Resonant tunneling through linear arrays of quantum dots

Resonant tunneling through linear arrays of quantum dots

M. R. Wegewijs and Yu. V. Nazarov

Faculty of Applied Sciences and Delft Institute of Microelectronics and Submicrontechnology (DIMES), Delft University of Technology, 2628 CJ Delft, The Netherlands

共Received 5 June 1998; revised manuscript received 7 May 1999兲

We theoretically investigate resonant tunneling through a linear array of quantum dots with subsequent tunnel coupling. We consider two limiting cases: 共i兲 the strong Coulomb blockade, where only one extra electron can be present in the array;共ii兲 the limit of almost noninteracting electrons. We develop a density-matrix description that incorporates the coupling of the dots to reservoirs. We analyze in detail the dependence of the stationary current on the electron energies, tunnel matrix elements and rates, and on the number of dots. We describe interaction and localization effects on the resonant current. We analyze the applicability of the approximation of independent conduction channels. We find that this approximation is not valid when at least one of the tunnel rates to the leads is comparable to the energy splitting of the states in the array. In this case the interference of conduction processes through different channels suppresses the current.

关S0163-1829共99兲04040-0兴

I. INTRODUCTION

In recent years, arrays of quantum dots have received an increasing amount of interest. With the progress of fabrica-tion techniques, quantum dot arrays are coming within the reach of experimental investigation.1If the electron levels in the individual dots are aligned, we encounter here a situation of resonant tunneling. In this regime, the transport in the array becomes sensitive to precise matching of the electron levels in the dots that can be controlled by external gates. This opens up new possibilities to control the transport and perform sensitive measurements even in the simplest case of two dots.2

Resonant tunneling in arrays of quantum dots and layered semiconductor heterostructures exhibit some similarities. The latter situation has been intensively studied in the con-text of possible Bloch oscillations.3 However, the Coulomb blockade dominates the properties of the arrays of quantum dots so that electron-electron interaction cannot be neglected as in the case of layered heterostructures.4,5 A way to cir-cumvent this difficulty is to perform an exact diagonalization of electron states in the array of coupled dots. Then one considers independent tunneling transitions between the re-sulting many-electron states.6 This we call the independent channel approximation. This is approximate because it disre-gards the simultaneous tunneling of an electron through mul-tiple conduction channels. Another approach is to restrict the basis to the resonant states of the uncoupled dots. Then the tunneling between the dots and the reservoirs is incorporated into a modified Liouville equation for the density matrix in this basis. For two quantum dots this has been done in Ref. 7 and here we extend this approach to the case of an array of an arbitrary number of dots.

In this paper we concentrate on an array of quantum dots where dots are connected in series and a tunnel coupling exists only between neighboring dots. This is the most inter-esting case because there is a unique path for the current and changes in any dot strongly affect the transport through the whole array. The first and last dot of the array are connected

to leads. We assume that the voltage bias is sufficiently high so that the energy change during the tunneling of an electron between a reservoir and the array is much larger than the energy uncertainty due to this tunneling. We also assume that the resonant electronic energies in the array lie well between the Fermi levels of the leads. This enables us to use the density-matrix approach. We consider two limiting cases of the electron-electron interaction within the array. In the first case we assume that the long-range Coulomb repulsion between electrons in different dots of the array is so strong that only one or no extra electrons are present in the array 共Coulomb blockade兲. This is to be contrasted with the case of ‘‘free’’ electrons. As we explain below共Sec. II兲, we do not disregard interactions completely in the latter case but rather account only for strong repulsion within each dot.

Using the density-matrix approach in the basis of local-ized states, we have obtained analytical results for the sta-tionary current. Our results hold for arbitrary values of the parameters共within the applicability of our model兲 character-izing the arraylike dot energies and tunnel couplings: no as-sumption about homogeneity of the array has been made. This may facilitate the comparison with experiments and the design of resonant tunneling devices. We report the effects of localization and Coulomb repulsion on the resonant current when the energy level of the first and the last dot are inde-pendently varied. We have also considered another picture of the transport using the approximation of independent con-duction channels in the array of dots. Using the density-matrix approach in the basis of delocalized states, we have calculated the occupations of the channels and their contri-butions to the current. We discuss in detail the range of va-lidity of this approximation. In the limit of both weak and strong coupling to one or both of the leads, we obtain results in agreement with the former more general calculations. However, there can be substantial deviations from the pre-dictions of this model when the tunnel rates and the coherent interdot couplings are comparable. To illustrate this, we study the dependence of the current on the transparencies of the tunnel barriers and find unusual features due to the

inter-PRB 60

ference of electrons during tunneling.

