4.2 Tight-binding approach
4.2.1 Results
We illustrate our method by calculating the WTDs for a number of scatterers.
First, we consider a QPC as introduced in Sec. 2.1.3. In the following examples we investigate a single-level quantum dot as well as a DQD. The two quantum dots can be arranged with the levels either in series or in parallel. In the latter case a magnetic ux can be enclosed. We also consider a bipartite chain, where a gap opens in the transmission spectrum as the length increases. We compare our numerical results to methods based on scattering theory [54, 55].
The Hamiltonian consists of three parts,
Hˆ = ˆH0+ ˆHS + ˆHT. (4.19) The leads are described byHˆ0 given in Eq. (2.7), whileHˆS characterizes the scatterer andHˆT the connection between the scatterer and the leads. If not otherwise stated, the latter is given by the expression
HˆT =−tL|1, Lih1| −tR|1, RihMs|+h.c., (4.20) which connects the rightmost (leftmost) site of the left (right) lead to the leftmost (rightmost) site of the scatterer with tunneling amplitudetL (tR).
Quantum point contact
The QPC is modeled as a link with hopping amplitude tQPC between the leads (see Sec. 2.1.3). We then have HˆS = 0 and HˆT = ˆHQPC. To begin with the two leads are unconnected, and we prepare the left lead with N0 =M/3 particles in the linear region of the dispersion relation, recalling that M is the number of sites in each lead. Specically, we ll the states with energies in the interval [−V /2, V /2], where V = 2¯t. This value of V is chosen as a trade-o between, on the one hand, staying within the linear region of the dispersion relation and, on the other hand, having a large energy window which reduces the computation time.
To ensure the universality of our results, we average over at least three calculations with dierent starting timest0 ∈[t1, t2], wheret1andt2 are the start and end times of the quasi-stationary regime as indicated in Fig. 2.2a. In addition,t0 must be chosen such that the WTD approaches zero before t2 is reached. The length of the quasi-stationary regime depends on the size of the leads. Typically we have t1 ≈ M/3τ0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Figure 4.2: WTD for a QPC and a single-level quantum dot in units of the mean waiting timehτi. Numerical results are shown as symbols and results obtained from scattering theory as solid lines. a) For the QPC the WTD shows a cross-over from Wigner-Dyson statistics at full transmission (blue curve) to Poissonian statistics with an approximately exponential WTD (green curve) for decreasing tunneling ampli-tudes tQPC. b) The WTD for a single-level quantum dot is shown for varying level positionsεD and oset vertically for clarity.
and t2 . M τ0, where τ0 =~/t. The WTD is obtained for discrete times¯ τm =mτ0, 0≤m ≤M, due to the discretization of the leads.
In Fig. 4.2a we show WTDs for three dierent values of the tunneling amplitude tQPC. We have rescaled the horizontal axis by the mean waiting time hτi. In the linear part of the dispersion relation, the size of the energy window V determines the mean waiting timeτ¯=h/V of the particles in the incoming many-particle state as shown in Ref. [54]. We have checked that the results in Fig. 4.2a do not depend on the value of V as long as the transport window remains in the linear part of the dispersion. The suppression of the WTDs at short times reects the fermionic statistics of the particles, which prevents two particles from being detected at the same time.
The calculations in Ref. [54] are based on scattering theory with semi-innite leads connected to the scatterer. One important prediction is that the WTD for a QPC should exhibit a cross-over from a Wigner-Dyson distribution at full trans-mission to Poisson statistics close to pinch-o. This is conrmed by our numerical results. Remarkably, at low transmissions the tight-binding results reproduce the small oscillatory features in the WTD with periodτ¯, also found in Ref. [54].
To make the comparison with scattering theory quantitative, we calculate the scattering amplitudes of our setup and evaluate the WTD using the method devel-oped in Ref. [54] (see also App. B.1). We see that our tight-binding calculations of the WTD in Fig. 4.2a are in excellent agreement with scattering theory using the transmission amplitude given in Eq. (2.33).
0.0 0.5 1.0 1.5 2.0 2.5
Figure 4.3: WTD for a serial DQD. Numerical results are shown as symbols and results obtained from scattering theory as solid lines. a) WTD for varying inter-dot coupling tD for both levels in the middle of the band. b) WTD for dierent level separations ε=|εL−εR|. The levels are shifted symmetrically with respect to zero such thatεL =−εR.
