The Conductance Oscillation in the SMT

The physical quantities of interest, such as current, conductance and higher order correlation functions, e.g. noise, can be obtained as the spectral density (density of states) is solved.

In experiments, the differential conductance has structure similar to the zero-bias spectral function.

1.2.1 Nanomechanical Oscillations in a single-C60 transistor

A single-C60 molecule is considered to have a single level of degeneracy. The vibrons may changed the single-level molecule into a multiple-level structure, as shown in Fig. 4. It can be seen that more and more vibron side-bands enter the transport window (the chemical

Figure 4: The density plot of the spectral function A(ω) at different bias, where the left denotes the small one and the right is the larger. The right figure shows the differential conductance as a function of the bias voltage and the gate voltage. The white triangle denotes the vibron sidebands. Reprinted from Ref.[1].

potential of the metal wire) as the bias is enhanced (the red arrow). As a result, the current abruptly increases when the chemical of the leads across the level energy of vibronic channel.

Usually the hole states are distributed above the chemical potential of the leads and the electron states below. Therefore, it is expected to obtain a symmetric conductance map as the coupling of the molecule to the leads is symmetric. The right figure shows the 2D distribution of the conductance vs. the bias (vertical-axis) and the gate voltage (horizontal-axis) for four C60 transistors of various sizes. The white arrow indicates the occurrence of vibron side-band[1].

1.2.2 Coulomb blockade and the Kondo effect in single -atom transistors In 2002, J. Park et al.[2] examined two related molecules containing a Co ion bonded to polypyridyl ligands to make a quantum dot and to observed the Kondo effect. In their work, a set of symmetric satellite Kondo peaks due to emission of the vibrons were observed as the bias matches multiples of the vibrational frequency. The model is given as follows.

The left pictures in Fig. 6 show the 2D conductance for three devices of various sizes. In the black region (the blockade zone in Fig. 6a), Co2+ and Co3+ represent the molecular ground state and the excited state. In the transport bias window, the white line denotes

Figure 5: Preparation of the Co polymer transistors. Reprinted from Ref.[2].

the vibron-side band. Fig. 6(b) shows the level-splitting of ground state for the Co polymer when a magnetic field of B = 6T is exerted in Fig. 6(a). The white triangle indicates the new energy levels after the magnetic field is turned on. This is the Zeeman splitting effect measured in the conductance. The splitting energy is linearly increased with the magnetic field (inset below Fig. 6(b)). At low temperature, the conductance peak starting from the Kondo effect is probed near eVb = 0 in the Coulomb blockade region, as seen in Fig.

7(b). The right picture shows the differential conductance vs. the bias at T = 1.5K, where a maximum Kondo peak is found at eVb = 0. As usual we may find maximum peak at Vb= 0. Due to the vibron assisting tunneling, we observe, furthermore, symmetric satellite maxima. The lower inset (c) shows that the conductance owns a logarithmic temperature dependence in the range of T = 30 to T = 20K. The lower-right figure (d): The splitting of the Kondo peak caused by a series of magnetic field.

1.2.3 Suspended Semiconductor Quantum Dot (Phonon Cavity)

Besides using molecules to confirm the phonon side-bands, E. M. Weig et al.[8]. designed a suspended quantum dot device embedded in a freestanding GaAs/AlGaAs membrane to verify the phonon blockade in the low bias region. There, the phonon of single frequency is applied by the suspended phonon cavity[31][32][33], as seen in Fig. 8. The conductance under different magnetic fields are shown on the RHS of Fig. 8. The right inset profiles the corresponding conductance vs. Vg at Vb = 0, where no conductance is found at Fig. 8(b).

reflecting the phonon blockade. At higher temperature (T = 350mK), the conductance

1/ 2

S= S=0

N N+1

Figure 6: (a)Coulomb blockade and the Kondo effect in Co polymer transistors. The right figure (b) presents that the quantum dot with a magnetic field of B = 6T . The below inset denotes the Zeeman splitting as a function of the magnetic field. Reprinted from Ref.[2].

that phonon blockade starts to be decreased because of thermal broadening of the Fermi function helping hole states transport through the dot system.

