k)*+k)Pk,-k
by changing
-
the sign of the wave
r(k)+-k)*P-k,kl
k>O
for XI
Jr =
< XL, where lr(k))*
+ It(k)/*
:x-$[cl - lr(k)l*)p
k,k -
= 1. Similarly, l+k)/*P-k,-k
+ r(k)*t(-k)/‘k.-k
+ r(k)+-k)*&k.k]
(6)
k>O
is obtained for x+ < x < x,.. The current density operator (4) requires only matrix elements pk,k and p&k. The expectation values of the current density (5) and (6) will be boundary conditions for the diffusion equations in the lead regions; Eqs. (5) and (6) connect the microscopic density matrix equation and the macroscopic diffusion equation. The information on the lead regions can be included in the density matrix equation of the active region as follows. Suppose an interface in the lead region to the active region. The interface should be macroscopically thin but much thicker than the mean free path of the carriers, so that the interface can be treated as a particle reservoir connected to the active region. The proper definition of the reservoir contains subtle problems [lo]. The following assumptions will help. First, only the exchange of the carriers is allowed as the interaction between the active region and the interface, in other words, the phonon-assisted carrier transfer is discarded. Second, the interaction is linear. Third, the same energy-dispersion relation applies to the active region and the interface. Fourth, there is no carrier reflection from the interface. These assumptions refer to an ideal reservoir [ 71. The Hamiltonian of the whole system is expressed by H=Ho+H,
=~f=i‘dk‘+k+
x(b~b;b~
+ k~o(f=k.K&
+ f@d@d
K
k + h.c.)
+
HI + &({n})
+ c
(fiU#I;b,y
+ h.c.)
k,K>O
,
(7)
where ak, bK and by are the annihilation operators of the carriers in the active region, the interface in the left lead region (x < XI), and the interface in the right lead region (x > x,.), respectively. The second term and the third term of Eq. (7) denote the interaction between the active region and the interfaces. The carriers in the interfaces are assumed to be in quasi-equilibrium described by the Fermi distribution function f_ (wk) and f+ (wk) with definite chemical potentials. Hamiltonian HI denotes the interaction of carriers in the active region with another reservoir whose Hamiltonian is HR. The reservoir is described by the variables (n}. This interaction causes the carrier dephasing and drives the carriers in the active region to the quasi-equilibrium states. The interaction HI is assumed to act only on the carriers in the active region; there is no phase relaxation in the interfaces by HI. Since the expectation value of the current density requires the density matrix elements pk,k and pk,-k, the projection operator method provides simple equations of motion for the density matrix. The density matrix equations up to second order of HI are reduced to
118
A. Tomita/Physics
. Cab) _ . PkK - i(Wk
-
wK)ppffp’
-
iu,‘,K[p:F)
p:“_“, = -( wk + W-&pi
Letters A 233 (1997) 115-120
- f-(WK)]
- wk&Kb),
,
(8)
Where wk and wkk’ denote carrier relaxation. approximation yield the relaxation terms
The time convolution-less
(TCL)
formalism
[ 1 l] and Markovian
where ~n( {n}) is the density matrix of the reservoir and the trace is taken over the reservoir variables {n}. If the scattering is non-Markovian, the carrier relaxation wk and Wkk’ are time-dependent. The non-Markovian scattering can be included by a straightforward extension of the present theory. The third equation of (8) implies that P&k -+ 0 is in the steady state. The current density can be calculated with Eqs. (5), (6) and the steady state solution of Eq. (8). In the following, we will calculate the current and conductivity in special cases. First, the Landauer formula [ 12,131 for the conductance is derived in the weak scattering limit. Neglecting the relaxation terms provides the steady state solution of Eq. (8) as
for k > 0. Eqs. (5),
J=;
c
(6), and (10) yield the expectation
$lti*[f-bk)
-
_f+(m-k)]
value of the current density as
(11)
,
k>O
where J = Jl = J,, because we neglect the carrier capture and recombination. Then considering f_ (Wk) f+( 0-k) = 8(bk - EF)CW yields J = (e*/h) lt126V for T = 0 K. The Landauer formula for the conductance is obtained, G= ;,t,*.