The outline of the paper is as follows. In Sec. II we intro-duce the density-matrix description of a multidot system coupled to leads which we apply to the Coulomb blockade case in Sec. III and to the ‘‘free’’ electron case in Sec. IV. In Sec. V we compare the results with those obtained from the independent channel approximation and we discuss the de-viations. We formulate our conclusions in Sec. VI.

II. ARRAY OF QUANTUM DOTS COUPLED TO RESERVOIRS

Let us first consider an array of N quantum dots共the ‘‘de-vice’’兲 without any contacts 共Fig. 1兲. We consider a quantum dot as some complex system with discrete many-electron states of which only two ground states participate in resonant transport. The one state is related to the other by the addition of an extra electron costing an addition energy which in-cludes intradot charging, i.e., the Coulomb interactions be-tween electrons in the dot. 共For simplicity, we disregard the electron’s spin degrees of freedom.8兲 We label the resonant states of dot i⫽1, . . . ,N by the number of extra electrons

n_i⫽0,1 and introduce fermionic operators aˆ_i†,aˆ_i which cre-ate and annihilcre-ate, respectively, an extra electron in dot i. The many-electron eigenstates of the array of uncoupled dots will be denoted by兩n1•••nN

典

⫽兩兵nk其

典

⫽兿k⫽1

N _aˆ k

†n_k

兩兵0其

典

. Let ␧i denote the energy for adding an electron to dot i and let u_{兩i⫺ j兩}⭓0 be the interdot charging energy due to the repulsive

Coulomb interaction between the pair of extra electrons in dots i⫽ j. A coherent interdot coupling with matrix element

ti⫽ti* accounts for the tunneling of electrons between dots i

and i⫹1. We obtain a Hubbard-type Hamiltonian Hˆ⫽Hˆ0 ⫹Hˆu for the array of coupled dots:

Hˆ0⫽

兺

i⫽1 N ␧iaˆi †_aˆ i⫹

兺

i⫽1 N_⫺1 ti共aˆi⫹1 † _aˆ i⫹aˆi †_aˆ i_⫹1兲, 共1a兲 Hˆ_u⫽

兺

i⬍ j⫽2 N

u_{兩i⫺ j兩}aˆ_i†aˆ_iaˆ_j†aˆ_j. 共1b兲 The density operator of the device ␴ˆ evolves according to the Liouville equation ⳵t␴ˆ⫽⫺i关Hˆ,␴ˆ兴 (ប⬅1) which

con-serves probability, i.e., Tr␴ˆ (t)⫽1. By expanding ␴ˆ in the many-electron eigenstates of the uncoupled dots, we obtain a 2N⫻2N density matrix␴⫽(␴_兵n_k其,兵n k ⬘其): ␴ˆ_⫽

兺

兵n_k其,兵n_k⬘其 ␴兵n_k其,兵n k ⬘其兩兵nk其

典具

兵nk

⬘

其兩.

We additionally introduce an N⫻N Hermitian matrix␳with expectation values of single-electron operators (i, j ⫽1, . . . ,N): ␳i j⫽

具

aˆj † aˆi

典

⫽Tr aˆj † aˆi␴ˆ .

Using the fermionic commutation relations, we find

␳ii⫽

兺

兵n_k⫽i其␴n1•••1i•••nN ,n₁•••1 i•••nN , 共2a兲 ␳i j⫽

兺

兵n_{k⫽i, j}其 共⫺1兲 n_i⫹1⫹•••⫹n_j⫺1 ⫻␴n₁•••1 i•••0j•••nN ,n₁•••0 i•••1j•••nN , 共2b兲

where 兵n_{k⫽i, j}其 indicates that we sum over n_k⫽0,1 for all dots k⫽1, . . . ,N except i, j. We will refer to␳as the aver-age occupation matrix with respect to electrons in individual dots whereas␴is the probability-density matrix with respect to many-electron states of the array.