1.2.4 Suspended Carbon Nanotube Quantum Dot

In 2006, S. Sapaz et al.[5] produced a suspended carbon nanotube quantum dot (CNT-QD), which was connected to metal wires and the step current was observed, as shown in Fig.

9. They deduced that the step current was from the longitudinal phonon waves in the carbon-nanotube. Besides, E. Onac et al.[6] adopted the same model and, furthermore, used a superconductor-insulator-superconductor device to measure the (a) current, (b) con-ductance, (c) noise and (d) differential noise. With current and noise, they calculated the Fano factor of the CNT-QD. They verify that in the Coulomb blockade regime, the inelastic co-tunneling yields a super-Poissonian noise, whereas poisson for the elastic co-tunneling.

Fig. 27(a) shows the Fano factor according to F = S/2e I, where I is the current given by Fig.10(c) and S is the noise shown in Fig.10(a).

In 2009, Leturcq et al.[39] took advantage of a suspended carbon nanotube (CNT) to

Figure 7: The Kondo effect in the Coulomb Blockade region (at eVb = 0). Reprinted with permission from Ref.[2]

generate a vibrating quantum dot for the observation of strong electron-vibron coupling effects on the current, as seen in Fig. 12. They demonstrate the vibron frequency is governed by the longitudinal stretching vibrons in the CNT, and more importantly, they shows strong evidence of the vibron-blockade behavior at low-bias. In addition, they probed the conductance peaks in the blockade regions. At high temperature, the absorption of the vibron energy becomes active, and vibron-mediated states assist the electron transport, even at the coulomb blockade area. This nontrivial phenomena, however, not reported in previous studies of this kind, is believed to associate with the higher-order co-tunneling of vibrons, especially, when the strong elecron-vibron interaction is considered. On the other hand, to understand the electron-vibron interaction (EVI) again becomes an hot topic in recent years.

1.3 Theoretical Development of the Vibron-assisted Tunnel-ing Problem

Theoretically, many efforts have been made to solve the electron transport the vibrating quantum dot or the SMT. Theoretical methods in this field include the scattering rate

Figure 8: The left depicts the suspended quantum dot in a freestanding phonon cavity (130nm thin GaAs/AlGaAs) membrane. The right insets show conductance maps under B = 500mT and B = 0. Inset (b) and (c) exhibit the conductance gap around V_b = 0 (linear response area). However, higer temperatue may reduce the suppress behavior. Reprinted from Ref.[8].

equation, the master equation and the non-equilibrium Green’s function. The rate equation (RE)[37], the master equation[12][46][36] and the NEGF[14][15] have successfully explained many transport experiments. Each of these theoretical approaches has its own advantages and limitations. Among theses approaches, the RE may rapidly yield the equation of motion for every state by directly replacing the density ρ (t^′) by ρ (t) In the report by Brandes[46], they calculated the main current of double quantum dots via the RE and concluded that the non-linear electric current stemmed from the shake-up effects of the vibrons. However, different treatment of methods owes it convenience, however, the convenience is often based on the decoupling assumption. For example, the RE can be quickly solved only if we omitted the influence of self-energy, which causes the error message of the energy shift and the level broadening of the quantum dot. That is, the RE is an effective method for predicting that the hopping (correlation) time is much smaller than time spent in the quantum dot, i.e.

valid for a weak-tunneling regime. To our knowledge, the non-equilibrium Green’s function is the most convincing tool to solve quantum transport at finite bias because it contains

Figure 9: The upper figure presents the staircase current as a function of source-drain voltage at a fixed gate voltage. The lower inset depicts the experimental setup of a suspended carbon nanotube quantum dot (CNT-QD). Reprinted from Ref.[5]

both the hopping ( correlation) and the self-energy of the interacting system. The early application of this technique on quantum transport were established by M. Wingreen and Y. Meir[15]. Later A. P. Jauho and his co-workers[14] extended this technique to other fields, like spin-dependent transport, superconductors and optical lattaice. For quantum transport, there is a basic current formula expressed as