(12)
Second, we will consider elastic scattering as the relaxation process. The elastic scattering will be dominant at low temperatures, where the phonon scattering rate is negligible. The elastic process scatters the carriers of wavenumber k only to the -k-state. Then the steady state solution of Eq. (8) satisfies
c K
Putting
(wk
-
wk
@K)*
+
wk’
(aa)bkK~*b,,
f-(UK)]
-
wk(@ - p’_,&>=o.
wk = W-k and jUk,K12= /fi-k,-K/* gives the expectation
value of the current density
(13)
A. Tomita/Physics Letters A 233 (1997) 115-120
g+2wk(Wk-kJ~)2+w~
$3”1 + 2w/g l
s
It(E)
1
127r (E
FiW -
.SF)~
+
(fiiw)2
where the values are taken at k = kF in the last equation
,vk,K,2[f-(%)
119
- f+(W-K)]
d&,
(14)
of ( 14), such as W = Wk, and (15)
The parameter g represents the coupling strength between the active region and the interface. The last equation of (14) shows that the carrier relaxation modifies the current density in two ways. The first is the factor which originates from the carrier exchange between k-states and -k-states. The carrier distribution (1+2w/g)-l, relaxes to the quasi-equilibrium Pkk = &k-k = fk. The second is the factor liW/[ (E - &F)~ + (hW)2], which appears in place of 6( E - EF) . This factor implies that the phase relaxation broadens the carrier energy spectrum. The integral shows that particles with different energies (phases) contribute to the expectation value of the current density. The quantum interference is thus canceled by the contribution from particles with different energies. If the transmission probability It( is varies slowly (the width of It( is much larger than W), it can be put outside the integral. Then the one-dimensional conductance at T = 0 K is (16) Eq. (16) describes the effect of the scattering on the conductance in terms of the product of the coupling coefficient g and the relaxation time 7 = I/W. For a wire, i.e. no potential structure in the active region, we obtain ltj2 = 1. If the scattering is strong, g7 < 1, Eq. (16) reduces to
G= ;g~.
(17)
We choose the coupling between the active region and the interface as the inverse of the transit time across the active region. Since L/uF refers to the transit time of the carrier with Fermi velocity UF, the coupling constant is written as 27dikF
2ruF IUkF,kFI
=
-
L
=
For the one-dimensional e2rN (T=---. in*
-
(18)
ML carrier density
N = kF we obtain the conductivity
CT= GL as (19)
This is a well-known conductivity for classical particles. We can obtain Eq. (19) for two- or three-dimensional conductors by choosing the definition of the coupling constant. However, the coupling constants are not so simple as Eq. ( 18) for higher dimensions. Eq. ( 16) provides both the quantum and classical conductance in the limits of g7 > 1 and g7 < 1, respectively. Since the mean free path of a particle IF is defined by OFT, the control parameter g7 is given by 2&/L, the ratio of the mean free path to the length of the active region. The present result has been obtained for slowly varying transmission probability. If the width of It( is
I20
A. TomitdPhysics
Letrers A 233 (1997) 115-120
comparable or smaller than W, the reduction of the quantum interference will be significant. We should then calculate the integral in Eq. (14) numerically. We have obtained a current through a nanostructure analyzing the effect of carrier scattering in a density matrix formulation for the coupling between the active region and the lead regions (interfaces). Recently, Gurevich et al. [ 141 have developed a quantum-linear-response theory for the effect of electron-phonon interaction on the current. They obtained the classical conductivity equation (19) by summing over a large number of channels. We have clarified the meaning of the summation by the last equation of ( 14): relaxation of the carrier distribution function and loss of the carrier coherence. We have also shown that the mean free path and the active region length determine the quantum-classical crossover in a one-dimensional conductor. The present theory can be extended to calculate a non-linear response. Therefore, it would be more favorable than a quantum-linear-response theory. The set of equations (8) conserves the current, in contrast to the conventional Pauli master equation [ 1.51. Furthermore, the theory has the advantage of efficient numerical calculation over the Wigner function and Green function methods. If n discrete wavenumbers are used for a numerical calculation, 2n matrix elements Pk,k and Pk.-k are required instead of the full n2 elements of Pk,k’. This reduction of matrix elements loses information of the carrier position. However, one can obtain the carrier density at a specific point by putting a probe. The probe is described by adding spatially localized interaction to the Hamiltonian equation (7). The probe may be a bound state of a quantum well in the active region. If the active region contains multiple quantum wells (MQW) , we can calculate the carrier distribution to the bound state of each well. Such a calculation will help us to understand the carrier transport in MQW laser diodes, where the carrier distribution has been calculated by the diffusion equation [ 161. However, Yamazaki et al. have found the importance of the quantum transport in the low temperature laser operation [ 171. A unified description of the carrier transport is thus desired. In summary, the present theory provides a simple description of the quantum transport under carrier scattering and dissipation. The theory could be useful for investigating quantum mechanical device operation as well as fundamental physics.
