Chapter 3 The Low-Frequency Noise
3.3 Theoretical Calculation of Zero-Frequency Noise
To our knowledge, there are two theoretical formulations that are used to explore the quantum transport in nanoscale systems, i.e. Rate equation method (RE) and the non-equilibrium Keldysh Green’s function (NEGF). In contrast to the weak perturbation method for the electron-electron coupling in Rate equation, the NEGF provides a general physical condition for particle transport through a non-equilibrium system, valid from the small
Figure 28: The plot of the differential conductance and differential noise vs. Vg for an photon-assisted tunneling model. Reprinted from Ref.[29]
bias to the large bias in the leads, where the influence of EVI is explicitly considered when applying the small polaron canonical transformation on the vibrating quantum dot system.
The relevant Green’s function can be solved via the Dyson equation and the Langreth rule. Among the NEGF approach, there are two different physical picture that are used to interpret the vibron-assisted process. Firstly, a concept of effective one-body tunneling scheme (with no fluctuations inside) is imposed to interpret the particle scattering in the QD system, i.e. replacing the electron-vibron interaction to a an average field. In practice, the relevant transport quantities such as current and the differential noise were discussed and reported by Chen et al [18], while the current correlation is not yet exposed before. In this work, we employ the mean-field approach to examine the zero-frequency noise, and then compare with an exact analytical solution to that solved by the analytic continuation.
Basically, the EVI not only breaks the symmetry properties of electrons and holes in the quantum dot system, but also yields a significant staircase phenomenon on the tunneling rate as well as on the bandwidth of a single state for the central system. The main difference between the MFT approach and LT are: The EVI correlation is regarded as scalar, and therefore the vibron correlation will not couple to the Fermi function of the leads, irrelevant to the chemical potential difference. This resulting correlation gives the 0th quantized state
Conductance
Diff Noise
Figure 29: The upper shows G = dI/dVb vs. Vg (gate volatge) and Vb (bias voltage, vertical axis). The lower density plot reveals the differential noise dS/dVb. Reprinted from Ref.
of the LT method, proving that the LT treatment is beyond than previous researches of this kind. Note that in this paper, we ignore higher order tunneling process from heating of vibration or mutual influence within the sub-electronic and sub-vibronic subsystems,[22]
we focus only on the lowest order electron-vibron interaction. Besides we neglect the spin degree of freedom and the influence of Coulomb interaction.
3.3.1 The JWM Transport Formula (from the perspective of the metal wire) The current from the left lead to the central region can be defined by:
JL(t) = 2e
Re
k,α∈L
Vkα,dG<d,kα t, t′
|t′→t, (56)
where the Green’s function G<d,kα(t, t′) ≡ i
c+kα(t′) X (t) d (t)
. The first-order expansion of Dyson series on Gd,kα gives
Gd,kα τ , τ′
=
dτ1Gdd(τ , τ1) F+(τ , τ1) Vkα∗ gkα τ1, τ′
. (57)
Figure 30: The vibrating quantum dot model. Here the equilibrium vibronic is coupled to the QD system. Reprinted from Ref.[11]
Performing the continuation rules[11] on Gd,kα(τ , τ′) and substituting the resulting G<d,kα(t, t′) into Eq.(56), the steady current is written as:
JL(t) = −2e
Im
t
−∞
dt1Grdd(t, t1) Σ<αǫL(t1, t) + G<dd(t, t1) ΣaαǫL(t1, t)
, (58)
where Σ>,<α (t, t1) =
α|Vkα|2g>,<kα (τ1, τ2), and G≶dd(t, t1) are defined as:
G≶dd(t, t1) ≡ F+≶
t, t′ G≶dd t, t′
, (59)
where F+>(t − t′) ≡ X+(t′) X (t) and F+<(t − t′) ≡ X (t) X+(t′) denote the greater and lesser vibron Green’s functions, and the lesser (greater) Green’s function for a free electron in the α lead is denoted by g<(>)kα (t − t′) . Moreover, the (self-consistent) Dyson expansion of the electron Green’s function Gdd is found as:[14][16]
Gdd τ , τ′
= G(0)dd τ , τ′
+
c1
dτ1
c2
dτ2