J = e

dω Γ^L(ω) Γ^R(ω)

Γ^L(ω) + Γ^R(ω)[fL(ω) − fR(ω)] A (ω) . (1) This formula, however, is well-defined in the EVI problem. Usually, th function Γ (ω) and f (ω) is related to the metal wire, can be solved by Fermi’s golden rule and the Fermi—Dirac distribution. The problem lies in the determination of the function A (ω). Jauho et.al [11]

solved this function by imposing the identical relation of A (ω) = 2 Im G^r(ω), and they directly decoupled the total system’s retarded Green’s function as

G^r,a(t) ≈ G^r,a(t) exp [−Φ (t)] , (2)

Φ (t) =

 λ ω0

2

1 − e^−iω0t

(N0+ 1) +

1 − e^iω0t N0

. (3)

Current

Noise

Conductance

Diff Noise

Figure 10: The density plot of (a) current, (b) conductance, (c) noise and (d) differential noise, where the vertical-axis is the bias and horizontal-axis is the gate voltage. Reprinted with permission from Ref.[6]

where the function G denotes the electron Green’s function and exp [−Φ (t)] is the vibron correlation. Later, Zhu and Balatsky extended this method to calculate the zero-frequency noise. At high temperature, this method gives the same tendency of the conductance as found in experiments, where the conductance peak does exists in the blockade area. How-ever, this method fails at low temperature, that is, no conductance shall be found in this limit. The erroneous prediction arises from the ill treatment of electron-vibron decoupling.

As a matter of fact, G^a(t) = [G^r(t)]^∗ but Φ^∗(t) = Φ (−t) = Φ (t), as a consequence, Eq.(2) is not appropriate for studying current—voltage characteristic of vibron assisted tunneling.

Instead, it is convenient to start from function G^<,>(t). Based on this, Chen and his co-workers solved this problem and quickly obtained the symmetric conductance, agreement with the experimental observations. However, in Chen’s work, since an averaged field con-cept has been executed to simplify the EVI self-energy, they missed an important thermal broadening information from vibron Green’s functions. In this thesis, we will examine the properties of the vibron correlation and address how to interpret the EVI transport question

Figure 11: (a) The dashed line represents the transport bias window and the dot line is for the Coulomb blockade area. (b): Fano factor vs. the applied bias voltage at different gate voltages. Reprinted from Ref.[30]

in a more physical way. More importantly, a conductance gap between the chemical poten-tial of the leads and the first vibron excited state, specifically, e |Vb| > 2ω⁰, is also examined.

Figure 12: Left: The inset depicts the experimental setup of a CNT-QD, where S, D, and TG stands for the source, the drain, and the gate voltage. The quantum dots are located below the gate and in the left and right leads. The frequency is decided by the longitudinal stretching mode. Right: Franck-Condon blockade in suspended carbon nanotube QD, where higher temperature yields phonon side-bands appearing outside the bias transport area (Coulomb Blockade region). Reprinted from Ref.[39]

CHAPTER II

THEORY OF QUANTUM TRANSPORT

Our formalism is based on the nonequilibrium Keldysh Green’s function method. The Keldysh Green’s function is used to deal with the system coupled to a time-dependent external fields. This function describes how the system evolves with time from the initial state. Before the interaction is on, the system and the environment are at their equilibrium, and the their physical quantities can be described by the Matsubara Green’s function. As the interaction is turned on, the transfer function of the system can be extended via the Dyson equation approach[14][16]. Next we briefly describe the mechanism behind the Keldysh Green’s Function.

2.1 The Keldysh Green’s Function

We consider a system under the Hamiltonian

H (t) = HS+ H^′(t) .