Single-level quantum dot
As our next application, we consider a single-level quantum dot. The quantum dot level is denoted as |Di and the corresponding energy is εD. In this case, the Hamiltonian of the scatterer reads
HˆS =D|DihD|, (4.21)
and the tunneling Hamiltonian takes on the form given by Eq. (4.20).
Figure 4.2b shows the WTD for a quantum dot with varying level position εD. Again our results agree very well with those obtained from scattering theory. The transmission amplitude is found following the procedure described in App. B.2 and reads
tk= 2itLtRsink
(εk−εD)¯t+ (t2L+t2R)eik. (4.22) By moving εD away from the center of the energy window, the overall transmission is lowered, and the peak of the WTD shifts to larger times. Because of particle-hole symmetry the results do not depend on the sign ofεD.
Serial double quantum dot
We now consider a system consisting of two single-level quantum dots in series.
The left (right) level |Li (|Ri) at energy εL (εR) is coupled to the left (right) lead and the levels are connected by the inter-dot tunnel coupling tD. The Hamiltonian of the scatterer reads
HˆS =εL|LihL|+εR|RihR| −tD(|LihR|+h.c.), (4.23)
and the tunneling Hamiltonian is given by Eq. (4.20).
In Fig. 4.3a we show the WTD with dierent inter-dot couplings tD and equal energy levelsεL =εR = 0. FortD > tL, tR, the curves exhibit an oscillatory behavior.
AstD is decreased, the oscillations are damped, and the WTD is shifted toward larger times.
To understand this behavior, we note that the dierence between the eigenener-gies of the Hamiltonian in Eq. (4.23) is
∆ε= q
ε2+ 4t2D, (4.24)
having dened ε=|εL−εR|. The energy splitting gives rise to coherent oscillations in the WTD with frequency ωosc = ∆ε (see also Sec. 4.3). Another interpretation is that when tD > tR, a particle is likely to oscillate back and forth between the left and right levels before exiting to the right lead.
To further corroborate this picture, we consider in Fig. 4.3b the WTDs with an increasing detuning of the levels. As expected from Eq. (4.24), the frequency increases as the two levels are dealigned. The decay of the WTD at long times is controlled by the tunneling rate to the right lead.
For the transmission amplitude, we nd in this case tk = 2i¯ttLtDtRsink
Q
α=L,R[(εk−εα)¯te−ik+t2α] +t2D¯t2e−2ik. (4.25) Calculations based on scattering theory are in good agreement with our tight-binding calculations as illustrated in Fig. 4.3.
Double quantum dot enclosing a magnetic ux
We now place the two quantum dots in parallel such that each one of them is coupled to both leads. In this setup, shown schematically in Fig. 4.4, a magnetic ux can be enclosed, inducing a variable phase for dierent paths through the sys-tem. The two levels with energies ε1(2) are coupled to the left (right) lead by the tunnel couplings tLi (tRi), i = 1,2. In addition, there is a direct link with tunnel-ing amplitude ∆ between the levels. The magnetic ux Φ through the central area causes a charge carrier to acquire a phase factor e±iφ/4 during each hopping event where φ = 2π(Φ/Φ0) and Φ0 = h/e is the magnetic ux quantum [153]. The plus (minus) sign in the exponential applies if the tunneling event occurs in the clockwise (counterclockwise) direction around Φ.
Note that a parallel double dot provides two independent transport channels when coupled to macroscopic leads such that the suppression of the WTD atτ = 0 would be lifted. However, in our one-dimensional description it remains a one channel problem.
The system Hamiltonian now reads
HˆS =ε1|1ih1|+ε2|2ih2| −∆(|1ih2|+|2ih1|). (4.26)
tL1 tR1
tR2 tL2
Φ ε1
ε2
∆
¯t
¯t t¯
¯t
Figure 4.4: Schematic sketch of a DQD enclosing a magnetic ux. A blue (red) arrow implies a phase change of eiφ/4 (e−iφ/4), where φ = 2π(Φ/Φ0) is given by the magnetic uxΦ. Direct tunneling between the dots is phase-neutral (green arrow).
