Dissipative double quantum dot - Non-Markovian systems

4.4 Non-Markovian systems

4.4.1 Dissipative double quantum dot

As a concrete application of our method, we consider charge transport through a DQD, where dephasing occurs due to coupling to a heat bath. Markovian dephasing in the weak-coupling limit has been discussed in Ref. [52]. Here we take these ideas further and examine the transition from Markovian to non-Markovian dephasing.

The approach bears similarities to the theory of DCB outlined in Sec. 2.3.

The system is shown in Fig. 4.12. The corresponding Hamiltonian reads

Hˆ = ˆHS+ ˆHT + ˆHL+ ˆHSB+ ˆHB. (4.69) Here HˆS = (ε/2)( ˆd^†_LdˆL−dˆ^†_RdˆR) + Ω( ˆd^†_LdˆR+ ˆd^†_RdˆL) (4.70) describes the left and right levels of the DQD with dealignment ε and coupling Ω. Tunneling between left (right) level and left (right) lead is accounted for by

HˆT = X

k,α=L,R

tk,αcˆ^†_k,αdˆα+ h.c., (4.71) and the Hamiltonian for the left (L) and right (R) lead is

HˆL = X

k,α=L,R

k,αˆc^†_k,αˆck,α. (4.72) The Markovian kernels in Eqs. (4.41) and (4.43) are derived from these three com-ponents of the Hamiltonian [165].

Non-Markovian eects are induced by the heat bath, which couples to the DQD via HˆSB = ( ˆd^†_LdˆL−dˆ^†_RdˆR)X

(gj/2)(ˆa^†_j + ˆaj), (4.73) 62

while

HˆB =X

~ωjˆa^†_jaˆj (4.74) describes the heat bath as an ensemble of harmonic oscillators with frequenciesωj.

Equation (4.69) corresponds to an open spin-boson problem, where the role of the spin is played by the two single-particle levels of the DQD that are coupled to the bath of bosons. Charges enter and leave the pseudo-spin states from the voltage-biased electrodes.

We derive a non-Markovian master equation for the populations of the DQD by tracing out the electronic leads and the heat bath. Here we present key points of the derivation that can be found in [176] for strong inter-dot Coulomb interaction.

Instead, we consider the opposite limit of negligible inter-dot interactions.

The equation of motion for the reduced density matrix ˆ

σ = (ˆσ00,σˆLL,σˆRR,σˆDD,σˆLR,ˆσRL)^T (4.75) of the system and the heat bath reads

dtσ(t) =ˆ Lσ(t)ˆ −ih

HˆB+ ˆHSB,ˆσ(t)i

, (4.76)

where L is given by Eq. (4.41) for λ = 0. Tracing out the bath degrees of freedom produces the term

Im{TrB[ˆσLR(t)]}. (4.77) It can be evaluated using the formal solution for σˆ_LR(t),

σLR(t) =iΩ Z t

dt⁰e⁻^(iε+Γ)(t⁻^t⁰⁾e⁻ⁱ^H^ˆ^B⁽⁺⁾^(t⁻^t⁰⁾[ˆσLL(t⁰)−ˆσRR(t⁰)]eⁱ^H^ˆ⁽⁻⁾^B ^(t⁻^t⁰⁾ +e^(−iε+Γ)te⁻^H^ˆ⁽⁺⁾^B ^tˆσ_LR(0)e^H^ˆ^B⁽⁻⁾^t,

(4.78)

recalling that Γ = (ΓL+ ΓR)/2. Here Hˆ_B⁽^±⁾= ˆHB±VˆB, where Vˆ_B =X

(g_j/2)(ˆa^†_j+ ˆa_j) (4.79) is the bath part ofHˆSB.