The time-independent Hamiltonian H can be split into HS = H0+ Hintra, where Hintra

denotes the interaction in the system HS, such as the Coulomb repulsion. A quantum statistical expectation value of an field operator O at time t is given by

O (t) = tr [ρ (t) O] = tr [ρ0OH(t)] = OH(t) , (4) where ρ (t) = U (t, t0) ρ₀U⁺(t, t0) and the quantity ρ₀ is the equilibrium density matrix given by

ρ₀= e^−βHS T r [e^−βHS]. Note here that U (t, t0) = exp

−
ⁱ

t

t^′dtH (t)

. At t > t0, OH(t) = U (t, t0) OU⁺(t, t0)

= UHS(t, t0) OHS(t) V_H⁺_S(t, t0) ,

Figure 13: The contour time path ct

Therefore, Eq.(4) can be rewritten as

O (t) = tr

OHS(t) is the time-dependent operator of the system, and the contour ct is depicted in Fig. 13Since Eq.(5) is expreseed in terms of the equilibrium Hamiltonian Heq, the contour-ordered Green’s function plays a similar role as the equilibrium Green’s function. Compared to the equilibrium Green’s Function, we have two additional contour ordered Green’s func-tions:

It implies that the contour ci stretches from t0 and passes through t1 and t1^′ and back to t0, The last equality in Eq.(7) combines the contours c1+ c^′₁, as plotted in Fig.14:

Figure 14: Parts of contour evolution operators canceling in Eq.(7). Reprinted from Ref.

2.1.1 Analytical Continuum

In general, the interaction complicates the evolving path in the contour-ordered Green’s function. The procedure of converting this complex-time Green’s function into a real-time one is called the analytic continuation or Langreth theorem, which is developed by Kadanoff, Baym[44] and Langreth.[17] The useful Langreth theorem is listed as follows:

Contour (Complex) Time Real Time C (τ , τ^′) =

cdτ1 C^≶(t, t^′) = dt1

A^r(t, t1) B^≶(t1, t^′) A (τ , τ1) B (τ1, τ^′) +A^≶(t, t1) B^a(t1, t^′)

C^r(a)(t, t^′) =

dt1A^r(a)(t, t1) B^r(a)(t1, t^′) . D (τ , τ^′) =

c1dτ1

c2dτ2 D^≶(t, t^′) = dt1

dt2[A^r(t, t1) B^r(t1, t2) × A (τ , τ1) B (τ1, τ2) C (τ2, τ^′) C^≶(t2, t^′) + A^r(t, t1) B^≶(t1, t2) C^a(t2, t^′) .

+A^≶(t, t1) B^a(t1, t2) C^a(t2, t^′) D^r(a)(t, t^′) =

dt1

dt2A^r(a)(t, t1) · B^r(a)(t1, t2) C^r(a)(t2, t^′) . C (τ , τ^′) = A (τ , τ^′) B (τ , τ^′) C^≶(t, t^′) = A^≶(t, t^′) B^≶(t, t^′) .

C^r(a)(t, t^′) = A^<(t, t^′) B^r(a)(t, t^′) +A^r(a)(t, t^′) B^<(t, t^′) +A^r(a)(t, t^′) B^r(a)(t, t^′) .

= ±θ (±t ∓ t^′) ·

[A^>(t, t^′) B^>(t, t^′) − A^<(t, t^′) B^<(t, t^′)]

D (τ , τ^′) = A (τ , τ^′) B (τ^′, τ ) D^≶(t, t^′) = A^≶(t, t^′) B^≶(t^′, t) . D^r(a)(t, t^′) = A^≶(t, t^′) B^a(r)(t^′, t)

+A^r(a)(t, t^′) B^≶(t^′, t)

2.2 The Single Molecular Transistor

Here we begin with a description of the system of interest, that is, the single-molecule transistor. The experimental setup of this vibrating single-molecule transistor is shown in Fig. 15. Here the spin degree of freedom and the influence of Coulomb interaction are omitted. The model Hamiltonian of the C60 and the metal wires reads

H_leads = 

kα

ε_kαc⁺_kαc_kα (8)

H_dot(c60) = εdd⁺d, (9)

Figure 15: The experimental setup of a vibrating single-molecule transistor (C60). The dot level is cotrolled by the gate voltage and the transport window is tuned by the bias voltage in the terminal.