References [ 1] [ 21 [3] [4] [5] [6] [7] 181 191
[lo] 11 l]
[ 121 [ 131 [ 141 [ 151 [ 161 [ 171
For example, Heterostrucures and quantum devices, eds. N.G. Einspruch and W.R. Frensley S. Feng, Phys. Lett. A 143 ( 1990) 400. I?N. Argryes, Phys. L.&t. A 171 (1992) 373. F. Sols, Ann. Phys. 214 (1992) 386. A. Imamoglu and Y. Yamamoto, Phys. Lett. A 191 (1994) 425. M. Biittiker, Phys. Rev. B 45 ( 1992) 3807. A. Shimizu and M. Ueda, Phys. Rev. Lett. 69 (1992) 1403. M. Ueda and A. Shimizu, J. Phys. Sot. Jpn. 62 ( 1993) 2994. A.M. Kriman, N.C. Kluksdahl and D.K. Ferry, Phys. Rev. B 36 ( 1987) 5953. Y. Isawa and T. Hatano, J. Phys. Sot. Jpn. 60 (1991) 3108; IBM J. Res. Dev. 32, No. 3 ( 1988). F. Shibata and T. Arimitu, J. Phys. Sot. Jpn. 49 ( 1980) 89 1. R. Landauer, IBM J. Res. Dev. 1 (1957) 223. R. Landauer, Philos. Mag. 21 (1970) 863. V.L. Gurevich, V.B. Pevzner and E.W. Fenton, Phys. Rev. B 51 ( 1995) 9465. W.R. Frensley, Rev. Mod. Phys. 62 ( 1990) 745. N. Tessler and G. Eisenstein, IEEE J. Quantum Electron. QE-29 (1993) 1586. H. Yamazaki, A. Tomita, M. Yamaguchi and Y. Sasaki, unpublished.