In addition, the tunneling Hamiltonian is HˆT = −tL1 eiφ/4|1ihML, L|+h.c.
−tL2 e−iφ/4|2ihML, L|+h.c.
−tR1 e−iφ/4|1ih1, R|+h.c.
−tR2 eiφ/4|2ih1, R|+h.c.
. (4.27) The transmission amplitude is again obtained by the method described in App. B.2.
An explicit expression can be found in publication (III).
In Fig. 4.5a we show WTDs for three dierent phase shifts: φ = 0, π/2, and π, without direct tunneling between the quantum dots, ∆ = 0. By varying the phase, we modify the interference between the two paths leading from the left to the right lead, changing it from being constructive (φ = 0) to being destructive (φ = π).
A particle coming from the left lead propagates through both quantum dots and interferes with itself in the right lead. For φ = 0, the interference is constructive and particles may perform coherent oscillations as seen in the WTD. At φ=π, the interference is maximally destructive and particle transfers through the DQD become increasingly rare. However, because the two paths have dierent amplitudes (since ε1 6=ε2), the transmission remains non-zero. The reduced transmission decreases the oscillations in the WTD as it approaches an exponential distribution corresponding to a Poisson process.
This picture changes qualitatively with a nite tunneling amplitude between the quantum dots, ∆ 6= 0, see Fig. 4.5b. Several paths through the systems are now possible so that the interference blockade is lifted and coherent oscillations are re-stored. In both cases, our tight-binding calculations are in excellent agreement with scattering theory.
Bipartite chain
As a last example we consider transport through a bipartite chain of variable length. This could be a simple model of an extended molecule suspended between
0.0 0.5 1.0 1.5 2.0
Figure 4.5: WTD for a parallel DQD enclosing a magnetic ux. Tight-binding calculations are shown as symbols, results based on scattering theory as solid lines.
a) Increasing the magnetic uxes from 0 toΦ0/2suppresses transport and shifts the WTD to larger times. b) This suppression can be lifted by allowing for tunneling between the dots.
two leads [159, 160]. The system Hamiltonian
HˆS =−
describesMD dimers consisting of two sites that are coupled by the tunneling ampli-tudev. Each dimer is in addition coupled to its neighbors with tunneling amplitude w < v and the outermost sites are connected to the leads by the tunneling Hamilto-nian in Eq. (4.20).
The transmission amplitudes are obtained numerically for dierent lengths of the chain using the method described in App. B.2. In Fig. 4.6a we show the energy-dependent transmission probability obtained for dierent values of MD. The trans-mission shows two bands around±v with a gap aroundε= 0 that becomes increas-ingly pronounced as the length of the chain 2MD is increased. Adding a dimer to the chain increases the number of peaks in the lower and upper bands by one. For MD → ∞ the peaks become dense within the regions ±v−w ≤ ε ≤ ±v +w, and the transmission peaks become rectangular as shown by the black curve.
In Fig. 4.6b we show results for the WTDs for dierent lengths of the chain. In the case MD = 1, we recover the result for a serial double quantum dot with ε = 0 andtD =v. Interestingly, as more dimers are added, the WTDs eventually converge to a universal curve (shown with a dashed line), which is independent ofMD.
0.6 0.4 0.2 0.0 0.2 0.4 0.6
Figure 4.6: a) Transmission probability for a bipartite chain of length 2MD. As the number of dimers MD increases, the gap around ε = 0 becomes clearly dened and the two bands around ±v become rectangular as indicated with a black line.
b) WTD for a bipartite chain of length2MD. Tight-binding calculations are shown as symbols, results based on scattering theory as solid lines. The curves are oset vertically for clarity. As the length of the chain is increased, the results converge towards the universal curve shown with a dashed line. This curve is obtained from scattering theory, taking a very long chain where the gap in the spectrum is fully developed.
