Transport in serial spinful multiple-dot systems: The role of electron-electron interactions and coherences

Quantum dots are nanoscopic systems, where carriers are confined in all three spatial directions. Such nanoscopic systems are suitable for fundamental studies of quantum mechanics and are candidates for applications such as quantum information processing. It was also proposed that linear arrangements of quantum dots could be used as quantum cascade laser. In this work we study the impact of electron-electron interactions on transport in a spinful serial triple quantum dot system weakly coupled to two leads. We find that due to electron-electron scattering processes the transport is enabled beyond the common single-particle transmission channels. This shows that the scenario in the serial quantum dots intrinsically deviates from layered structures such as quantum cascade lasers, where the presence of well-defined single-particle resonances between neighboring levels are crucial for device operation. Additionally, we check the validity of the Pauli master equation by comparing it with the first-order von Neumann approach. Here we demonstrate that coherences are of relevance if the energy spacing of the eigenstates is smaller than the lead transition rate multiplied by ħ.

][23][24][25] Electron transport through these structures [26][27][28][29] has been widely considered for level spectroscopy, where a variety of interesting physical phenomena could be observed 4,[30][31][32] .Generally, one assumes, that specific resonances between energy levels dominate the transport through quantum dot systems.The central idea is the resonance between neighboring levels, which provides specific conditions for transport.In double-dot structures, this implies specific resonances, so that current flow occurs only for specific points if the energy levels of the individual dots are modified by appropriate gates 1 .For multiple dot systems, this concept would require a path of states in resonances connecting both contacts, which is unlikely to happen.In contrast, energy relaxation by scattering is needed.An example for such a scattering are Auger processes as discussed in Ref. 33.However even in this case, distinct resonance conditions are required, which prevents current flow in a general case, where the levels are not fully controlled, but depend on tiny details such as the location of charged impurities.
Electron-electron interaction is naturally occurring in all electronic devices and affects transport both by scattering (such as the Auger term) and level shifts.For systems with many degrees of freedom, such as bulk or layered systems, the continuum of states justifies in many cases an effective meanfield description, so that one can work with effective singleparticle levels with renormalized energies.In this case the resonances occur for different parameters, but the essential principle remains.A very successful example for this concept are quantum cascade lasers, whose operation is based on a clever design for the alignment of such single-particle levels 34 .For quantum-dot systems, however, any mean-field model is very questionable, as one replaces the interaction between quantized charges by the interaction of a charge with an averaged quantity.
The reproducibility of serial quantum dot structures is limited because of sensitivity to growth conditions.Experimentally it is a challenge when one requires similar dots with same level structures coupled in series as it is in an ideal model system.Gate electrodes can be used to control size and shape variations in quantum dots by adjusting the energy levels.However, it is a rather difficult task to fabricate quantum dots with intrinsically digital fidelity with small fluctuations in their size, shape and arrangement. 35,36 this work the relevance of different parts of the electronelectron interaction is analyzed for coupled quantum dots.The focus is mainly the effect of these different parts of the interaction term on aspects of electron transport through quantum dot devices.We show that the Coulomb interaction between electrons opens up a large variety of different channels, which go far beyond simple pictures.We identify two main causes: (i) Coulomb scattering provides a generic possibility for energy relaxation within the dot system.(ii) The multitude of many-particle states provides an enormous amount of different possible current paths.Our findings demonstrate that this provides significant deviations a simple picture of few resonances.

