Electron tunneling through a coupled system of electron and phonon

Chinese Journal of Physics 60 (2019) 307–312

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Electron tunneling through a coupled system of electron and phonon

T

⁎

Li-Ling Zhoua,b, , Xue-Yun Zhoua,b, Rong Chenga,b a b

Department of Physics, Jiujiang University, Jiujiang 332005, China Jiangxi Province Key Lab of Microstructure Functional Materials, China

ABS TRA CT

We study the electron tunneling through a single level quantum dot in the presence of electron–phonon interaction. By using the Green’s function and canonical transformation methods, we calculated exactly the current. It is found that the current vs dot level exhibits satellite peaks even without occurring of phonon-assisted tunneling processes, and properties of the current are affected heavily by the strength of electron–phonon interaction and phonon temperature.

1. Introduction Due to advances in nanotechnology, investigation on electron transport through a nanoscale junction has been of great interest recently [1–18], and a variety of experimental and theoretical results have been obtained, which may help to make predictions for improved designs of nanoscale junctions. On this topic, the hybrid system consisting of a quantum dot (QD) contacted to metallic leads acting as macroscopic charge reservoirs has been investigated extensively [3–6,9,10,12–16]. Because of the small size of the QD, electron transport is influenced significantly by nuclear vibrational degrees of freedom, local excitations of substantial energy and represented by phonons, which has been observed in a variety of experiments [19–27]. The electron–phonon interaction (EPI) in QD leads to phonon-satellites in spectral functions of QD electron, and then vibronic structures appear in transport characteristics. Theoretically, based on the nonequilibrium Green function method, some work has been done on investigating transport behavior through a vibrating QD. The analytical expression for the current is not exact in general. In order to get this expression approximations [28–32] have been made in these works to decouple the EPI. However, the spectral density of the QD electron is not able to be given properly by these simple decoupling schemes, which take no accounts of vibrational effect on electronic self-energies. In this letter, we study the electron tunneling through a vibrating QD coupled to two normal leads (N-QD-N). We consider the problem where only a single electron (or hole) is present in the QD, of which the analytical expression for current can be given exactly. The system under our consideration can be described by the following Hamiltonian (hereafter e, ℏ = 1): (1)

H = HL + HR + HD + HT .

∑k ϵα, k cα†, k cα, k

d†d

a†a

λ (a†

a) d†d

+ ω0 + + represents the left and right normal-metal leads, HD = ϵd describes the dot Here, Hα = L, R = state. Hybridization between leads and dot is depicted by HT = ∑α, k (tα cα†, k d + tα*d†cα, k ) . The fermion operators cα†, k (cα,k) and d† (d) create (destroy) an electron in the α-lead and dot, respectively. The boson operator a† (a) creates (annihilates) a phonon mode of frequency ω0 in the QD. λ is the electron–phonon interaction strength, and tα the electron hopping amplitude between the QD and αlead. The current from the left (right) normal metal to the QD can be calculated from the time evolution of the occupation number

⁎

Corresponding author. E-mail address: [email protected] (L.-L. Zhou).

https://doi.org/10.1016/j.cjph.2019.04.023 Received 15 February 2019; Accepted 29 April 2019 Available online 24 May 2019 0577-9073/ © 2019 The Physical Society of the Republic of China (Taiwan). Published by Elsevier B.V. All rights reserved.

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a

b

Fig. 1. The current I (a) vs the QD level ϵd and the spectral function (b) of QD electron for different EPI strength λ. Other parameters are chosen as V = 0.3ω0 , TL = TR = Tph = 0, ΓL = ΓR = 0.05ω0 , and the unit I0 = eΓ/ℏ .

operator NL / R = ∑k cL†/ R, k cL / R, k : IL / R (t ) = −e〈dNL / R/ dt 〉. In steady state, there is I = IL = −IR . Following Ref. [33] we get

I= ΓL / R

ie ΓLΓR ℏ Γ

∫ 2dεπ [fL (ε ) − fR (ε )][Gr (ε ) − Ga (ε )].