Now we include leads L and R connected to the first and last dot of the device by a tunnel barrier, respectively 共Fig. 1兲. The leads are considered here as electron reservoirs at zero temperature with a continuum of states filled up to their Fermi levels ␮L and␮R, respectively. We assume that the

energy levels of the device are located well between the chemical potentials of the leads, i.e., ␮LⰇ␧i,tiⰇ␮R 共large

bias兲, and that the level widths are much smaller than the bias, i.e., ␮_L⫺␮_RⰇ⌫_L,R 共discrete states兲. Under these as-sumptions an evolution equation for the density matrix ␴of the device can be derived9,10by incorporating the details of the lead states into tunnel rates. Due to the high bias, elec-trons only tunnel through the barrier from lead L to dot 1 with a rate ⌫L⫽2␲DL兩tL兩2 and from dot N to lead R with

rate ⌫R⫽2␲DR兩tR兩2. We assume that the density of states DL,R in leads is constant and that the tunneling matrix

ele-ments tL,R between lead and dot states depend only weakly

on the energy. Due to the destructive interference of an elec-tron tunneling between a discrete state in the device and the continuum of states in a reservoir, the rate for tunneling in one direction (L→1,N→R) is constant in time whereas the rate of the reversed process (L←1,N←R) is zero.11 In gen-eral, transitions between discrete many-electron states a,b of a device with Hamiltonian Hˆ induced by tunneling to and from reservoirs can be included in a modification of the Liouville equation:10 ⳵t␴ab⫽⫺i关Hˆ,␴ˆ兴ab⫺ 1 2

冉

兺

_a⬘ ⌫a→a⬘⫹

兺

_b⬘ ⌫b→b⬘

冊

␴ab ⫹

兺

a⬘b⬘ ⌫ab←a⬘b⬘␴a⬘b⬘. 共3兲 FIG. 1. Linear array of N quantum dots coupled to leads L and

R. The energy levels of the uncoupled dots are given by full lines when relevant for resonant transport and dashed lines when irrel-evant.

The first term of the modification describes the separate de-cay of states a→a

⬘

and b→b

⬘

due to共in general兲 different tunneling events between the device and the reservoir with rates ⌫a→a⬘ and⌫b→b⬘, respectively. The second term

de-scribes the joint generation of states a←a

⬘

and b←b

⬘

due to a single tunneling event occurring with rate ⌫_{ab←a⬘b⬘}: the coherence lost by the simultaneous decay of states a

⬘

,b

⬘

is transferred to states a,b. When we assume that there is a unique path for the current, then each state a is generated from a unique state a

⬘

by the tunneling of an electron to or from a reservoir: ⌫_{aa←a⬘b⬘}⫽⌫_a←a⬘␦_a⬘b⬘. The modified Liouville equation conserves probability: summing the equa-tions for the diagonal elements gives⳵_tTr␴ˆ (t)⫽0, so

兺

兵n_k其 ␴兵nk其,兵nk其⫽1. 共4兲

We consider the following two cases for resonant trans-port through the array: 共i兲 The ‘‘free’’-electron 共F兲 case where interdot Coulomb repulsion is negligible, i.e., u_{兩i⫺ j兩} ⬅0; up to N electrons can populate the array and all 2N

many-electron states lie between the chemical potentials ␮L

Ⰷ␮R and participate in resonant transport.共ii兲 The Coulomb

blockade共CB兲 case where the interdot Coulomb repulsion is so strong that the smallest charging energy is large relative to the bias, i.e., u_N⫺1Ⰷ␮_L⫺␮_R; many-electron states with more than one electron are highly improbable and can be neglected for resonant transport. We point out that in general the presence of electrons in the array modifies the rates for tunneling to or from the leads by Coulomb repulsion.9 How-ever, in the limiting cases considered here there is either no repulsion共F兲 or no other electron present in the array 共CB兲 so two parameters ⌫L and⌫R suffice to incorporate the

de-tails of the electronic states in the respective leads. The cur-rent flowing from dot N to reservoir R is determined by the average occupation and the tunnel rate:

IN共t兲

e ⫽⌫R␳NN共t兲. 共5兲

With the leads included, the dynamics of the average occu-pation matrix ␳ should be calculated from the full density matrix␴which evolves according to an equation of the type 共3兲. However, in both cases considered here we can derive a dynamical equation for the matrix ␳ which is solved more easily.

III. CURRENT IN THE COULOMB BLOCKADE CASE When the long-range Coulomb repulsion is so strong that at most one electron can be present in the array of dots, we can restrict the set of many-electron basis states to

兩0•••0

典

,兩0•••1

i

•••0

典

, i⫽1, . . . ,N.