G(0)dd (τ , τ1) ΣT (τ1, τ2) Gdd τ2, τ′
, (60)
where the contour-ordered self-energy reads
ΣT (τ1, τ2) =
kα∈L,R|Vkα|2gkα(τ1, τ2)
TcX+(τ1) X (τ2)
= F+(τ1− τ2) ΣT (τ1− τ2) . (61)
The self-energy contains all correlations about lead electrons and the vibrons. F+(τ1− τ2) ≡
TcX+(τ ) X (τ′) is the vibron Green’s function and ΣT(τ1− τ2) =
kα∈L,R|Vkα|2gkα(τ1, τ2) (62) is the self-energy due to the electron coupling. The retarded (advanced) self-energy can be easily found as
The Fourier transform further gives the retarded and the lesser self-energies Σr,<α (ω) =
where fα<,>(εkα) denotes the electron and the hole Fermi functions in the α lead, and Γα(ω) = 2π
kα|Vkα|2δ (ω − εkα) shows the tunneling rate without EVI. Also, the Fourier transform of Eq.(59) leads to
G≷dd(ω) =
n
p∓nG≷dd(ω + nω0) . (67) The weighting factor pn, which indicates the probability of the electron interacting with n vibrons, is
pn= e−2g(N0+12)enω0/2kBTIn
2g
N0(N0+ 1) , (68)
where g = (λ/ω0)2, N0 is the Bose function, and In is the modified Bessel function. As to G≷(ω) in Eq.(67), this propagator can be evaluated using the Keldysh formulation[14],
that is,
G≷dd(ω) = Σ≷d (ω)... Grdd(ω)...2 (69)
= ∓i
n,α
p∓nΓ≷α(ω − nω0)... Grdd(ω)...2, Gr,add (ω) =
ω − ε0− Σr,aT (ω)−1
, (70)
where Γ≷α(ω) = Γα(ω) fα≷(ω) and Gr,add is derived from Eq.(60).
Eq.(18) can be recast into the effective expression derived by Wingreen and Meir[15], where the current is expressed as the product of the transport window and the tunneling function. Performing some algebra on Eq.(18), we get:
Jα= ie h
dω
fα<(ω) − fα<(ω)
T (ω) , (71)
T (ω) = ΓL(ω) ΓR(ω)
ΓL(ω) + ΓR(ω)A (ω) , (72)
A (ω) = i
n pn
G>dd(ω − nω0) − G<dd(ω + nω0)
, (73)
where T (ω) (A (ω)) denotes the EVI tunneling (spectral) function for the vibrating QD (or SMT). Substituting the results of Eq.(69) back into Eq.(72), the EVI tunneling function T (ω) is written as
T (ω) = ΓL(ω) ΓR(ω)
ΓL(ω) + ΓR(ω)· (74)
∞ n,n′=−∞,α
p−npn′Γ>α[ω + (n + n′) ω0] + pnp−n′Γ<α[ω + (n + n′) ω0] [ω + nω0− ε0(ω + nω0)]2+ [W (ω + nω0) /2]2 ,
where ε0= ε0(Vg) − Re ΣrT (ω) denotes the renormalized level position of the quantum dot state. W (ω) = −2 Im Σrd(ω) represents the life-time broadening (bandwidth).[19] it is worth mentioning that T (ω) possesses some symmetries such as T (ω, Vg, Vb) = T (ω, Vg, −Vb), and
T (ω = ε0− ∆ω, −Vg, Vb) = T (ω = ε0+ ∆ω, Vg, Vb) . 3.3.2 Zero-Frequency Noise formula
Now we proceed to calculate the noise. Generally, the noise is the current correlation defined as Sαα′(t, t′) ≡ {δIα(t) , δIα′(t′)} = {Iα(t) , Iα′(t′)} − 2Jα2, where δIα(t) =
Iα(t) − Iα(t) together with the stationary fact of Jα = Iα(t) = Iα′(t′) are performed for previous description. Substituting the current operator of Eq.(56) into the definition of noise and taking the Fourier transform Sαα′(ε) =∞
−∞d (t − t′) eiε(t−t′)Sαα′(t − t′) with ε → 0[22][14][26], we obtain the current correlation from the zero-frequency noise (see Appendix E). integrand in Eq.(58), Eq.(75) is rewritten as
S = 2e2
3.3.3 Thermal Noise and Shot Noise
The advantage of Eq.(76) is that all the EVI effects are kept in the dot system T (ω), not in the leads. Besides, the first term, which vanishes at zero temperature, means the thermal noise Sth. As the bias voltage is smaller than kBT , i.e. the equilibrium system, the second term (shot noise) disappears, and the equilibrium noise is Sth = 4kBT GT, aka the Johnson-Nyquist formula. Here, the differential conductance is defined as GT =
2e2
is referred to the thermal broadening function in the leads[27]. Considering the inverse case of e |Vb| > kBT , the second term of Eq.(76) dominates the non-equilibrium noise, which is basically proportional to the average current and inversely proportional to 1 −T (ω), saying that the fluctuation of the electrical current due to the discreteness of the charge carriers
in mesoscopic devices behaves the fluctuation of the occupation number. If the particles randomly transmitted, i.e. uncorrected, then the shot noise is SP = 2e I, this is the shot noise which was firstly proposed by Schottky and was observed in vacuum diodes.