(2)

ΓL

|2 δ (ε

ΓR .

r,a

+ = 2π ∑k |tL / R − ϵL / R, k) is the linewidth and assumed to be independent of energy, Γ = G (ε) are the Fourier Here, transformation of the retarded and advanced components of the Green function for QD electron: G (t , t ′) = −i〈T {d (t ) d† (t ′)} 〉, governed by H, and fL / R (ε ) = 1/{exp[(ε − μL / R )/ κB TL / R] + 1} is the Fermi distribution function with TL/R and μL/R being respectively the temperature and chemical potential of the left (right) lead. In order to obtain the current I, we have to solve the Green functions Gr,a(ε). Here, we discuss this problem in a similar way as in Ref. [34], where shown is the current of a quantum well in presence of EPI. We give the detail calculation procedure because it is important in analysis of the transport properties. First, calculate gr(ε), the Fourier transformation of the retarded Green function g r (t , t ′) = −iθ (t − t ′) 〈 {d (t ), d† (t ′)} 〉 for the isolated QD. gr(t, t′) is defined as the same as Gr(t, t′), but governed by HD. ∼ Before proceeding a canonical transformation: HD = UHD U † need to be made, with the unitary operator † † U = exp[λ / ω0 (a − a) d d], to decouple the electron–phonon interaction. Under this transformation the electron operator d and phonon operator a are transformed to d → d˜ = UdU † = dX ,

(3)

a → a˜ = UaU † = a − λ / ω0 d†d,

(4)

− a)]. Replacing d and a in HD with d˜ and a˜ respectively, we obtain straightforwardly the transformed where, X = exp[−λ / ω0 ∼ Hamiltonian HD for isolated QD: ∼ HD = ϵ˜d d†d + ω0 a†a, (5) (a†

with ϵ˜d = ϵd − λ2 / ω0 , λ2/ω0 being the electron self-energy. By using this transformation we can now solve the Green function gr (t , t ′) explicitly.

g r (t , t ′) = −iθ (t − t ′) 〈 {d (t ), d† (t ′)} 〉HD = −iθ (t − t ′)[〈d (t ) d† (t ′) 〉HD + 〈d† (t ′) d (t ) 〉HD ].

(6)

The subscript HD indicates that this Green function is governed by the Hamiltonian for isolated QD. There are two terms in the square 308

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a

b

c

Fig. 2. The spectral functions of QD electron (a) for varying EPI strength λ, with Γ = 0.1ω0 , Tph = 0, (b) at different phonon temperature Tph, with λ = 1.4ω0 , Γ = 0.1ω0 , (c) for different electronic coupling strength Γ between the QD and leads, with λ = 1.0ω0 , Tph = 0 .

Fig. 3. The current I vs the QD level ϵd for different phonon temperatures Tph. The EPI strength is λ = 1.4ω0 and other parameters are the same as in Fig. 1.

bracket in Eq. (6). Consider, for example, the first term, which we evaluate as

〈d (t ) d† (t ′) 〉HD = e βΩ Tr{e−βHD eiHD t d (0) e−iHD t eiHD t ′d† (0) e−iHD t ′} ∼

∼

∼

∼

†

∼

= e βΩ Tr{e−βHD eiHD t d˜ (0) e−iHD t eiHD t ′d˜ (0) e−iHD t ′} † = 〈d˜ (t ) d˜ (t ′) 〉∼ H .

(7)

D

Everything is now changed to the transformed representation. In the third line, we inserted the factor 1 = U †U into the trace and used ∼ ∼ the cyclic property of the trace. In the transformed Hamiltonian HD, there is no interaction between the electron part He = ϵ˜d d†d and ∼ † phonon part Hph = ω0 a a . Then Eq. (7) can be greatly simplified,

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Fig. 4. The current I vs the bias V for different EPI strength, with the QD level fixed at ϵd = 0.5ω0 + λ2 / ω0 . The other parameters are the same as in Fig. 1.

Fig. 5. The current I vs the bias V for different phonon temperature, with λ = ω0 , ϵd = 1.5ω0 , TL, R = 0, and ΓL, R = 0.05ω0 . †

† † ∼ 〈d˜ (t ) d˜ (t ′) 〉∼ HD = 〈d (t ) X (t ) X (t ′) d (t ′) 〉HD † ∼ =〈d (t ) d† (t ′) 〉∼ He 〈X (t ) X (t ′) 〉Hph

=[1 − nF (ϵ˜d)] e−iϵ˜d (t − t ′) 〈X (t ) X † (t ′) 〉∼ Hph ,

(8)

˜d on the isolated QD. Since the QD Hamiltonian describes a where nF (ϵ˜d) = 〈d† (0) d (0) 〉∼ He is the average occupation number of energy ϵ fixed particle of energy ϵd interacting with phonons with frequency ω0, we set nF = 0 for the case of a single particle [35,36]. Similar manipulation can be applied to the other term, and we find 〈d↑† (t ′) d↑ (t ) 〉HD = nF (ϵ˜d) e−iϵ˜d (t − t ′) 〈X † (t ′) X (t ) 〉∼ Hph .