The average occupations of individual dots are simply equal to the nonzero probability densities of these states and

␳i j⫽␴0•••1

i•••0,0•••1j•••0 .

Conservation of probability共4兲 suggests that we additionally define an average occupation of the many-electron vacuum state:

␳00⫽␴0•••0,0•••0.

This quantity is positive 0⭐␳00⭐1 and satisfies a conserva-tion law:

␳00⫹

兺

k⫽1 N

␳kk⫽1. 共6兲

In the restricted basis, the matrix elements of the Hamil-tonian Hˆ which describe the coherent part of the evolution of the state only involve Hˆ0 关Eq. 共1a兲兴. Modifying the Liouville equation according to Eq. 共3兲, we obtain a dynamical equa-tion for the average occupaequa-tion matrix,

⳵t␳00⫽⫺⌫L␳00⫹⌫R␳NN, 共7a兲

⳵t␳ii⫽i共ti_⫺1␳ii_⫺1⫹ti␳ii_⫹1⫺ti_⫺1␳i_⫺1i⫺ti␳i_⫹1i兲

⫹⌫L␳00␦i1⫺⌫R␳NN␦iN, 共7b兲 ⳵t␳i j⫽i共␧j⫺␧i兲␳i j⫹i共tj⫺1␳i j⫺1 ⫹tj␳i j⫹1⫺ti⫺1␳i⫺1 j⫺ti␳i⫹1 j兲⫺ 1 2⌫R␳iN␦jN, 共7c兲 where j⬎i⫽1, . . . ,N and t0⫽tN⬅0. On the right-hand side

of Eq. 共7a兲 the negative contribution describes the decay of the vacuum state due to the tunneling of an electron from lead L to dot 1 with rate⌫Lwhereas the positive contribution

describes the generation of this state due to the tunneling of the共only兲 electron in the device from dot N to lead R. In Eq. 共7c兲 there is only a negative contribution due to the tunneling of an electron out of dot N to reservoir R with rate⌫R. There

is no negative contribution with ⌫L because we have

incor-porated the Coulomb blockade: we disregard the decay of many-electron states with one electron to a state with two electrons which occurs when an electron tunnels from lead L

into dot 1. We can eliminate␳₀₀from Eqs.共7兲 using Eq. 共6兲 and obtain N2 equations for the average occupations.

Equations共7兲 can be used to describe nonstationary trans-port with a typical relaxation time scale ⌫_L,R⫺1. Here we are interested in the stationary limit⳵t␳⫽0 only. In general, the

solution of Eq. 共7兲 can be obtained by inverting a matrix of dimension N2. However, since the system under consider-ation is an array with subsequent tunnel coupling, most of the equations only relate matrix elements of ␳on neighbor-ing dots. One can obtain the solution by iteratively express-ing all matrix elements in terms of ␳00 and finally making use of the normalization constraint共6兲. The resulting station-ary current in general reads

I_NCB e ⫽ 1 1 ⌫L ⫹_⌫1 R F_N⫹ ⌫R 4t_N2_⫺1FN⫺1 . 共8兲

Here the dimensionless expression FN depends only on

␧1, . . . ,␧N through the differences␧i j⬅␧i⫺␧j,i⬍ j and on t1, . . . ,tN⫺1. Note the curious property of Eq.共8兲: FNenters

the expressions for both IN and IN⫹1. This helps when

de-riving expressions for the current through an array with an increasing number of dots. For N⫽2 we reproduce the result of Stoof and Nazarov:12

I₂CB e ⫽ 1 1 ⌫L⫹ 1 ⌫R

冋

2⫹

冉

␧12 t1

冊

2

册

⫹⌫R 4t₁2 . 共9兲

For N⫽3 we obtain for the current through a triple dot sys-tem, I₃CB e ⫽ 1 1 ⌫L⫹ 1 ⌫R

冉

3⫹␧12␧13⫹2t1 2_⫺t 2 2 t1t2 ␧12␧13⫹t1 2_⫺t 2 2 t1t2 ⫹ ␧12⫹2␧23 t2 ␧12⫹␧23 t2

冊

⫹ ⌫R 4t₂2

冉

2⫹ ␧12 t1

冊

. 共10兲

One can show FN(␧1, . . . ,␧N,t1, . . . ,tN_⫺1)⫽N for ␧i j

Ⰶti⫽t. This property will be used later on and it is derived

in another way in Sec. V A.