When the C60 molecule is connected to a gold wire, the van der Waals potential energy appears. This potential energy may be described as the harmonic potential,

Hph = p²₀ 2m0 +1

2m0ω²₀x²₀

= ω0b⁺b. (10)

We consider an electron-hole electric force will depart in the direction from the source to C60. As the electron is added to the molecule, the hole in the wire then attracts the electron in C60, forming a constant electric field and pulling C60 towards the source wire.

This behavior is described by

Hdb = qEx0d⁺d (11)

= λ

b⁺+ b

d⁺d, λ = qE/√ 2mω0.

It is worth noting that the electric field causes the original harmonic potential to shift x0, but it does not change the harmonic potential much. In the second quantization picture, Eq.(11) interprets an electron in the transport process, where λ reflects the EVI coupling strength. The system will emit (or absorb) a vibron at the same time. However, it only emits vibrons at zero temperature.

Basically, there are interactions among the QD system, the wire, and the vibrons. For convenience, two of them are usually dealt with as one quasi-particle. In previous studies, a quasi-particle polaron may result from the combination of vibron and QD. Here, we first combine the vibron and the non-interacting wire, and then perform the EOM expansion on the coupling of QD and the wire. In this case, the model is simplified as the coupling of multi-channel leads and a QD system with simple energy levels. That is to say, the electrons jump to the QD system from the chemical potentials and then to the wire.

2.3 The Current and the Spectral function

In this section, we derive the standard EVI current formula through the SMT.

2.3.1 Discrete Spectrum: The Anderson-Holstein Model

The electron transport between the leads and the central region is considered as shown in Fig. 15. In our study only one level of the dot system is considered, and electrons vibrate at a single frequency ω0. The EVI system is studied theoretically through a non-perturbative canonical transformation H = e^SHe^−S with S = (λ/ω0) d⁺d (b⁺− b) (the detailed deriva-tion is shown in Appendix A). Under this unitary transformaderiva-tion, the Hamiltonian now reads:

HT = Hcen+ Hlead+ HT,

Hcen≡ ε0(Vg) d⁺d + ω0b⁺b, (12) Hlead= 

kα,α∈L,R

εkαc⁺_kαckα, (13)

Ht= 

kα,α∈L,R

Vkαc⁺_kαdX + h.c, (14)

where

X ≡ exp



− λ ω0

b⁺+ b

, (15)

The operators d⁺(d) and c⁺_kα(ckα) represent the creation (annihilation) operators of electron in the QD (or SMT) and the α lead, respectively. The operator b⁺(b) is the creation (annihilation) operator of the vibron. ε ≡ ε (V ) − ∆ is the dot level energy controlled

by the gate voltage, with the canonical energy shift ∆ = λ²/ω0. The coupling strength of EVI is denoted by λ and the tunneling matrix element between the QD (or SMT) and the α lead is defined as Vkα. Here εkα is the energy of the electron in α lead, which remains unchanged because of the absence of vibron field in the α lead.

2.3.2 The Jauho’s Current Formula (from the Perspective of the Quantum Dot)

The current from the left lead to the central region can be defined[11] as JL(t) = 2e

Re 

k,α∈L

Vkα,dG^<_d,kα t, t^′

|t^′→t (16)

where the Green’s function G^<_d,kα(t, t^′) ≡ i

c⁺_kα(t^′) X (t^′) d (t)

together with its conjugate property are applied above. Note that the electron-hole interaction between the metal leads and the dot is mutual, both terminals are vibrating from the perspective of the quantum dot, hence c⁺_kα and X evolves at the same time. Gd,kα(τ , τ^′) can be solved by