(Academic
Press. San Diego. CA, 1994)
cm
18 August
1997
d
PHYSICS
me
Physics Letters A 233 ( 1997)
ELSJZVIER
Quantum-classical
LETTERS
A
115-120
crossover in carrier transport Akihisa Tomita ’
Opto-Electronics Research Laboratories. NEC Corp., 34 Miyukigaoku, Tsukuba, Ibaraki 30.5, Japan Received 29 January
received 29 May 1997; accepted for publication 29 May 1997 Communicated by L.J. Sham
1997; revised manuscript
Abstract Relaxation effects on the carrier transport have been studied using a density matrix approach. The analysis shows that the transport characteristics are determined by the ratio of the mean free path IF to the active region length L. The Landauer formula and the classical conductivity are obtained in the limits 2&/L < 1 and 2&/L > 1. @ 1997 Elsevier Science
B.V. PACS: 72.10.-d; 73.40.-c; Keywords: Carrier transport;
85.42+m Scattering;
Quantum
transport
device
Recently, an increasing number of studies [ l] have been reported on the control of carrier transport by a microscopic potential, such as quantum interference switching devices, resonant tunneling diodes, and multiple quantum well barrier structures in quantum well lasers. These devices are based on the quantum coherence of the carrier wave. In actual devices, however, the carrier transport is affected by the carrier relaxation arising from carrier-phonon scattering and carrier-carrier scattering. The relaxation forces carriers to quasiequilibrium. It also broadens the energy spectrum and erases the phase memory. These effects will reduce quantum interference. If the relaxation is strong, the carriers act as classical particles, and the quantum devices will no longer work. The performance of the quantum devices are limited by the carrier relaxation, especially at high temperatures. This issue has been attracting researchers’ attention [ 2-51. Non-equilibrium fluctuations have been also discussed [ 6-81. This Letter will investigate the effect of the carrier relaxation on the transport within a density matrix formalism including the coupling with reservoirs. The theory is simple and can provide a general treatment of the interaction. It will show a quantum-classical crossover and a criterion for the appropriate description of the carrier motion. In other words, we will answer the question whether we can treat a wire of length L as a classical conductor. The intuitive answer would be yes, if the wire is longer than the mean free path. The density matrix analysis will show that our intuition is correct. The model of a quantum transport device is depicted in Fig. 1. The device consists of three parts: an “active” region between lead regions, which connect the active region and power supplies. The device function is realized by the carrier action in the active region; the carrier may tunnel the barriers, or may interfere with itself, or may emit a photon. Therefore, predicting the device performance will require detailed information ’ E-mail: [email protected]. 0375-9601/97/$17.00 @ 1997 Elsevier Science B.V. All rights reserved. PIISO375-9601(96)00430-l
A. Tomita/Physics
116
Letters A 233 (1997) 115-120
ACTIVE REGlON LEAD REGION
I 3 I
I I LEAD l REGION
: j
POWERlA_ SUPPLY
POWER SUPPLY
I I I I
I
I 5 :-
-___; I.
INTERmCE
I
i
x. c
l :
,NTERFr(CB
Fig. 1. Model of a quantum transport device. The device consists of an active region between two lead regions, which are connected power supplies. The potential near the end of the active region (xl < x < .I_, x+ < n < xr) is assumed to be flat.
to
in the active region, such as energy distribution of the carriers. The carrier dynamics should be considered quantum mechanically. A density matrix provides a proper description of the carriers in the active region. The lead regions act as reservoirs in the density matrix equation. The detailed definition of the reservoirs will be given later. On the other hand, the current is all we need to know in the lead regions. The carrier motion is described by a diffusion equation. The carrier flux in the active region provides a boundary condition for the diffusion equation; the active region can be treated as a source and a sink of carriers. Treating all the regions by quantum-mechanically would be intractable, though it is possible in principle. We will define a current density operator and its expectation values in the active region. The scattering state wave functions [9] provide a good starting point at which no carrier relaxation takes place. A density matrix p is obtained with the bases of the scattering states. Only the one-dimensional problem will be considered here for simplicity. The carriers are assumed to move in the x direction. The potential is assumed to be flat near the ends of the active region (XI < x < I_, x + < x < x,), where the wave number k is a good quantum number. The wave function for the particle incident from the right side of the active region of the wave number k (k < 0) is written as t(k) exp( ikx)
for xl < x < x-
exp(ik’x)
for x+ < x < xr
+ r(k) exp( -ik’x)
(1)
and that from the left side of the active region (k > 0) exp(ikx)
+ r(k) exp( -ikx)
for x1 < x < xfor x+ < x < xr
(2)