We consider Eq.共8兲 in more detail for equal interdot cou-plings t_i⫽t and several different configurations of the levels ␧i. At resonance␧i j⫽0 we find ␳ii⫽ 1 ⌫R⫹ ⌫R 4t2共1⫺␦iN兲 1 ⌫L⫹ 1 ⌫R N⫹⌫R 4t2共N⫺1兲 and the current is maximal:

I_N,maxCB e ⫽ 1 1 ⌫L⫹ 1 ⌫R N⫹⌫R 4t2共N⫺1兲 . 共11兲

Clearly the current is reduced when the array increases in size: the number of states participating in transport relative to the number of states of the array decreases⬀1/N due to the Coulomb blockade. In an actual system we expect a positive deviation from this decrease to occur when the spatial size of the array exceeds some range over which the Coulomb re-pulsion cannot exclude the occupation of a second dot in the array. Away from resonance, i.e.,␧i j⫽O(␧)Ⰷt, the

conduc-tion of the device decays exponentially with the size of the array N due to the localization of the electron in one of the individual dots: I_NCB/e⬀⌫_R(␧/t)⫺2(N⫺1). To illustrate this we vary only the last level关Fig. 2共a兲兴, i.e., we consider the so-lution of Eq.共7兲 for ␧i⫽␧N␦iN:

I_NCB e ⫽ 1 1 ⌫L ⫹_⌫1 R

冋

N⫹共N⫺1兲

冉

␧N t

冊

2

册

⫹⌫R 4t2共N⫺1兲 .

As we increase the number of dots, the curve keeps its Lorentzian shape with respect to␧Nand always depends on

both t and⌫R since the electron is localized only just before

tunneling out of the array:

I_NCB I_N,maxCB ⫽ N→⬁ 1⫹

冉

⌫R 2t

冊

2 1⫹

冉

⌫R 2t

冊

2 ⫹

冉

␧N t

冊

2.

Now we vary only the first level关Fig. 2共b兲兴, i.e., we consider the solution of Eq. 共7兲 for ␧i⫽␧1␦i1:

I_NCB e ⫽ 1 1 ⌫L⫹ 1 ⌫R

再

N⫹

冋冉

␧1 t

冊

2 ⫹

冉

⌫_2tR

冊

2

册

k

兺

⫽1 N⫺1 k

冉

␧1 t

冊

2(N⫺1⫺k)

冎

. For large N the normalized current vanishes when the detun-ing exceeds the tunnel coupldetun-ing 共since an electron is local-ized in the first dot兲 whereas near resonance the peak takes on a parabolic shape which is independent of the tunnel rate ⌫R: I_NCB I_N,maxCB ⫽ N→⬁

再

1⫺

冉

␧1 t

冊

2 , ␧1/t⬍1 0, ␧1/t⬎1.

For the case where the energies are configured as a ‘‘Stark ladder’’ of total width␧, we have plotted the current in Fig. 3共b兲. The localization of electrons clearly dominates the cur-rent since the tails of the curcur-rent peak decrease rapidly with increasing N as in Fig. 2共a兲.

IV. CURRENT IN THE ‘‘FREE’’-ELECTRON CASE When interdot Coulomb repulsion is altogether disre-garded, all 2N many-electron states 兩兵nk其

典

of the array of

dots must be taken into account. Modifying the Liouville equation with Hˆ⫽Hˆ0 according to the general prescription 共3兲, we obtain the following set of 22N _{equations for the} density matrix:

⳵t␴n₁•••n_N,n 1 ⬘•••nN⬘⫽⫺i关Hˆ0,␴兴n1•••nN,n1⬘•••nN⬘ 共12a兲 ⫺1₂⌫L关共1⫺n1兲⫹共1⫺n1

⬘

兲兴␴n₁•••n_N,n 1 ⬘•••nN⬘⫺ 1 2⌫R共nN⫹nN

⬘

兲␴n1•••nN,n1⬘•••nN⬘ 共12b兲 ⫹⌫Ln1n1

⬘

␴0n₂•••n_N,0n 2 ⬘•••nN⬘⫹⌫R共1⫺nN兲共1⫺nN

⬘

兲␴n1•••nN⫺11,n1⬘•••nN⬘⫺11. 共12c兲

Here Eq.共12a兲 describes transitions between the nonorthogo-nal states of the device with a fixed number of electrons due to the tunneling between neighboring dots. In contrast to the Coulomb blockade case, both tunneling into (⌫L) and out of

the array (⌫R) give negative contributions 共12b兲.