Gd,kα τ , τ^′

=

c

dt1Gdd(t, t1) V_kα,d^∗ gkα t1, t^′

(17)

Performing the continuation rules[11] on Gd,kα(τ , τ^′) and substituting the resulting G^<_d,kα(t, t^′) into Eq.(16), we obtain:

JL(t) = −2e

Im


t

−∞

dt1G^r_dd(t, t1) Σ^<_αǫL(t1, t) + G^<_dd(t, t1) Σ^a_αǫL(t1, t)

, (18)

The main task is to calculate the self-energy Σα. 2.3.3 Self-Energy

In frequency space, the appliance of the LT on the self-consistent Dyson equation of the elec-tron Green’s function in the QD (or SMT)[14][16] leads to G^r,a_dd (ω) =

ω − ε0− Σ^r,a_T (ω)−1

,

where the contour-ordered self-energy ΣT induced by the tunneling process reads ΣT(ω) = F⁺(ω) ΣT (ω), they carry all the same time information on the lead electrons and vibrons.

( )

^{, '}

Figure 16: The diagrammatic representation of the perturbative expansion of the electron Green’s function, where the electron-phonon interaction is depicted by the wave curve; the solid line denotes for the fermion line.

In Eq.(18), the partial part of the tunneling self-energy in the current formula can be found as g^<(>)_kα (t − t^′) is the lesser (greater) Green’s function for the free electron in the α lead. The Fourier transform of the self-energies in Eq.(19) lead to:

Σ^r,a_α,n(ω) = the quantum dot system without the EVI. The factor pn denotes the weighting func-tion of the interacfunc-tions between the electron and n vibrons, which is found as pn = e^−2g(^N0+12)e^nω0/2kBTIn

2g

N0(N0+ 1) [16], where N0 and In are the Bose function and the modified Bessel function.

Following Eq.(18), we obtain the EVI current formula, where the first term in the RHS of Eq.(18) explains the in-tunneling current Jin, where electrons entering the single-energy-level quantum dot system through the non-interacting wire with multi-channel. The second term in the RHS in Eq.(18) describes the out-tunneling current Jout, where the electron in the central region tunnels out of the central region via two channels, the transmission scheme is depicted in Fig.17.

Figure 17: Jauho’s transport picture. Reprinted with from Ref.[11]

Note here that the source wire emits vibrons when electrons transfer from one end to the other in a non-equilibrium system at low temperature. In addition, the quantum dot will couple with the multi-channel leads and forms step bandwidth (decay), which decides not only the lifetime of electron in the system but also the probability distribution. We have so far derived the diagrammatic formula and the Jauho’s transport formula. Now we shall analyze the spectral function.

2.3.4 The Spectra Function Ad(ω)

From the definition of the spectral function Ad(ω) = −2 Im G^r_dd(ω), we obtain:

Ad(ω) =



n,α[pnΓαf_α^<(ω + nω0) + p−nΓαf_α^>(ω + nω0)]

(ω − ε⁰)²+ [W (ω) /2]² , (23) where ε0 = ε0(Vg) − Re Σ^rT (ω) denotes the renormalized level position, and W (ω) =

−2 Im Σ^rT(ω) represents the life-time broadening (bandwidth) of the dot state. A compari-son with the conventional JWM’s formula in Ref.[14] assures that the life-time broadening

of a dot state W (ω) in Eq.(23) equals to the summation of out-tunneling rates between the leads and the system, i.e. Eq.(21); Eq.(18) is therefore self-consistent and meaning-ful. Note that G^<_dd(ω) in Eq.(18) can be quickly solved via using the Keldysh equation

of a dot state W (ω) in Eq.(23) equals to the summation of out-tunneling rates between the leads and the system, i.e. Eq.(21); Eq.(18) is therefore self-consistent and meaning-ful. Note that G^<_dd(ω) in Eq.(18) can be quickly solved via using the Keldysh equation

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