Master equation approach

Master equations have numerous applications describing stochastic processes in physics and beyond [86, 87]. In quantum transport in particular, the master equation approach has been very successful in modeling electron transport in low-transmission structures like quantum dots [88].

The typical setup consists of a small quantum system coupled to two particle reservoirs. The master equation is the equation of motion for the system's density matrix ρˆS which is obtained by tracing out the degrees of freedom of the reservoirs.

In the Markovian limit, the time scales considered are larger than the relaxation time of the reservoirs so that memory eects can be neglected. The master equation is then local in time and given by the Lindblad equation [86]

d

dtρˆS =Lρˆ=i[ ˆH,ρˆS] +X

k

Γk

ˆ

ckρˆSˆc^†_k− 1 2

ρˆSˆc^†_kcˆk+ ˆc^†_kcˆkρˆS

. (2.39)

The rst term on the right hand side describes the unitary evolution of the system, whereas the second term captures dissipative processes involving the reservoirs. The Liouville superoperatorLhas been introduced, andˆck,cˆ^†_kare the Lindblad operators.

If, on the other hand, memory eects cannot be neglected, one needs to go beyond the Markov approximation and take into account the history ofρˆS. This is the case considered in Sec. 4.4, where the coupling of an electronic system to a bosonic bath induces non-Markovian correlations.

2.2.1 Full counting statistics

To calculate the FCS, we resolve the density matrix with respect to the number n of particles that have been transferred from the system to one of the particle reservoirs during the time interval[0, t], [89, 90]. For negativen, more particles have come into the system from the reservoir than have gone out.

The n-resolved density matrix ρˆ⁽ⁿ⁾_S obeys the master equation dρˆ⁽ⁿ⁾_S (t)

dt = X∞ n⁰=−∞

L⁽ⁿ⁻ⁿ⁰⁾ρˆ⁽ⁿ_S⁰⁾(t), (2.40) with the initial condition

ˆ

ρ⁽ⁿ⁾_S (0) = ˆρ^stat_S δ_n,0. (2.41) Hereρˆ^stat_S is the stationary solution to Eq. (2.39),

Lρˆ^stat_S = 0. (2.42)

Choosing the initial condition Eq. (2.41) amounts to assuming that the system, after having been prepared in some initial state in the past, has relaxed to the stationary state before counting starts at t = 0. L⁽ⁿ⁾ describes all processes that transfer n

particles from the system to the reservoir under consideration, and the full Liouvillian Here we have introduced the transformed density matrix,

ˆ It obeys the transformed master equation

dρˆS(λ, t)

where the transformed Liouvillian reads M(λ) = X such that the MGF becomes

χ(λ, t) = Tr

e^M(λ)tρˆ^stat_S

. (2.49)

From this result the mean particle number follows as hni= ∂

where we have used that ˆ

ρS(λ= 0, t) = e^M(0)tρˆ^stat_S =e^Ltρˆ^stat_S = ˆρ^stat_S , (2.51) due to the denition of the stationary solution. We see thathni is linear in time so that the mean current hIi=hn˙i is constant as it should be in the stationary state.

A common approximation is the sequential tunneling limit, which is valid if the coupling between the reservoirs and the system is so weak that processes with more than one particle tunneling at the same time can be neglected. The transformed Liouvillian then reads

M(λ) = L⁽⁰⁾+e^−iλL⁽⁻¹⁾+e^iλL⁽¹⁾, (2.52) and for the current one has

hIi=Tr

We now discuss the example of sequential tunneling through a metallic island coupled to two leads, taken from Ref. [91]. Other studies of the problem can be found in Refs. [92] and [93]. A metallic island is coupled to a left (L) and right (R) lead by rates Γ^α_N→N±1, α=L, R, as shown in Fig. 2.3a. It is considered big enough that the average level spacing is negligible and the density of states is continuous.

Electrons on the island are conned in all three dimensions so that the Coulomb interaction between them cannot be neglected. A state with N electrons on the island has an energy of

EN = e²

HereC is the sum of all capacitances between the island and its surroundings, and Q0 is a charge oset that can usually be controlled by applying a gate voltage.

In addition, we have dened the charging energy EC = e²/2C. For an additional electron to tunnel onto the island, the energy

EN+1−EN =EC has to be paid. This energy can be provided by the bias voltage V that is applied between the leads. However, if the island is occupied byN electrons andeV is smaller than the energy dierence Eq. (2.55), no current can ow through the system at zero temperature. This phenomenon is called Coulomb blockade.