Further-more, there are tunnel processes which induce a transition between two pairs of many-electron states and give a posi-tive contribution 共12c兲 to the coherences. Only a subset of Eq.共12兲 forms a closed system of equations for the diagonal and some nondiagonal elements of ␴ 共the remaining equa-tions only couple a closed set of nondiagonal elements which are irrelevant兲. From this subset we can derive an even sim-pler dynamical equation for the average occupation matrix␳. Let us first derive the coherent part of this equation by con-sidering ‘‘free’’ electrons in the array of dots without the leads. Because the Hamiltonian consists of only one-electron operators (H0⫽兺kl⫽1

N

H0klak

†

al), the commutator in the

Heisenberg equation of motion for␳i j⫽

具

aj

†

ai

典

is readily

ex-pressed in other one-electron operators and the matrix ele-ments of Hˆ0: i⳵t␳i j⫽

具

关aj † ai,Hˆ0兴

典

⫽

兺

l_⫽1 N H0il具a†_jal

典

⫺

兺

k_⫽1 N

具

a_k†ai

典

H0k j ⫽关H0,␳兴i j, 共13兲

where i, j⫽1, . . . ,N . This equation can also be derived by taking the average 关defined in Eq. 共2兲兴 of the coherent part 共12a兲. The contributions which describe the coupling to the leads are found by adding the average of the incoherent con-tributions 共12b兲 and 共12c兲 to the right-hand side of Eq. 共13兲. We obtain the following closed system of only N2 equations, which describes the dynamics of the average occupation number matrix:

FIG. 2. Normalized resonant current for the Coulomb blockade case with N⫽2,3,4,5,6 共full curves downwards兲 and N⫽⬁ 共thick full curve兲. 共a兲 Variation of the last level, i.e., ␧i⫽␧N␦iN.

Increas-ing N does not alter the Lorentzian form of the curve.共b兲 Variation of the first level, i.e.,␧i⫽␧1␦i1. Electrons are localized in the first

dot after tunneling through the left barrier resulting in the exponen-tial decay of the current tails with N. For N⫽2, the curves in 共a兲 and共b兲 coincide.

FIG. 3. Normalized current through an array of N⫽2,3,4,5,6 dots with energies configured as a Stark ladder ␧i⫽i/(N⫺1)␧ of

varying width ␧. 共a兲 ‘‘Free’’-electron case: the curves for N⫽2,3

共indicated by the arrow兲 coincide for an isotropic array ti⫽t with

⳵t␳ii⫽i共ti⫺1␳ii⫺1⫹ti␳ii⫹1⫺ti␳i⫹1i⫺ti⫺1␳i⫺1i兲 ⫹⌫L共1⫺␳11兲␦i1⫺⌫R␳NN␦iN, 共14a兲 ⳵t␳i j⫽i共␧j⫺␧i兲␳i j⫹i共tj⫺1␳i j⫺1 ⫹tj␳i j_⫹1⫺ti_⫺1␳i_{⫺1 j}⫺ti␳i_{⫹1 j}兲⫺ 1 2⌫L␳1 j␦i1 ⫺1₂⌫R␳iN␦jN, 共14b兲

where j⬎i⫽1, . . . ,N and t₀⬅t_N⬅0. These equations closely resemble those for the Coulomb blockade case: the coherent contributions are exactly the same, but in contrast to Eq.共7b兲 the average occupations␳11,␳NNin Eq.共14a兲 are

not coupled by incoherent transitions to some vacuum state. Furthermore, in Eq. 共14b兲 there is a negative contribution due to the tunneling of an electron into dot 1 with rate⌫L,

which is absent in Eq.共7c兲 due to the Coulomb blockade. Despite the close resemblance to Eq. 共7兲 the 共calculation of兲 stationary solution of Eqs. 共14兲 for the general case is far more complicated. An analytical expression seems neither

feasible nor instructive and we consider only a few represen-tative cases here. Assuming equal couplings ti⫽t, one finds

that the resonant current peak (␧i j⫽0) is independent of N: I_N,maxF e ⫽ 1 1 ⌫L ⫹_⌫1 R ⫹⌫L 4t2⫹ ⌫R 4t2 . 共15兲