II. SYSTEM
The system under consideration is schematically shown in Fig. 1.It consists of three serial quantum dots sandwiched between metallic contacts.This can be realized by a specific gate structure confining a two dimensional electron gas 23,25,37 , carbon nanotubes, 38 or nanowires with heterostructures. 39It also serves as a model system for longer arrangements such as quantum dot superlattices 40 or possible dot-based QCLs. 41nly the energy levels of the right/left dot are directly connected to the contacts with a continuum of levels.For an applied bias, standard considerations of coherent transport, such as the transmission formalism, predict current flow, if there is a state connecting both leads.Neglecting weak cotunneling effects, this is only the case, if three levels align.For the situation in Fig. 1 one would thus not expect any elastic transport  I.
through the structure.Neglecting phonon scattering, the current would thus be zero, even if the energies E 3 and E 4 are varied.However in Ref. 42 it was shown, that for this situation, electron-electron scattering enables a current flow.If the levels 1 and 4 are occupied, an energy-conserving scattering event creates the simultaneous transitions 1→2 and 4→3 (for , with subsequent tunneling to the right contact via the state 5.The inset of Fig. 1 shows the current as a function of left Fermi level for the described system.As it is shown the current is decreasing drastically when the left Fermi level comes close to the excited state of the first dot: Then the level E 2 becomes occupied by an electron, and consequently the Auger scattering process is prevented due to the Pauli blocking.This is a distinct feature of the Auger driven transport in multiple dots, which discriminates it from other possible scattering processes. The magnitude of parameters used in our calculations relate to a particular nanostructure which is made from a nanowire containing three InAs wells (thickness 40 nm) embedded between InP barriers (thickness 1.5 nm).Similar structures have been recently fabricated 39,43,44 .However, it is not the aim to model a specific structure here, but to demonstrate a general property of transport in multiple quantum dot systems.

A. Hamiltonian
The system Hamiltonian consists of two parts: where ĤL−T describes the Hamiltonian of the contacting leads and their coupling to the dots.
where n labels single electron dot states as depicted in Fig. 1, σ denotes the electron spin, and E klσ is the single particle energy of the lead = L/R in state k. a nσ and c k σ are the annihilation operators for states in the quantum dots and the leads, respectively.The Hamiltonian of the dots is given by where Ω nm describes the coupling between states n and m in neighboring dots, which are estimated by a standard tightbinding superlattice model, see Ref. 45.
is the standard Coulomb part of the Hamiltonian.For a system that has more than one confined electron, this part plays an important role and we will discuss the different matrix elements V mnkl in the following.

B. Terms of the ee-interaction
In general the Coulomb matrix elements read where ϕ * m (r) is the spatial part of the single particle state m.ε r = 15 (for InAs) and ε 0 are the relative and vacuum permittivity, respectively.In the following we neglect all terms, where either m and l or k and n belong to different wells, as their overlap would be vanishingly small.Furthermore, terms connecting levels of next-nearest neighboring wells are small and neglected as well.The remaining terms can be categorized into Intradot and Interdot interactions and are separately treated below.For the estimation of Coulomb matrix elements we apply wavefunctions confined to boxes in the individual dots.

Intradot Interaction
For intradot interaction all the levels mnkl are considered to be in the same dot.By employing the normalization condition for the wave function, the direct elements can be estimated as: where d is the typical distance within the dot and for this specific structure that we assumed here d ≈ 10 nm.Another set of interaction matrix elements that has to be taken into account are V mnmn , which act as exchange terms for equal spins and scattering terms for different spins.A numerical calculations based on box-wavefunctions provides V mnmn = U ex ≈ 2 meV.

Interdot Interaction
The direct interaction between two states in the neighboring dot can be written in the same way as Eq. ( 6).where d ≈ 40 nm is the approximate distance between the centers of the dots.The terms with different combinations of indices, are estimated by a Taylor expansion of 1 |r−r | around the centers of the respective dots R l , R m , see Ref. 46.We find Here s mn is the dipole matrix element which is found to be s 21 = −8 nm for our model wave functions.
The above equation can be interpreted as a dipole-charge interaction for U dc and a dipole-dipole scattering term for U sc .The latter one is responsible for Auger process addressed above and crucial for current flow in our system.Thus it is taken into account in all simulations.
The values for U dc and U sc , have been also calculated numerically based on box-wavefunctions with less than 10% difference from the estimates given here.We actually used rounded values for all quantities in order to allow an easy recognition of scales in the plots.The specific values are displayed in Table I.