Summary
We have presented a method for obtaining the WTD for non-interacting fermions on a nite tight-binding chain. As applications, we have calculated the WTDs for a QPC and several dierent quantum dot structures. Our approach reproduces the Wigner-Dyson distribution expected for a fully transmitting QPC, and it agrees well with predictions based on scattering theory at transmissions below unity. In addition, we can associate oscillations in the WTDs to internal energy scales of the quantum dot structures. For quantum dots in series, the oscillations are clearly related to the energy splitting of the hybridized states. For quantum dot structures enclosing a magnetic ux, we nd that the WTD carries signatures of the interference between dierent traversal paths. Finally, for a bipartite chain, the WTDs converge towards a universal curve as the length of the chain is increased. The agreement with existing approaches is an important check of our method. In particular, it raises the hope that similar tight-binding calculations may be generalized to include interactions.
It would also be interesting to investigate the WTDs for tight-binding chains with periodic drivings in the spirit of Refs. [53, 134, 154, 155, 161].
4.3 Master equation approach
In this section we consider the master equation approach to electronic WTDs [52]. We focus on the unidirectional transport regime, which is reached when the bias voltage is the largest energy in the system. We begin by introducing the approach and compare it to the tight-binding results of the previous section, and continue by considering various detector models and their eect on the WTD.
For an unidirectional process the transformed Liouvillian M(λ) of the trans-formed ME, Eq. (2.46), can be written as
M(λ) =L(0)+eiλJ. (4.29) Here J = L(1) describes transfer processes from the system to the drain electrode, andL0captures both the unitary evolution of the system as well as transfer processes from the source electrode to the system.
To nd the WTD, we use that in the stationary state the ITP is given by Π(τ) =P(0, τ) = Tr[ˆρ(0)S (τ)]. (4.30) From Eq. (2.45) it follows that
ˆ
ρ(0)S (τ) = ˆρS(λ→i∞, τ), (4.31) sincen≥0. We then obtain the ITP from the MGF in the limitλ→i∞, where the counting factoreiλ vanishes,
Π(τ) = χ(λ→i∞, τ) = Tr
eM(λ→i∞)τρˆstatS
=Tr
eL0τρˆstatS
. (4.32) The mean waiting time follows from Eq. (4.31) as
hτi=− 1
Π(τ˙ = 0) =− 1
Tr[L0ρˆstatS ] = 1
Tr[JρˆstatS ], (4.33) where we have used that L0ρˆstatS = (L − J)ˆρstatS = −JρˆstatS because of Eq. (2.42).
With Eq. (4.4) the WTD then reads [52]
W(τ) = Tr
JeL0τJρˆstatS
Tr[JρˆstatS ] , (4.34) which follows from Tr[L0•] = Tr[(L − J)•] = −Tr[J •], since Tr[L•] = 0 due to probability conservation. This result implies that the WTD vanishes at τ = 0 for only one transport channel since then J2 = 0.
Sometimes it is useful to consider the WTD in Laplace space rather than in the time domain in order to obtain tractable expressions,
Wf(z) = If an analytical inverse Laplace transformation is not feasible it can be done with the help of numerical methods [162].
0.0 0.5 1.0 1.5 2.0
Figure 4.7: a) Single-level quantum dot at energyεD coupled to left and right reser-voirs with transition ratesΓLandΓR. In the unidirectional regime, particles can only enter from the left reservoir and exit towards the right one. b) Comparison of the WTDs obtained by the master equation and tight-binding approach for a single-level quantum dot. For small tunneling amplitudes,tL, tR¯t, and the level in the center of the transport window the two approaches show good agreement.
Single level quantum dot
As a rst illustration of the method, we consider a single-level quantum dot coupled to two leads as shown in Fig. 4.7a. Electrons enter the level from the left electrode at a rate ΓL and leave to the right with a rate ΓR. In principle, the level can be occupied by two electrons with opposite spin. However, we assume that the Coulomb interaction between them is so strong that the doubly occupied state can be neglected. The electron spin can then be omitted.
The density matrix of the dot is diagonal and has two non-zero elements: ρˆ0
(empty level) andρˆ1(full level). Combining the elements into a vector,ρˆS = (ˆρ0,ρˆ1)T, the transformed Liouvillian takes the matrix form
M(λ) =
−ΓL eiλΓR
ΓL −ΓR
. (4.36)
From Eq. (4.34) the WTD then follows as W(τ) = ΓRΓL
ΓR−ΓL
e−ΓLτ −e−ΓRτ
=Γ2τ e−Γτ, ΓL= ΓR= Γ,
(4.37)
which in Laplace space corresponds to
Wf(z) = ΓLΓR
(z+ ΓL)(z+ ΓR). (4.38)
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Figure 4.8: a) Serial double quantum dot coupled to left and right reservoirs by transition rates ΓL and ΓR. The levels εL and εR are coupled by the rate Ω. In the unidirectional case, particles can only enter from the left reservoir and leave to the right, while internal transitions are possible in both directions. b) WTD for a serial double quantum dot with and without Coulomb blockade. In addition the result of the tight-binding (TB) approach is shown.