We assume that the bath reaches a local equilibrium between tunneling events, ˆ

σLL(t)'ρˆL(t)⊗σˆ_B⁽⁺⁾, ˆ

σRR(t)'ρˆR(t)⊗σˆ_B⁽⁻⁾, (4.80) with two distinct equilibrium states depending on whether the left or right level is occupied,

σ_B⁽^±⁾ = e^−β^H^ˆ^B^(±)

TrB[e^−β^H^ˆ^B^(±)]. (4.81)

Hereβ = (kT)⁻¹ is the inverse bath temperature and the approximation is valid to lowest order in Ω².

By eliminating, σˆLR and σˆRL we nd a non-Markovian master equation of type Eq. (4.69) for the reduced density matrix of the system,

ρS(n, t) = [ˆρ0(n, t),ρˆL(n, t),ρˆR(n, t),ρˆD(n, t)]^T. (4.82) ˆ

ρS(n, t) contains the probabilities for the DQD to be empty, the left or right dot occupied, or to be doubly occupied. The kernel of the master equation for the transformed density matrix Eq. (4.59) reads

fW(λ, z) = are given by the bath-correlation functions

g_±(t) =D

eⁱ^H^ˆ^B⁽⁻⁾^te⁻ⁱ^H^ˆ^B⁽⁺⁾^tE

(±), (4.85)

which are evaluated with respect toσˆ_B⁽^±⁾.

Equation (4.85) is a correlation function of the same type as Eq. (2.76) and can be evaluated by the same methods [183]. We write it as

g_±(t) =e^−F⁽^∓^t), (4.86) where the exponent is given by

F(t) =

HereJ(ω) is the spectral function of the heat bath, J(ω) = X

|gj|²δ(ω−ωj). (4.88) In fact, Eq. (4.87) is the same as Eq. (2.77), where the role of the spectral function is played by

ωReZ(ω) RK

. (4.89)

Here we consider an ohmic bath with a spectral density given by

J(ω) = 2αωe^−ω/ω^c, (4.90)

whereαis the strength of the coupling to the DQD andωcis a high-frequency cut-o that necessary in order to ensure the convergence of Eq. (4.87). We then have [184]

F(t) =αln(1 + (ωct)²) + 2αln β

πtsinh πt

+ 2αiarctan(ωct), (4.91) which provides us with all the information necessary to evaluate the WTD.

Using Eq. (4.65), we nd the WTD in Laplace space as Wf(z) = ΓLΓR(z+ ΓL+ ΓR)²eΓ+(z)

(z+ ΓL)(z+ ΓR)(ΓL+ ΓR)[zeΓ₋(z) +{z+ ΓL+ ΓR}{z+eΓ+(z)}]. (4.92) This expression reduces to Eq. (4.42) when the bath and the system are decoupled, α= 0, since then the inter-dot tunneling rates read

eΓ_±(z) = 2Ω² Z _∞

dte⁻^ztRe

e⁻^(iε+Γ/2)t

= 2Ω²(z+ Γ)

(z+ Γ)²+ε². (4.93) We obtain results in the time domain by numerically performing an inverse Laplace transformation as described in Ref. [162].

Results

In Fig. 4.13a we plotW(τ)for strong inter-dot couplingΩΓL,ΓRfor a weakly coupled heat bath at temperatureT. In this regime coherent oscillations are observed in the WTD as explained in Sec. 4.3. The oscillations are gradually washed out as the temperature of the heat bath is increased, which leads to stronger dephasing of the electrons in the DQD. The eect of the heat bath is thus similar to that of a dephasing rate (see Sec. 4.3.1).

Next, we consider the WTD for variable coupling to the heat bath. In Fig. 4.13b, we take zero bath temperature and Ω = Γ_L = Γ_R. As the coupling is increased, the heat bath tends to localize electrons on the quantum dots and the inter-dot tunneling rate becomes suppressed. For large couplings, tunneling events are rare and uncorrelated, and the transport process essentially becomes Poissonian.