C. Transport Calculation
We diagonalize the Hamiltonian (3) ĤD and obtain the many-particle states |a , |b , . . . of the triple dot.Transitions between these states with different particle numbers are possible by electrons entering or leaving towards the leads, which is modeled by a standard Pauli Master equation following Refs.47-49.Using the notation of Ref. 50, the transition rates are given by where T ba (kσ l) is the tunneling matrix element in the basis of the many-body states.Considering these transition rates the master equation for the probability P b of finding the system in state |b can be written as, We determine the stationary state of this equation and finally obtain the current entering from the left lead which is the main output presented in the paper.

III. RESULTS
In the following we show calculations for varying levels E 3 and E 4 .This is motivated as follows: Even in a nominally identical triple-dot system, growth imperfections, charged impurities and the electrostatic environment will modify the energy levels in a way which is difficult to predict.We fix the excitation energy in the left dot, E 2 − E 1 , as a reference point and the (upper) level E 5 of the right dot can be shifted by a source drain bias.However, the remaining levels E 3 and E 4 are less controllable, albeit there is a possibility of gating.Thus it is a central question of practical interest, for which range of parameters E 3 , E 4 current flow occurs.
If we neglect the Coulomb interaction at all, there is only a very small current unless three levels align in a row (e.g. for E 1 ≈ E 4 ≈ E 5 , a case we never consider below).Thus, scattering-assisted processes dominate, which can occur due to the Coulomb scattering matrix element V 2341 42 .This provides an Auger process, where one electron relaxes from level 4 to level 3, while transferring its energy to an electron being excited from level 1 to level 2. Therefore U sc = −0.2meV is used in all calculations below, while the impact of the other Coulomb terms are neglected in some calculations in order to demonstrate their relevance.

A. Restricting to one spin direction
At first we restrict to one spin direction.Fig. 2 shows the current as a function of E 3 and E 4 , when all the interaction matrix elements except the scattering elements are zero.
Only for specific values of E 3 and E 4 the electrons are able to pass through the system.This reflects the conservation of energy both in the electron-electron scattering and for tunneling, E 3 = E 5 = 20 meV and E 4 = E 1 = 40 meV.Otherwise, transport is blocked.Thus any slight changes in the geometry of the dots and the configuration of energy levels would prevent from current flow.On the other hand such a selective situation would be optimal for devices, which rely on well defined transitions, such as quantum cascade lasers.On the other hand by considering the full e-e interaction with all the interaction matrix elements, a much richer scenario for the current is observed, as shown in Fig. 3.The multiple peak structure provides current flow for a wider range of parameters.This scenario is reflected by the addition energies, where a particle can tunnel into or out of the triple dot.In Fig. 4 we display the differences in energy between all possible two and three-particle states (values on x-axis).In order to specify their relevance for single-particle transitions, we plot the respective transition probabilities for electrons to enter from either contact, while changing the state of the system between these states.For the case restricting to Coulomb scattering, there are only three distinct energies, where lead electrons can enter, which correspond to the energies of the levels 1, 2, and 5.In contrast, a larger variety of excitation energies is relevant if the full Coulomb interaction is taken into account.This is fully consistent with the differences between FIG. 4. Addition energies (horizontal) for the two-particle states in the spin-polarized system.The vertical axis shows the respective coupling strength for electronic transitions from either contact.In the left panels a point is drawn for each transition.In order to resolve multiple transitions, the right panel sums Lorentzians with full width at half maximum of 5 meV with a peak given by the points in the left panel.
the conduction plots of Figs 2 and Fig. 3.