(9)

Following Ref. [36], we obtain the phonon part of Eqs. (8) and (9), ∞

〈X (t ) X † (t ′) 〉∼ Hph =

∑

Ln e−inω0 (t − t ′),

n =−∞ ∞

〈X † (t ′) X (t ) 〉∼ Hph =

∑

Ln e−inω0 (t ′− t ), (10)

n =−∞

where,

Ln = e−g (2Nph + 1) e nω0 /(2κB Tph) In (2g Nph (Nph + 1) ),

(11)

)2 ,

with g = (λ / ω0 Nph = 1/[exp(ω0 /(κB Tph )) − 1], Tph the phonon temperature, and In(z) the nth Bessel function of complex argument z. At zero temperature, Ln reduces to e−gg n / n! for n ≥ 0 and 0 for n < 0. Now, gr(t, t′) is given explicitly, of which the Fourier transformation is 310

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L.-L. Zhou, et al. ∞

g r (ε ) = with

0+

1 − nF nF ⎞, Ln ⎛ + + ˜ ˜ ε − ϵ − nω + i 0 ε − ϵ + nω0 + i0+ ⎠ d 0 d ⎝ n =−∞

∑

⎜

⎟

(12) r

a positive infinitesimal. There is only a single electron in the system we consider, then g (ω) reduces to (nF = 0 ) ∞

g r (ε ) =

∑ n =−∞

Ln . ε − ϵ˜d − nω0 + i0+

(13)

Using the Dyson equation, we obtain directly the full Green functions of the QD coupled to leads,

G r (ε ) = Ga * (ε ) = {g r (ε )−1 − Σr (ε )} −1,

(14)

Σr (ε )