This result was previously obtained by Frishman and Gurvitz for tunneling through multiple-well heterostructures5 using essentially the same approach as for the derivation of the modified Liouville equation10which we have used关Eq. 共3兲兴. By solving Eqs. 共14兲 for N⫽2 we reproduce the result ob-tained by Gurvitz13,10from Eqs. 共12兲,

I₂F e⫽ 1 1 ⌫L ⫹_⌫1 R ⫹_⌫1

冉

␧12 t1

冊

2 ⫹⌫L 4t₁2⫹ ⌫R 4t₁2 , 共16兲

where ⌫⫽⌫L⫹⌫R. For N⫽3, solving Eqs. 共14兲 gives the

current through a triple dot:

I₃F e⫽ 1 1 ⌫L ⫹_⌫1 R ⫹_⌫1 L

冉

␧12 t1

冊

2 ⫹_⌫1 R

冉

␧23 t2

冊

2 ⫹_⌫1

冉

t1 t2 ⫺t2 t1

冊

2 ⫹⌫L 4t₁2⫹ ⌫R 4t₂2⫺

冋

1 ⌫L ␧12 t1 ⫺_⌫1 R t1 t2 ␧23 t2 ⫺_⌫1

冉

t1 t2 ⫺t2 t1

冊

␧13 t2

册

2 1 ⌫L ⫹_⌫1 R

冉

t1 t2

冊

2 ⫹_⌫1

冉

␧13 t2

冊

2 ⫹ ⌫ 4t₂2 . 共17兲

If one would first calculate␴, then the subset of 20 relevant equations of Eqs. 共12兲 共containing in total 64 equations兲 needs to be solved. For the case where the energies are con-figured as a ‘‘Stark ladder’’ of total width␧, we have plotted the current in Fig. 3共a兲. Comparison with Fig. 3共b兲 shows that the current for the ‘‘free’’-electron case is less sensitive to localization effects than in the Coulomb blockade case.

V. APPROXIMATION OF INDEPENDENT CONDUCTION CHANNELS

It is instructive to consider an approximate approach of independent conduction channels to our problem. In Sec. V A this approximation is introduced assuming that both res-ervoirs are weakly coupled to dots in the array. In this case the time ⌫⫺1 needed for tunneling to or from a reservoir is much longer than typical time t⫺1 of the evolution of a co-herent state in the array. Therefore, an electron completes many coherent oscillations in the array before tunneling. Somewhat surprisingly, this approach can also be used in the opposite limit of very strong coupling to the reservoirs as is shown in Sec. V B. In Sec. V C we compare the results of the independent channel approximation and discuss some pe-culiar features of the more general results obtained in Secs. III and IV.

A. Weak coupling to the leads

We first consider the array of dots without the leads. Let us denote the N localized共delocalized兲 eigenstates of a single electron in the array of uncoupled 共coupled兲 dots by 兩i

典

(兩i

⬘典

), where i⫽1, . . . ,N and the single-electron vacuum by 兩0

典

. By transforming to the basis of delocalized states that diagonalizes Hˆ0 关Eq. 共1a兲兴, we obtain new fermionic opera-tors

aˆ_i

⬘

†⫽

兺

k⫽1

N

具

k兩i

⬘典

aˆ_k†, aˆ_i

⬘

⫽

兺

k⫽1

N

具⬘

i兩k

典

aˆk.

Since in a delocalized state there is a nonzero probability for finding an electron in both dot 1 and dot N, such a state can be regarded as a conduction channel which carries a current. The new operators add an electron to channel 兩i

⬘典

and re-move an electron from channel 兩i

⬘典

, respectively. By ex-panding the density operator of the array ␴ˆ in the ‘‘many-channel’’ basis states 兩n1•••nN

⬘ 典

⫽兩兵nk其

⬘典

⫽兿k_⫽1

N

aˆ_k

⬘

†nk_兩兵₀

典

_, we obtain a new density matrix␴

⬘

. As in Sec. II, we define an average occupation matrix for the channels ␳_{i j}

⬘

⫽

具

aˆ

⬘

_j†aˆ_i

⬘典

which can be expressed in the density matrix ␴

⬘

. The ad-vantage of the new basis is that the dynamical equations for the average occupations ␳_ii

⬘

are decoupled from the

nondi-agonal elements when we only take Hˆ0 into account: ⳵t␳i j

⬘

⫽⫺i(␧i

⬘

⫺␧

⬘

j)␳i j

⬘

, where␧i, j

⬘

are the energies of the

delocal-ized states i, j . In the Coulomb blockade case the effect of

Hˆ_u is easily translated since the basis transformation pre-serves the trace of␳: the total occupancy of the channels is restricted to values ⭐1 whereas in the ‘‘free’’-electron case all N channels can be occupied共Fig. 4兲.