At zero temperature, the Fermi functions goes forward the step functions, and the weighting factors are given by pn = e−ggn/n! for n ≧ 0 and pn = 0 for n < 0. Applying these rules on Eq.(76) leads to
S = 2e2 h
µL µR
dω
2πT (ω) [1 − T (ω)] , (77)
T (ω) = ΓLΓR
ΓL+ ΓRe−2g
n,n′,α
g(n+n′)
n!n′! · (78)
Γαθ [ω − (n + n′) ω0− µα]
[ω − nω0− ε0(ω − nω0)]2+ WT2=0(ω − nω0) /4 + Γαθ [µα− ω − (n + n′) ω0]
[ω + nω0− ε0(ω + nω0)]2+ WT2=0(ω + nω0) /4
,
where the life-time broadening is given by WT=0(ω) = e−g
n,α
gn
n!Γα[θ (ω − nω0− µα) + θ (µα− ω − nω0)] , (79) different from previous studies with a mean-field approximation. For simplicity, we ig-nored the energy dependence of Γα (wide-band approximation) here and in the following.
Furthermore, it can be checked that Eq.(76) also satisfies the symmetric relations such as S (Vg, Vb) = S (Vg, −Vb), and S (Vg, Vb) = S (−Vg, Vb). Next we study the stationary proper-ties of the tunneling function, the current, and its zero-frequency noise according to Eq.(74), Eq.(71) and Eq.(76).
In general, there exist two sources of correlations in the mesoscopic device: The first is the Pauli principal for the non-interacting electrons, and the other is the EVI effect stemming from the vibrons, both effects are revealed in Eq.(78). Before study this tunneling coefficient, first of all, it is convenient to assume that the energy level of the QD is aligned with the chemical potential of leads. When applying an external bias to the leads, it is expected that the chemical potential in the leads would asymmetrically deviate from the level energy of the dot, that is, µL(R) = ε0(Vg = 0) ± eVb/2 and ε0(Vg = 0) = µL+µ2 R. The
Figure 31: The density plot of the tunneling function. The left (right) inset in the middle figure depicts the distribution of the left (rigth) out-tunneling rate ΓoutL(R).
tunneling probabilities are depicted for high- and low-lying levels at ε0 = ω0 and ε0 = −ω0
in Fig. 31(a)(c) and for medium level at ε0 = 0 in Fig. 31(b), with the bias eVb = 2ω0 and the tunneling rate ΓL= ΓR= 0.4ω0.
Fig. 31shows that the amplitude of tunneling function are distributed asymmetrically with respect to ε0(±Vg) symmetrically for ε0(eVg = 0), implying the broken symmetry between electrons and holes in the QD (or SMT)[18]. In addition, it is found that the satellite peaks in the LT tunneling function are nonuniformly broaden, and its amplitude is much smaller than the MFT one[18]. This is because a staircase change of the vibron-assisted tunneling rate has been considered as the particle tunneling through the junction. Owing to the vibron emission and absorption, the statistical probabilities of finding n vibrons in occupied states and available states are different, breaking the electron-hole symmetry in the QD (or SMT). Generally, the electron states are located below the chemical potential of the leads, while the hole states are above. Such a non-uniform tunneling phenomenon results in a step-like tunneling rate (Wα) in energy space, with the symmetric centers at
Figure 32: The current profile as a function of the gate voltage. The upper is wih the MFT method and the lower with the LT method. The upper inset in each figure denotes the total out-tunneling rate (or the level-broadening), where the blue (red) curve denotes for the left (right) out-tunneling rate. The lower inset depicts the corresponding differential conductance. The density plot of the conducatnce is accompnied attached.