The wave number k for x1 < x < x_ (or k’ for x+ < x < x,) specifies the particle state in the whole active region (we here discard bound states). The scattering states ( 1) and (2) are orthonormal [ 91. Rigorously, one should calculate the scattering states self-consistently by taking into account the space charge potential as well as the heterostructure potential. We assume here that the scattering states have already been obtained, and will not concern ourselves with their calculation. The current density operator Jk,k/ is defined by the scattering state wave functions as
Suppose we want to measure the current density near the end of the active region (xl < x < x- or x+ < x < x,). The wave function (1) or (2) can be safely applied. We assume the measurement is done over a length larger than the de Broglie wavelength. Then the orthogonality of the function exp(ikx) reduces the current density operator to
A. Tomita/Physics
(1 -
efi
Jk,k’ = m*L
I Itf
Irl*wk,k’
Letters A 233 (1997) 115-120
(k,k’ > 0)
k&y
(k, k'
< 0)
-rt’k&-kf
(k > 0, k’ < 0)
r* tk&_k’
(k < 0, k’ > 0)
(4)
for XI < x < x_. The current density for x + < x < xr can be obtained numbers. The expectation value of the current density is obtained by Tr(pJ): Jl = ;x$[(l
- ir(k)l*)p
111
k,k - It(-
P-k,-k
-
r(k)*+k)Pk,-k
by changing
-
the sign of the wave
r(k)+-k)*P-k,kl
k>O
for XI
Jr =
< XL, where lr(k))*
+ It(k)/*
:x-$[cl - lr(k)l*)p
k,k -
= 1. Similarly, l+k)/*P-k,-k
+ r(k)*t(-k)/‘k.-k
+ r(k)+-k)*&k.k]
(6)
k>O
is obtained for x+ < x < x,.. The current density operator (4) requires only matrix elements pk,k and p&k. The expectation values of the current density (5) and (6) will be boundary conditions for the diffusion equations in the lead regions; Eqs. (5) and (6) connect the microscopic density matrix equation and the macroscopic diffusion equation. The information on the lead regions can be included in the density matrix equation of the active region as follows. Suppose an interface in the lead region to the active region. The interface should be macroscopically thin but much thicker than the mean free path of the carriers, so that the interface can be treated as a particle reservoir connected to the active region. The proper definition of the reservoir contains subtle problems [lo]. The following assumptions will help. First, only the exchange of the carriers is allowed as the interaction between the active region and the interface, in other words, the phonon-assisted carrier transfer is discarded. Second, the interaction is linear. Third, the same energy-dispersion relation applies to the active region and the interface. Fourth, there is no carrier reflection from the interface. These assumptions refer to an ideal reservoir [ 71. The Hamiltonian of the whole system is expressed by H=Ho+H,
=~f=i‘dk‘+k+
x(b~b;b~
+ k~o(f=k.K&
+ f@d@d
K
k + h.c.)
+
HI + &({n})
+ c
(fiU#I;b,y
+ h.c.)
k,K>O
,
(7)
where ak, bK and by are the annihilation operators of the carriers in the active region, the interface in the left lead region (x < XI), and the interface in the right lead region (x > x,.), respectively. The second term and the third term of Eq. (7) denote the interaction between the active region and the interfaces. The carriers in the interfaces are assumed to be in quasi-equilibrium described by the Fermi distribution function f_ (wk) and f+ (wk) with definite chemical potentials. Hamiltonian HI denotes the interaction of carriers in the active region with another reservoir whose Hamiltonian is HR. The reservoir is described by the variables (n}. This interaction causes the carrier dephasing and drives the carriers in the active region to the quasi-equilibrium states. The interaction HI is assumed to act only on the carriers in the active region; there is no phase relaxation in the interfaces by HI. Since the expectation value of the current density requires the density matrix elements pk,k and pk,-k, the projection operator method provides simple equations of motion for the density matrix. The density matrix equations up to second order of HI are reduced to
118
A. Tomita/Physics
. Cab) _ . PkK - i(Wk
-
wK)ppffp’
-
iu,‘,K[p:F)
p:“_“, = -( wk + W-&pi
Letters A 233 (1997) 115-120
- f-(WK)]
- wk&Kb),
,
(8)
Where wk and wkk’ denote carrier relaxation. approximation yield the relaxation terms
The time convolution-less
(TCL)
formalism
[ 1 l] and Markovian
where ~n( {n}) is the density matrix of the reservoir and the trace is taken over the reservoir variables {n}. If the scattering is non-Markovian, the carrier relaxation wk and Wkk’ are time-dependent. The non-Markovian scattering can be included by a straightforward extension of the present theory. The third equation of (8) implies that P&k -+ 0 is in the steady state. The current density can be calculated with Eqs. (5), (6) and the steady state solution of Eq. (8). In the following, we will calculate the current and conductivity in special cases. First, the Landauer formula [ 12,131 for the conductance is derived in the weak scattering limit. Neglecting the relaxation terms provides the steady state solution of Eq. (8) as
for k > 0. Eqs. (5),
J=;
c
(6), and (10) yield the expectation
$lti*[f-bk)
-
_f+(m-k)]
value of the current density as
(11)
,
k>O
where J = Jl = J,, because we neglect the carrier capture and recombination. Then considering f_ (Wk) f+( 0-k) = 8(bk - EF)CW yields J = (e*/h) lt126V for T = 0 K. The Landauer formula for the conductance is obtained, G= ;,t,*.