To describe this behavior quantitatively, we consider a master equation approach.

For sequential tunneling, the master equation describes transitions in whichN changes by one,

˙

pN =−(ΓN→N+1+ ΓN→N−1)pN + Γ_N−1→NpN−1+ ΓN+1→NpN+1. (2.56) HerepN is the probability to ndN electrons on the island. Changes inN can occur by processes involving each of the leads so that we have,

Γ_N→N±1 = Γ^L_N→N±1+ Γ^R_N→N±1. (2.57)

N Γ

Γ

Γ

Γ

V

C

, R

C

, R

a) b)

Figure 2.3: Model of a metallic island coupled to two leads. a) Particles tunnel between the island and lead α = L, R with the tunneling rate Γ^α_N→N±1, changing the particle numberN on the island. b) Electric circuit model. The tunnel junction between leadα and the island consists of a capacitanceC_α in parallel to the tunnel resistance Rα. The island is located between the junctions and the circuit is biased by a voltageV.

Equation (2.56) is a classical or Pauli master equation because the density matrix, ˆ

ρ= diag (. . . , pN−1, pN, pN+1. . .), (2.58) is diagonal so that no quantum-mechanical superpositions occur. In fact, Eq. (2.56) describes a classical stochastic process, and the only quantum mechanical property involved is the fact that transitions between the island and the leads happen through tunneling processes. In contrast, master equations that include quantum-mechanical superpositions between states are also called generalized master equations.

We are interested in the stationary solution, p˙N = 0 for all N. In this case, the relation

ΓN→N+1pN = ΓN+1→NpN+1 (2.59) follows from Eq. (2.56), which means that the ow between probabilities is constant.

Together with the normalization Tr[ˆρ] =P

NpN = 1, the stationary solution of the problem is determined.

The leads are held at dierent chemical potentials µ_α such that a bias voltage V = (µL−µR)/e is applied across the island. In the sequential tunneling limit the transition rates are obtained from Fermi's golden rule,

Γ^α_N→N±1 = 2π X

fN±1,iN

D

fN±1

Hˆ_T^αiNE²Wi_Nδ εf_N±1 −εi_N

. (2.60)

Here the sum goes over all initial and nal states iN and fN±1 with N and N ±1 particles on the island, respectively. WiN is the probability of stateiN to occur, and

the delta function provides for energy conservation. The transfer HamiltonianHˆ_T^α is

We assume energy-independent tunneling rates, a constant density of states in the leads and the island, as well as local thermal equilibrium in each of the leads and on the island. The transition rates then read

Γ^α_N→N±1 = 1

and β = (kT)⁻¹ is the inverse temperature. Here Rα is the tunnel resistance of junctionα =L, R.

The chemical potentialµD of the island is determined by the properties of the left and right tunnel junctions. To this end, tunneling is neglected, and the junctions are considered as capacitances CL and CR. The voltages VL and VR, dropping between the left lead and the island and between the island and the right lead, are then given by

We can now calculate the transition rates as a function of the bias voltage V, given that the resistances and capacitances of the junctions are known. The current through the system is then given by the current through the left tunnel junction,

IR(V) = eX

N

Γ^R_N→N−1(V)−Γ^R_N→N+1(V)

pN, (2.65)

or alternatively by IL(V) = −IR(V) because of current conservation. Note that Eq. (2.65) has the form of Eq. (2.53).

The outlined approach can explain a rich class of phenomena such as Coulomb diamonds and the I-V characteristics of single-electron transistors [94]. In Ch. 5 we apply it to the conductance spectrum of a Pb island grown on a Si substrate covered by 3 monolayers of NaCl. The spectrum was obtained by scanning tunneling spectroscopy measurements with the STM tip and the substrate constituting the leads. The capacitances as well as the residual charge were inferred from steps and peaks in the conductance spectrum as explained in Ref. [95], while the tunnel resistances were given by the overall current amplitude.

The measurement was part of a broader investigation in which Pb islands were separated from the substrate by dierent barriers. For the other samples the tunnel

V Z

(ω)

C

, R

Figure 2.4: DCB circuit. A tunnel junction with capacitanceCT and resistance RT

in series with an impedance Z(ω) is biased by the dc voltage V.

resistance between the island and the substrate was of the order of RK = h/e² ' 26 kΩ, such that the sequential tunneling approximation does not apply. Those spectra were modeled by the DCB eect instead, which we introduce in the following section.