µL and µR. The left (right) profile in Fig. 31(b) exhibits the vibron-assisted tunneling rate from the vibrating QD (or SMT) to the α lead, where the solid curve stands for the LT method, and the dashed line for the MFT one.[18] Note that the MFT tunneling rate remains constant, which coincides with the 0th vibron mode of the LT one, no matter whether the bias voltage is changed or not. This owes to the fact that the average-field approximation, i.e. VkαX → VkαX, is relevant to the time evolution of the vibron field, thus F+><(t − t′) → X2= p0.
Fig. 32 shows the profile of the current as a function of the gate voltage. Compared to the MFT results (Up), we find that the LT current is strongly suppressed at Vg/ω0 = ±0.75 (Down). This can be understood as follows: In Eq.(18), the current is expressed as the tun-neling function T (ω), multiplied by the transport window, fα<(ω)−fα<(ω). Basically, T (ω) is composed of G>(ω − nω0) and G<(ω + nω0), corresponding to the hole and electron oc-cupation density, respectively. At zero temperature, pn= 0 for n < 0, fα≶(ω) → θ (±µα∓ ω) and ... Gr(ω)...2 = A (ω) /WT=0(ω), and therefore the tunneling function T (ω) is expanded
as the summation of rectangular functions multiplied by the sideband peaks A (ω ± nω0).
In the weak coupling limit, i.e. A (ω) → 2πδ (ω), the current is found as:
J = e
ΓLΓRe−3g ΓWT=0(ε0)
∞ n,n′=0
g(n+n′)Γα
n!n′! {[θ (µL− ε0− nω0) − θ (µR− ε0− nω0)]
· θ
ε0− n′ω0− µα
+ [θ (µL− ε0+ nω0) − θ (µR− ε0+ nω0)]
· θ
µα− ε0− n′ω00
, (80)
For the nth transport channel, electrons are allowed to transmit within µα < ω < µα−nω0, and holes within µα + nω0 < ω < µα. In addition, there exists a broadening function WT=0(ε0) which renormalizes the current distribution (see Fig. 31(b2)). Note it stays the same (WT=0(ε0) → Γ) in the MFT method. This effect results from the fact that the bias-associated information are included into the self-energy as the particle transport from the α lead to the α lead, that is, the analytic continuation on Eq.(19). According to Eq.(79), WT−1=0(ε0) behaves as multiple decreasing steps at the resonant energies and symmetric about eVg = 0, and hence the height of the electron (hole) current is suppressed along with the decrease (increase) of the gate voltage. Based on this, the peak-structure current at eVg/ω0= ±0.75 disappears, resulting in a remarkable conductance gap in Fig. 32(b).
3.3.4 Zero-frequency Noise
Now let we study the energy dependence of the shot noise. Fig. 33(a) to Fig. 33(d) shows the zero-frequency noise S vs Vg at various bias voltages, and the insets denote the dif-ferential noise. For comparison, we further plot the MFT noise as red dashed lines. For eVb/ω0 = 0.02 (nearly zero dc bias), the current approaches zero. However, because the electrons tunneling through the QD can absorb or emit photons (here n = 0), the thermal noise changes significantly. It is noticed that no noise change occurs around eVg = nω0
(n = ±1, ±2, and so on) because the current stream is forbidden to flow. In other words, satellite peaks of the differential noise dS/dVb do not occur in this area, as is shown in the inset of Fig. 34(a). These phenomena are different from previous theoretical predic-tions reported by Sun et al.[29] and Balatsky et al.[9] but agrees with the experimental observations[30] of the single resonant peak (see Fig. 28 and Fig. 29).
Figure 33: Zero-frequency noise vs. the gate voltage under four different bias voltage (form left to rgiht, eVb/ω0 = 0.02, 1.5, 2.5 and 3). The insets denote the corresponding differential noise. The black line represents the LT method and the gray line for the MFT one. Noe that the shot noise is normalized by 2e
Fig. 34(a) depicts the detailed profiles for dSth/dVb (red curve) and dSch/dVb (blue curve) vs. Vg, where the vibron-free case is labeled by dashed lines. Here, a sudden decline appears at Vg = 0 for the thermal noise because the PAT process is slightly forbidden by the Pauli exclusion principle[29].
3.3.5 Probing the EVI Coupling Strength
At T = 0, the thermal noise vanishes, and a remarkable peak structure of dSch/dVbis left, as seen in Fig. 34(a). This is different from the vibron-free case with a double-peak structure.