(12)
Second, we will consider elastic scattering as the relaxation process. The elastic scattering will be dominant at low temperatures, where the phonon scattering rate is negligible. The elastic process scatters the carriers of wavenumber k only to the -k-state. Then the steady state solution of Eq. (8) satisfies
c K
Putting
(wk
-
wk
@K)*
+
wk’
(aa)bkK~*b,,
f-(UK)]
-
wk(@ - p’_,&>=o.
wk = W-k and jUk,K12= /fi-k,-K/* gives the expectation
value of the current density
(13)
A. Tomita/Physics Letters A 233 (1997) 115-120
g+2wk(Wk-kJ~)2+w~
$3”1 + 2w/g l
s
It(E)
1
127r (E
FiW -
.SF)~
+
(fiiw)2
where the values are taken at k = kF in the last equation
,vk,K,2[f-(%)
119
- f+(W-K)]
d&,
(14)
of ( 14), such as W = Wk, and (15)
The parameter g represents the coupling strength between the active region and the interface. The last equation of (14) shows that the carrier relaxation modifies the current density in two ways. The first is the factor which originates from the carrier exchange between k-states and -k-states. The carrier distribution (1+2w/g)-l, relaxes to the quasi-equilibrium Pkk = &k-k = fk. The second is the factor liW/[ (E - &F)~ + (hW)2], which appears in place of 6( E - EF) . This factor implies that the phase relaxation broadens the carrier energy spectrum. The integral shows that particles with different energies (phases) contribute to the expectation value of the current density. The quantum interference is thus canceled by the contribution from particles with different energies. If the transmission probability It( is varies slowly (the width of It( is much larger than W), it can be put outside the integral. Then the one-dimensional conductance at T = 0 K is (16) Eq. (16) describes the effect of the scattering on the conductance in terms of the product of the coupling coefficient g and the relaxation time 7 = I/W. For a wire, i.e. no potential structure in the active region, we obtain ltj2 = 1. If the scattering is strong, g7 < 1, Eq. (16) reduces to
G= ;g~.