= −iΓ/2 . Then the current I can be readily obtained by substituting Eq. (14) into Eq. (2). The with the retarded self-energy spectral function of the QD electron is A (ε ) = i [G r (ε ) − Ga (ε )], which depends on the EPI strength λ, phonon temperature Tph, phonon frequency ω0, and linewidth Γ. Now, we have calculated the Gr(ε), and hence A(ε), exactly. If the polaron transformation (3) and (4) are applied to the Hamiltonian H, instead of HD, to obtain the full Green function Gr(ε) some approximation has to be made to decouple the EPI as in Refs. [17,28]. On the basis of Eq. (2), we investigate properties of the current I. In the following numerical investigation, we set ω0 = 1 as energy unit, μR = 0 as the reference energy, and then the bias voltage eV = μL . The effect of the EPI on the current is shown in Fig. 1(a). Compared with the EPI-free case, λ = 0, for which the I exhibits a single resonant peak at ϵd = 0.5V , with finite EPI the resonant current peak is shifted to a higher QD level energy by λ2/ω0, the electron self-energy, and the current shows a series of peaks locating at ϵd = 0.5V + λ2 / ω0 − mω0 (m ∈ N). These properties can be well explained with help of the spectral function of the QD electron A(ϵ) shown in Fig. 1(b). With finite EPI, the resonant spectral peak at ϵd is shifted to left by λ2/ω0, and spectral sidepeaks appear at ϵ = ϵd − λ2 / ω0 + mω0 . The sidepeaks implies that in the ground state, the particle in QD has a nonzero probability of occupying states having m phonons, and the particle energy is not always ϵd − λ2 / ω0 , but fluctuates among these different values ϵd − λ2 / ω0 + mω0 [36]. At Tph = 0 and V < ω0, no phonon-emitting or -absorbing tunneling process takes place. Every time a spectral peak lies between μL and μR, a current peak appears. It is clear that the current peak at ϵd = 0.5V + λ2 / ω0 − mω0 in Fig. 1(a) is contributed by the spectral peak at ε = ϵd − λ2 / ω0 + mω0 shown in Fig. 1(b), via which an electron can tunnel through the QD. Therefore the I is influenced by the λ in a similar way as the A(ε). In the following we discuss in detail the characteristics of the A(ϵ) and the I when the EPI is present in the QD. Due the EPI, satellite peaks may appear in the spectral function beside the resonant peak at ϵd − λ2 / ω0 . Fig. 2(a) shows that at zero temperature, only for ε > ϵd − λ2 / ω0 do spectral sidepeaks appear, and that the λ exerts a big effect on the weight of each spectral peak. For λ/ ω0 < 1, the biggest peak is m = 0, and higher m peaks get smaller rapidly. For λ/ω0 ≥ 1, the peak height increases with m up to a value of m = ⌊ (λ / ω0)2⌋, and then begin to decrease. Notice that when ⌊ (λ / ω0)2⌋ = (λ / ω0)2 = m ≥ 1, there are two biggest peaks, locating at ε = ϵd − λ2 / ω0 + mω0 and ϵd − λ2 / ω0 + (m − 1) ω0 repectively, since Lm equals Lm − 1 at zero phonon temperature. At elevated phonon temperature, peaks can also arise at ε < ϵd − λ2 / ω0 , see Fig. 2(b), which grow with increasing Tph. Because of the factor e nω0 /(2κB Tph) in Eq. (11) the peak at ϵd − λ2 / ω0 − mω0 is much smaller than the one at ϵd − λ2 / ω0 + mω0 . The spectral peaks are widened and lowered by increasing Γ, as shown in Fig. 2(c), and smoothed out if the Γ is large enough. As expected, at elevated phonon temperature extra peaks arise at ϵd = 0.5V + λ2 / ω0 + lω0 (l ∈ N*) in the I-ϵd curve, see Fig. 3. The reason is spectral peaks appear at ε = ϵd − λ2 / ω0 − lω0 , see Fig. 2(b). Fig. 3 also shows that with rising phonon temperature, current peaks at 0.5V + λ2 / ω0 + lω0 raise rapidly and the height difference between big and small peaks narrows. These can be well understood with the help of Eq. (11), which decides the spectral function. In Fig. 4, we present the numerical results of current I as a function of the applied voltage V for different EPI strength, at zero temperature, TL, R, ph = 0, and weak dot-leads coupling, ΓL, R = 0.05ω0 . In our calculation, we take μR = 0, μL = V , ϵd = 0.5ω0 + λ2 / ω0 . One can see that with increasing V, current begins to arise at a bias voltage of V = 0.5ω0 despite of the EPI strength, and that the current reaches the same saturation value for different λ. But the larger the λ, the larger the bias Vsat, at which the current saturates. This phenomenon can be well explained by Fig. 2(a). For ϵd = 0.5ω0 + λ2 / ω0 , spectral peaks appear at ε = (m + 0.5) ω0 . So, current occurs when the bias increases to pass 0.5ω0. After this, a step comes every time an new spectral satellite peak enters the bias window, which indicates the opening up of new tunneling channels, the larger the peak, the longer the step. For example, at V = 1.5ω0 every curve in Fig. 4 exhibits one step because of the entering of the m = 1 spectral peak at ε = 1.5ω0 . At V = 1.5ω0 , an electron can tunnel through the QD via the m = 0 or the m = 1 spectral peak, and it can also pass the QD in a phonon-assisted process, in which it enters the QD via the m = 1 peak and jumps to the m = 0 peak by emitting one phonon before it leaves. More spectral satellite peaks come out for larger λ, see Fig. 2(a). So, we see the curve of λ / ω0 = 1.4 displays more steps than the other two, and reaches its top plateau at higher voltage. The spectral weight of each peak changes with λ as shown in Fig. 2(a), but the total weight of all spectral peaks keeps constant. This is why the saturation current values are the same for different EPI strength. If the phonon temperature increases, the I-V properties change, as shown in Fig. 5. For the same reason behind steps in Fig. 4, steps appear in Fig. 5. However, we see the positive saturation current reduces with increasing Tph and, at the same time, negative current emerges for negative V. The reason is at elevated phonon temperature spectral peaks emerge at ε = ϵd − λ2 / ω0 − lω0 , and grow rapidly with rising Tph, which causes decline of the sum weight of spectral peaks lying between μR = 0 and μL = Vsat . While, the difference between the maximum and minimum current value of a I–V curve is constant for varying Tph, because of the constant total spectral weight. In summary, we have investigated the electron tunneling through a QD, coupled to two normal leads, in the presence of local electron–phonon interaction. An exact analytical expression of the electric current I is given. The numerical results show that the current I vs QD level εd exhibits satellite peaks even when no phonon-emitting or -absorbing process takes place, and that the resonant 311

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current peak is shifted to a higher QD level energy by the electron self-energy λ2/ω0. At zero system temperature and small bias voltage V, smaller than the phonon frequency ω0, except for the resonant peak at εd = 0.5V + λ2 / ω0 , the I exhibits satellite peaks at εd = 0.5V + λ2 / ω0 − lω0 , l is a positive integer. The reason is that in the ground state of the coupled system of electron and phonon, the electron has a nonzero probability of occupying states having l phonons. For weak EPI, λ/ω0 < 1, the biggest current peak appears at εd = 0.5V + λ2 / ω0 , while it locates at εd = 0.5V + λ2 / ω0 − ⌊ (λ / ω0)2⌋ω0 for λ/ω0 ≥ 1. At elevated phonon temperature, extra satellite current peaks also arise at εd = 0.5V + λ2 / ω0 + lω0 , leading to a smaller saturation current value. Declarations of interest None. Acknowledgments We gratefully acknowledge the financial support from the National Natural Science Foundation of China under Grant no. 11764023 and the Scientific Project of the Jiangxi Education Department of China under Grant no. GJJ170944. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36]

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