Such behavior can be explained by the following: For eVb < ω0, only the 0th channel makes the contribution, and the tunneling function of Eq.(78) reduces to
T0(ω) = ΓLΓRe−g (ω − ε0)2+
ΓL+ΓR 2 e−g 2
. (81)
Figure 34: (a) dSth/dVb (red curve) and dSch/dVb (blue curve) vs. Vg, the dashed lines denotes λ = 0 (no EPI) and the solid lines for λ = 1.5ω0. (b) The renormalized noise function S/S0 vs. Vb, where S0 is the noise without EPI. The black line denotes the LT method and the gray line as the MFT one. The inset reveals the original noise function S, an additional blue-dashed line is the noise without EPI. (c) The Fano factor vs. Vb. The left inset here denotes the corresponding noise and the right inset shows the Fano factor vs. Vg at Vb = 6ω0. (d) shows the renormalized factor (F/F0) vs. Vb, we can see that they depart from each other at Vb= 2ω0.
Substituting Eq.(81) into Eq.(77) and taking the derivative over Vb, we obtain
On the other hand, for the symmetric electron coupling ΓL = ΓR = Γ, the single peak structure of dS/dVb appears as λ > 0.83ω0. This is useful in verifying the coupling strength of the EVI in the experiments.
Next, we examine the noise for eVb> ω0. As shown from Fig. 32(b) to Fig. 32(d), large bias voltages yield a staircase noise with the steps occurring at the resonant energies and symmetric about eVg = 0. This is in contrary to the MFT results, where the heights of the noise are close, analogous to that observed in current.
Thermal noise becomes significant with finite temperature. Performing the weak cou-pling limit on the first term of Eq.(76), we obtain
Sth = 2e2 are not apparent in the LT curve because the LT shot noise is much larger than its thermal noise. As one plots Sthvs. Vb, we find that thermal broadening peaks occurs at eVb = 2nω0. Nonetheless, due to the staircase structure of the vibron-assoicated tunneling rate W (ε0), LT thermal noise decays faster than the MFT one along with the increasing bias, as shown in Fig. 32(b). In the large bias limit, e.g. µL ≫ µR, fL< = fR> = 1, fL> = fR< = 0, the thermal noise vanishes and the shot noise are found as
S = 2e I
Noise
Figure 35: The density plot of the Fano factor as studied in a CNT-QD. Reprinted from Ref.[5].
where γα → e−gΓα for the MFT method and γα→ Γα for the LT one. It is noticeable that both approach result in the same current expression in large bias limit, that is, I = e
ΓLΓR Γ
(see also in the left inset of Fig. 34(c)), but not for the shot noise (see the inset in Fig. 34 (b)), which implies that the shot noise gives additional and complementary information on the current-voltage characteristic. Substituting I and Eq.(85) back into the definition of the Fano factor and considering the primary correlation of n = n′, the Fano factors become FMF T ≈ 1−2ΓΓL2ΓReg
∞ n=−∞
pnΓL+ p−nΓR2
and FLT ≈ 1−2ΓΓL2ΓR
∞ n=−∞
pnΓL+ p−nΓR2
, respectively. Apparently, FLT shows the higher value than FMF T due to the absence of eg. In the absence of EVI effect (λ → 0), Eq.(85) reduces to S0= 2e I
1 − 2ΓLΓR/
ΓL+ ΓR2
, and we obtain F = 0.5 for the symmetric electron-electron coupling. This is in agreement with that observed in experiments[5], as shown in Fig. 35.
CHAPTER IV
SUMMARY AND FUTURE WORKS
By applying the small polaron transformation and the non-equilibrium Green’s function (NEGF) technique, we examine the joint effects due to the vibron-assisted tunneling rate.
We conclude that:
(1) As the vibrons coupled to the electron tunneling process, the relevant vibron corre-lation will break the electron-hole symmetry in the non-interacting terminals, making the tunneling rate change in a quantized feature of the vibration frequency.
(2) The electrons through a SMT can be remodeled into to a single-level quantum dot coupling to multi-channel leads, and the current is described as a sum of all tunneling flux via various channels.
(3) The conductance gap between the chemical potential of the leads and the first vibron excited state is in agreement with recent experimental results and is recognized as an occurrence of a virtual state.
(4) We can reproduce our results to that worked with the rate equation method, only if we replaced the single particle spectral function as delta function. That is, the NEGF approach gives more general information than the RE method, such as the energy shift and the level broadening.