We see that the WTD grows asτ for short times whereas in the scattering approach it goes asτ2 [54]. This is due to the assumption that the bias voltage is much larger than the energy-scales of the system. The mean separation time between incoming electrons τ¯=h/eV is thus very short, such that after each transfer process another particle is immediately available.
This result can be compared to our numerical results for the single level quantum dot in the tight-binding approach (Sec. 4.2.1). To this end we have to relate the tun-neling ratesΓL,ΓRin the master equation description with the tunneling amplitudes tL, tR in the tight-binding model. This is done in App. B.3 where we show that
Γα ≈ 4|tα|2 vF
, α=L, R, (4.39)
wherevF = 2¯t.
In Fig. 4.7b we compare the results obtained by the two approaches. In order to get a good agreement the tunneling amplitudes have to be small compared to the bias voltage in the tight-binding model, tL, tR 2¯t. Note that the level energy εD
does not enter in Eq. (4.37) because it is assumed far from the Fermi levels of either of the leads. Accordingly, we get the best agreement with the tight-binding approach when the level is in the center of the transport window, εD = 0.
Serial double dot
In a two-level system, in addition to the levels being occupied individually, quan-tum mechanical superpositions between them can occur. Here we investigate what
eect this has on the WTD for a DQD formed by two single-level quantum dots in series. The system is shown in Fig. 4.8. Electrons enter from the left electrode at a rate ΓL and exit towards the right one with the rate ΓR. The coupling between the dots isΩ and the energy separation between the levels is ε=εL−εR.
As for the single dot, we assume strong Coulomb interactions on the levels such that only single occupation is possible and the spin of the electrons is irrelevant. In addition, Coulomb interaction between particles residing on dierent levels has to be taken into account. As treating this interaction is complicated in general, we only consider two limiting cases here. In a rst instance we assume the interaction to be zero so that both levels can be occupied independently. The density matrix then has six non-zero elements, that give the probability to nd both levels empty, the left or right dot occupied, both dots occupied, as well as two o-diagonal elements corresponding to the bonding and anti-bonding superposition of the left and right level. We arrange these elements in a vector
ˆ
ρS = (ˆρ0,ρˆL,ρˆR,ρˆ2,ρˆLR,ρˆRL)T, (4.40) keeping in mind that the normalization reads ρˆ0+ ˆρL+ ˆρR+ ˆρ2 = 1.
The Liouville superoperator is derived from a microscopic description of the sys-tem and reads in our case [163]
M(λ) = On the other hand, in the strongly interacting limit, inter-dot Coulomb interac-tions are assumed to be so strong that both levels cannot be occupied at the same time. We then haveρˆ2 = 0 and the transformed Liouvillian reduces to
MCB(λ) =
from which the WTD in Laplace space follows as WfCB(z) = ΓLΓR(2z+ ΓR)Ω2
(z+ ΓL){z(z+ ΓR) [(z+ ΓR/2)2+ε2] + 4Ω2(z+ ΓR/2)2}. (4.44) In Fig. 4.8b we show the WTD for both cases in the time domain. As in the tight-binding approach, we observe oscillations with frequency∆ =√
4Ω2+ε2/~, which is the Rabi frequency of the isolated two-level system. This behavior can be understood by noticing that in the limit of vanishing tunneling rates (ΓL,ΓR → 0), W(z) has imaginary poles at z = ±i∆, corresponding to coherent oscillations between the
4Ω2+ε2/~, which is the Rabi frequency of the isolated two-level system. This behavior can be understood by noticing that in the limit of vanishing tunneling rates (ΓL,ΓR → 0), W(z) has imaginary poles at z = ±i∆, corresponding to coherent oscillations between the