B. Transport of electrons with two spin directions
In general, due to the Kramer's degeneracy theorem 51 , each energy level is doubly degenerate in the absence of an external magnetic field.Let us first consider the case where all the interaction matrix elements except the scattering elements are zero.In a simple picture, one could expect, that the current doubles compared to the case with only one spin direction (Fig. 2).The main panel of Fig. 5 shows indeed a strong increase of the current.
However, a much more prominent increase is observed for a variety of parameter combinations where new transport channels open up.
In order to understand the reason of the different behavior in Fig. 5, we study the eigenstates of the system for two points in the parameter range as indicated by the squares in current plot.The upper arrangement is for E 3 =19 meV and E 4 =30 meV, where a significant current is observed for the two-spin case, but current is blocked for the spin polarized case.
The five one-particle eigenstates of the Hamiltonian are illustrated in the green diagram in Fig. 5.There is a strong coupling between E 1 and E 4 as well as between E 3 and E 5 thus the superposition of each of these two states creates two new states with two new energies, which are referred to as E 1,4 , E 4,1 , E 3,5 and E 5,3 (the left side of the Fig. 5).For the upper arrangement the conservation of energy is satisfied, (E 2 −E 1,4 ) ≈ (E 1,4 −E 3,5 ), which allows for Coulomb scattering.However the state E 1,4 appears as the initial state in both parts for an Auger process.Thus the Pauli principle can only be satisfied if both spin directions are allowed.Thus for this configuration of the energy levels, in the system with two different spin direction, the electrons are able to transfer through the triple dot although the current is blocked in a spin polar- FIG. 6.Current through the two spin direction system and all the possible interactions in the system.ized system.Spin is definitely more than a factor of two here!
The lower arrangement in the left side of the Fig. 5 corresponds to E 3 =15 meV and E 4 =30 meV.Here no such resonance occurs and thus current flow is neither observed in Fig. 5 nor in Fig. 2. Moreover for the system with all the interaction matrix elements, we find a much richer scenario for the current as shown in Fig. 6.Here current is flowing for a wider range of parameters.This follows the trend seen for the single spin case.However, the number of possible single particle excitations is further enhanced.

IV. DISCUSSION AND CONCLUSION
The main results are collected in Fig. 7, which displays Figs.2,3,5,6 in logarithmic scale.For the single spin case restricting to scattering interaction a single peak is seen, which is easily predicted by standard single particle states.Such resonances are well known and are commonly used for device design.However taking into account two spin directions and all interactions, provides an entirely different picture, where current flow is spread over a wide range of parameters where unexpected paths become of relevance.In this article we related this observation to the multitude of many-body states as well as the enhanced possibilities for electron-scattering if the spin degree of freedom allows for double occupancy.As a consequence, designing devices based on multiple quantum dots is questionable, if one restricts to single-particle models even if the mean-field is included.On the other-hand, the multitude of channels makes such devices less sensitive to size fluctuations and random charges.

FIG. 1 .
FIG. 1. Sketch of the triple dot structure under biasing conditions.For symmetric structures a sixth level in the third dot below E 5 is expected.It is not considered here, as it hardly contributes to the current flow.Inset: Current as a function of the left Fermi level restricting to one spin direction taking only into account the Coulomb scattering term as in Ref. 42. Parameters as in TableI.

FIG. 2 .
FIG. 2. Current through the spin polarized system using U = U ex = U n = U dc = 0 displaying a single distinct peak where E 3 ≈ E 5 and E 4 ≈ E 1 .

FIG. 5 .
FIG. 5. Main panel: Current through the system with two spin directions using U = U ex = U n = U dc = 0. Left panel: Energy spectrum for the one-particle states at two different parameters of E 3 and E 4 indicated by the squares in the main figure.

FIG. 7 .
FIG. 7. Current in logarithmic color scale, where red/blue is highest/lowest current.The current is in units of: (a) 4 nA (b) 0.35 nA (c) 12 nA (d) 0.5 nA.

TABLE I .
Parameters used in this work if not mentioned otherwise.They refer to an InAs-InP nanowire structure.All energies are in meV.