(5) At high temperature, the holes may occupy higher energy levels above the chemical potential due to the absorption of vibrons, and thus we can measure conductance peaks in the Coulomb blockade region, in agreement with the recent CNT-QD experiments.
(6) When coupling to the vibrational modes, the zero-frequency noise still can be de-composed into the standard formation of thermal noise (Johnson—Nyquist noise) and the
shot noise, the same with that without EVI.
(7) In contrast to previous theoretical works of this kind with an anti-symmetric struc-ture in noise and conductance, we obtain symmetric results, both in differential conductance and differential noise, fulfilling with recent CNT-QD experiments.
(8) We demonstrate that the differential noise could be a feasible tool for probing the EVI coupling strength.
(9) The first vibron excited state changes the noise, which is proportional to λ4, not previous prediction λ2. This is in agreement with the First principal calculation.
Finally, we have to admit that, in comparison with recent theoretical work done by Felix von Oppen’s group, our method is insufficient to interpret the Franck-Condon blockade at low bias. Usually this behavior takes places in the strong electron-vibron interaction area. The lacking of suppression in our calculation is due to the usage of the equilibrium vibron assumption through the whole calculation. In the weak EVI, our results matches the experiments, including the conductance gap between the chemical potential od the leads and the position of the vibron sidebands. However, as the EVI is increased, the theoretical and experimental results are far from each other. We believe the difference lies on the assumption of the equilibrium vibron bath is insufficient to interpret the influence of vibrational motion of the atoms. In fact, it makes sense that the time scale of the vibron relaxation is comparable to the flow of the electrons into and out of the molecule. In the NEGF’s calculation, the shorter life-time of the vibrons means more explicit calculation of the self-energy in vibron Green’s function. We will briefly describe the numerical recipe of this project in the Appendix F.
In conclusion, we hope that this new transport scheme together with the noise formula could provide useful insights into the quantum transport field, and more studies, such as non-equilibrium vibrons bath, can be explored, allowing a systematic description of inter-esting physical effects in the future.
REFERENCES
[1] H. Park, J. Park, A. K. L. Lim, and E. H. Anderson, A. P. Alivisatos, and P. L.
McEuen, Nature 407, 57 (2000).
[2] J. Park, A. N. Pasupathy, J. I. Goldsmith, C. Chang, Y. Yaish, J. R. Petta, M. Rinkoski, J. P. Sethna, H. D. Abruña, P. McEuen and D. C. Ralph, Nature 417, 722 (2002).
[3] L. H. Yu and D. Natelson, Nano Lett. 4, 79(2004).
[4] S. Sapmaz, P. Jarillo-Herrero, J. Kong, C. Dekker, L.P. Kouwenhoven, and H.S.J. van der Zant, Phys. Rev. B 71, 153402 (2005).
[5] S. Sapmaz, P. Jarillo-Herrero, Y.M. Blanter, C. Dekker, and H.S.J. van der Zant, Phys.
Rev. Lett. 96, 026801 (2006).
[6] E. Onac, F. Balestro, B. Trauzettel, C. F. J. Lodewijk, and L. P. Kouwenhoven, Phys.
Rev. Lett. 96, 026803 (2006).
[7] R. Leturcq, C. Stampfer, K. Inderbitzin, L. Durrer, C. Hierold, E. Mariani, M.G.
Schultz, F. von Oppen, and K. Ensslin, Nature Physics 5, 327 - 331 (2009).
[8] E.M. Weig, R. H. Blick, T. Brandes, J. Kirschbaum, W. Wegscheider, M. Bichler, and J.P. Kotthaus, Phys. Rev. Lett. 92, 046804.
[9] J.-X. Zhu and A.V. Balatsky, Phys, Rev. B 67, 165326 (2003).
[10] M. Galperin, A. Nitzan, and M. A. Ratner, Phys, Rev. B 73, 045314 (2006).
[11] A.P. Jauho, N.S. Wingreen and Y. Meir, Phys. Rev. B 50, 5528 (1994).
[12] A. Mitra, I. Aleiner, and A. J. Millis, Phys. Rev. B 69, 245302 (2004).
[13] A. Zazunov, D. Feinberg, and T. Martin, Phys. Rev. B 73, 115405 (2006); A. Zazunov and T. Martin, ibid. 76, 033417 (2007).
[14] H. Haug and A.P. Jauho, Quantum Kinetics in Transport and Optics of Semiconductors
[14] H. Haug and A.P. Jauho, Quantum Kinetics in Transport and Optics of Semiconductors