The DQD can be tuned to an interesting regime, where tunneling between the quantum dots becomes the rate-limiting step in the transport process. Choos-ing tunnel couplChoos-ing Ω or the dealignment of the quantum dot levels ε such that eΓ₋(z),Γe+(z)ΓL,ΓR, Eq. (4.92) can be approximated by

Wf(z)' Γe+(z)

z+Γe+(z). (4.94)

This result oers the possibility of directly probing spectral properties of the heat bath through the detection of the electron waiting time, since the bath correlation functions enter the bath-assisted hopping rate eΓ+(z).

0 1 2 3 4 5

Figure 4.13: Electronic waiting time distributions for a dissipative double quantum dot. a) Coherent oscillations in the WTD for weak couplings to the heat bath (α = 0.01) are washed out with increasing bath temperature due to dephasing of the electrons in the DQD. b) Beyond the weak-coupling limit the heat bath tends to localize electrons on the quantum dots. c) Electronic waiting time distributions for strongly dealigned levels. At low bath temperatures (blue lines), there is a strong asymmetry between positive and negative dealignments. The asymmetry diminishes at high bath temperatures (red lines) as the emission and absorption of energy from the bath becomes equally likely. Red and blue curves are obtained from Eq. (4.92), while the black curves follow from the approximate result Eq. (4.94).

In Fig. 4.13c we focus on the emission and absorption of energy to and from the heat bath as electrons tunnel from the left to the right quantum dot. At low tem-peratures, there is a clear asymmetry between the WTDs for positive and negative dealignments, since the heat bath mainly contributes to the transport for positive detunings by absorbing energy from tunneling electrons. At high temperatures, this asymmetry disappears as the heat bath in addition can assist the tunneling process at negative detunings through the emission of energy. Figure 4.13c shows that Eq. (4.94) provides an excellent approximation to the exact results based on Eq. (4.92).

Summary

We have presented a theory of electron waiting times for non-Markovian gener-alized master equations that generalizes earlier approaches to waiting time distri-butions in electronic transport. As an illustrative example, we considered electron transport through a double quantum dot, by which we examined non-Markovian dephasing mechanisms beyond the weak-coupling limit. We hope our method may pave the way for future investigations of memory eects and electron waiting times, similar to how full counting statistics and nite-frequency noise in non-Markovian quantum transport have been popular research topics in recent years.

Chapter 5 Dynamical Coulomb blockade in nano-sized electrical contacts

In this Chapter, we employ the DCB eect introduced in Sec. 2.3 as a means to probe and characterize the electrical contact between at metallic nano-scale is-lands and their supporting substrates. We consider measurements of the electri-cal conductance of individual Pb islands on metallic, semimetallic, semiconducting, and partially insulating substrates obtained by low-temperature scanning tunneling microscopy (STM). In these measurements a suppression of the dierential tunnel conductance is observed at low voltages. We attribute this observation to the DCB eect since the tunnel current between the STM tip and the islands is highly sensi-tive to the impedance of the electrical contact between the island and the supporting substrate. We are able to characterize the nano-sized electrical contacts and extract the resistances and capacitances of the islandsubstrate contacts from the measured dierential conductance spectra.

5.1 Experiment

The experiments were performed by Christophe Brun, I-Po Hong and François Patthey in Wolf-Dieter Schneider's group at EPFL Lausanne. Figure 5.1 shows STM images of Pb(111) islands grown on substrates of Cu(111), Si(111)-7×7, highly oriented pyrolytic graphite (HOPG), hexagonal (h-)BN/ Ni(111), and NaCl/Ag(111).

The island areas range from 10 nm² to 10⁴nm² with island heights between 2 and 60 monolayers (MLs). The substrate crystals were prepared according to standard procedures: The h-BN ML was epitaxially grown on Ni(111) [185]. The Si(111) crystal was heavily n-doped and prepared to form a Si(111)-7×7 reconstruction.