(17)
We choose the coupling between the active region and the interface as the inverse of the transit time across the active region. Since L/uF refers to the transit time of the carrier with Fermi velocity UF, the coupling constant is written as 27dikF
2ruF IUkF,kFI
=
-
L
=
For the one-dimensional e2rN (T=---. in*
-
(18)
ML carrier density
N = kF we obtain the conductivity
CT= GL as (19)
This is a well-known conductivity for classical particles. We can obtain Eq. (19) for two- or three-dimensional conductors by choosing the definition of the coupling constant. However, the coupling constants are not so simple as Eq. ( 18) for higher dimensions. Eq. ( 16) provides both the quantum and classical conductance in the limits of g7 > 1 and g7 < 1, respectively. Since the mean free path of a particle IF is defined by OFT, the control parameter g7 is given by 2&/L, the ratio of the mean free path to the length of the active region. The present result has been obtained for slowly varying transmission probability. If the width of It( is
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A. TomitdPhysics
Letrers A 233 (1997) 115-120
comparable or smaller than W, the reduction of the quantum interference will be significant. We should then calculate the integral in Eq. (14) numerically. We have obtained a current through a nanostructure analyzing the effect of carrier scattering in a density matrix formulation for the coupling between the active region and the lead regions (interfaces). Recently, Gurevich et al. [ 141 have developed a quantum-linear-response theory for the effect of electron-phonon interaction on the current. They obtained the classical conductivity equation (19) by summing over a large number of channels. We have clarified the meaning of the summation by the last equation of ( 14): relaxation of the carrier distribution function and loss of the carrier coherence. We have also shown that the mean free path and the active region length determine the quantum-classical crossover in a one-dimensional conductor. The present theory can be extended to calculate a non-linear response. Therefore, it would be more favorable than a quantum-linear-response theory. The set of equations (8) conserves the current, in contrast to the conventional Pauli master equation [ 1.51. Furthermore, the theory has the advantage of efficient numerical calculation over the Wigner function and Green function methods. If n discrete wavenumbers are used for a numerical calculation, 2n matrix elements Pk,k and Pk.-k are required instead of the full n2 elements of Pk,k’. This reduction of matrix elements loses information of the carrier position. However, one can obtain the carrier density at a specific point by putting a probe. The probe is described by adding spatially localized interaction to the Hamiltonian equation (7). The probe may be a bound state of a quantum well in the active region. If the active region contains multiple quantum wells (MQW) , we can calculate the carrier distribution to the bound state of each well. Such a calculation will help us to understand the carrier transport in MQW laser diodes, where the carrier distribution has been calculated by the diffusion equation [ 161. However, Yamazaki et al. have found the importance of the quantum transport in the low temperature laser operation [ 171. A unified description of the carrier transport is thus desired. In summary, the present theory provides a simple description of the quantum transport under carrier scattering and dissipation. The theory could be useful for investigating quantum mechanical device operation as well as fundamental physics.
References [ 1] [ 21 [3] [4] [5] [6] [7] 181 191
[lo] 11 l]
[ 121 [ 131 [ 141 [ 151 [ 161 [ 171
For example, Heterostrucures and quantum devices, eds. N.G. Einspruch and W.R. Frensley S. Feng, Phys. Lett. A 143 ( 1990) 400. I?N. Argryes, Phys. L.&t. A 171 (1992) 373. F. Sols, Ann. Phys. 214 (1992) 386. A. Imamoglu and Y. Yamamoto, Phys. Lett. A 191 (1994) 425. M. Biittiker, Phys. Rev. B 45 ( 1992) 3807. A. Shimizu and M. Ueda, Phys. Rev. Lett. 69 (1992) 1403. M. Ueda and A. Shimizu, J. Phys. Sot. Jpn. 62 ( 1993) 2994. A.M. Kriman, N.C. Kluksdahl and D.K. Ferry, Phys. Rev. B 36 ( 1987) 5953. Y. Isawa and T. Hatano, J. Phys. Sot. Jpn. 60 (1991) 3108; IBM J. Res. Dev. 32, No. 3 ( 1988). F. Shibata and T. Arimitu, J. Phys. Sot. Jpn. 49 ( 1980) 89 1. R. Landauer, IBM J. Res. Dev. 1 (1957) 223. R. Landauer, Philos. Mag. 21 (1970) 863. V.L. Gurevich, V.B. Pevzner and E.W. Fenton, Phys. Rev. B 51 ( 1995) 9465. W.R. Frensley, Rev. Mod. Phys. 62 ( 1990) 745. N. Tessler and G. Eisenstein, IEEE J. Quantum Electron. QE-29 (1993) 1586. H. Yamazaki, A. Tomita, M. Yamaguchi and Y. Sasaki, unpublished.
(Academic
Press. San Diego. CA, 1994)