NaCl was thermally evaporated onto Ag(111) at substrate temperatures between 300 K and 500 K [186]. The at islands were grown by evaporation of Pb from a W lament onto the substrates whose temperatures were stabilized between 130 K and 300 K to control the island size. Pb was chosen as a metallic material because of the well-known growth of Pb lms or islands on Si(111) [187, 188], Cu(111) [189],

Pb/Si(111)-7x7 Pb/NaCl/Ag(111) b) e)

50 nm 50 nm

Pb/HOPG d)

50 nm

Pb/BN/Ni(111)

50 nm

c) Pb/Cu(111)

50 nm

Figure 5.1: STM topographic images of at Pb islands on dierent substrates.

a) metal Cu(111); island heights between 2 and 4 atomic monolayers (ML). b) semiconductor Si(111)-7x7; 4-7 ML. c) semi-metal HOPG; 7-10 ML. d) metal covered by one insulating ML, h-BN/Ni(111); 6-60 ML. e) metal covered by two insulating MLs, NaCl/Ag(111). The images were obtained with (from a to e) a voltage of 1.0 V, −1.0 V, −0.5 V, −1.0 V, and 3.0 V, and a corresponding tunnel current of 0.1 nA,−0.1 nA,−0.1 nA,−0.05nA, and 0.02 nA.

and HOPG [190]. On HOPG, h-BN/Ni(111), and NaCl, islands grow directly on top of the substrates, whereas on Si(111) and Cu(111) a 1 ML wetting layer forms rst followed by the growth of single-crystal Pb islands [191].

Conduction measurements were performed in a home-built STM with PtIr tips operated under ultra-high vacuum at a temperature of T = 4.6 K [192]. Focus was put exclusively on islands supported by just a single substrate terrace such that the electrical contact between the island and the substrate is essentially uniform across the whole contact area. The dierential conductance was measured in an open feedback loop using a lock-in technique with a peaktopeak modulation voltageVpp

between0.5 mV and5mV. The typical current was of the order of 1 nA and voltage in ranges between[−20,+20]mV and[−200,+200]mV were used. The conductance spectra are independent of the injected power in a range between10⁻¹¹ and 10⁻⁹ W as veried experimentally.

Figure 5.2 displays a selection of dierential conductance spectra measured on individual Pb islands of varying sizes on top of the dierent substrates. The ex-perimental data was corrected for a background contribution due to quantum-well states [193, 194], and rescaled by the tunneling resistance RT between the islands and the STM tip, such that the normalized spectra approach unity at large volt-ages. The temperature was below the critical temperature of bulk lead (Tc= 7.2 K) so that one would expect the Pb islands to display superconducting properties for voltages below the superconducting gap of bulk Pb, |eV|/2 < ∆ = 1.3 meV [193].

However, recent experiments have shown that both Tc and ∆ decrease below their bulk values for small systems [193, 195], and superconducting gap features give a negligible contribution to the conductance under the described experimental condi-tions. On larger voltage scales, the quasi-particle transport between the islands and their normal-state substrates is the same as for islands in their normal-state, as we

checked by performing experiments at temperatures aboveTc.

While Fig. 5.2a shows that the dierential conductance for the islands on Cu(111) remains essentially at independently of the island size, in Figs. 5.2b-e we observe a suppression of the dierential conductance at low voltages, which becomes in-creasingly prominent as the island size is decreased. The normalized dierential conductance is reduced below unity when the applied voltageeV is smaller than the charging energy EC =e²/CΣ, where CΣ is the total capacitance of the system. As the islands become smaller, both the suppression and the charging energy increase.

For the islands on Cu(111), the electrical contact has a very low resistance such that the spectrum is essentially ohmic. In this case the small features in the spectra are ascribed to the reduced electronphonon scattering of the quantum-well states below the Debye energy ED ' 10 meV of Pb [196, 193]. In Figs. 5.2e-f, results for Pb islands on Ag(111) covered with 2 and 3 MLs of insulating material (NaCl) between islands and substrates are shown. As several insulating MLs are introduced, the spectra begin to display qualitatively dierent features, as is explained below.

Dans le document Quantum transport in nano-scale conductors: entanglement entropy, waiting time distributions, and dynamical Coulomb blockade (Page 75-84)