Now we include the coupling to the reservoirs by a Golden Rule approach. Each channel 兩i

⬘典

is connected to lead L with modified matrix element t_Li⫽

具

i

⬘

兩1

典

tLand to lead R with t_Ri⫽

具

i

⬘

兩N

典

tR: an electron in lead L can tunnel

through any channel to lead R. The rate for tunneling into channel 兩i

⬘典

is proportional to the probability to find the electron in dot 1 whereas the rate for tunneling out of chan-nel兩i

⬘典

is proportional to the probability to find the electron in N: ⌫L i_⫽⌫ L円

具

i

⬘

兩1

典

円2, ⌫R i_⫽⌫ R円

具

i

⬘

兩N

典

円2. 共18兲

These probabilities depend on the energies␧_iof the localized states and the matrix elements ti which couple them. When

we assume that the uncertainty in the energy during the tun-neling of an electron between leads and dots is much smaller than the level splitting, we can disregard correlations due to the simultaneous tunneling of one electron through multiple channels. Thus for weak coupling we can include the leads by only modifying the equations for the diagonal elements of ␳

⬘

, i.e., the channel occupations. The channels can be treated as independent and their contributions to the current can sim-ply be added.

In the Coulomb blockade case关Fig. 4共a兲兴 the interdot re-pulsion couples the occupations of the channels and restricts their sum兺i_⫽1

N _␳ ii

⬘

⬍1 . The tunneling of an electron into the empty device with rate⌫_Li fills a channel兩i

⬘典

. This electron must tunnel out with rate ⌫_Ri before another electron can tunnel into the device and occupy one channel. The equa-tions for the occupaequa-tions are

⳵t␳00

⬘

⫽

兺

i_⫽1 N ⌫R i ␳ii

⬘

⫺

冉

兺

i_⫽1 N ⌫L i

冊

␳00

⬘

, ⳵t␳ii

⬘

⫽⌫L i ␳00

⬘

⫺⌫R i ␳ii

⬘

.

Using the implied conservation of probability 关兺_iN_⫽0␳_ii

⬘

(t) ⫽1兴, the stationary solution 关limt→⬁⳵t␳ii

⬘

(t)⫽0兴 is easily

found. Summation of the contributions of the independent channels gives the stationary current:

I_NCB e ⫽i

兺

⫽1 N ⌫L i_␳ ii

⬘

共⬁兲⫽ i

兺

⫽1 N ⌫L i 1⫹

兺

i⫽1 N ⌫L i ⌫R i ⫽ 1 1 ⌫L ⫹_⌫1 R FN . 共19兲

This is just the expression for the current through one double barrier: the tunneling time is just the sum of the tunneling times of the individual barriers where the time for tunneling through the right barrier is increased by a factor

FN⫽

兺

i_⫽1 N 兩

具

i

⬘

兩1

典

兩2 兩

具

i

⬘

兩N

典

兩2⬎

兺

i⫽1 N 兩

具

i

⬘

兩1

典

兩2

兺

i⫽1 N 兩

具

i

⬘

兩N

典

兩2 ⫽1. 共20兲

In the ‘‘free’’-electron case关Fig. 4共b兲兴 the occupations of the channels are not coupled and their sum is restricted only by 兺_iN_⫽1␳_ii

⬘

⬍N . The device is equivalent to N independent double barriers in parallel where the occupancy of channel i obeys ⳵t␳ii

⬘

⫽⌫L i 共1⫺␳ii

⬘

兲⫺⌫R i ␳ii

⬘

.

From the stationary solution关lim_t→⬁⳵_t␳_ii

⬘

(t)⫽0兴 the current is readily found: I_NF e ⫽i

兺

⫽1 N ⌫L i ␳ii

⬘

共⬁兲⫽

兺

i⫽1 N 1 1 ⌫L i⫹ 1 ⌫R i . 共21兲

The tunneling time through each double barrier is the sum of the tunneling times (⌫_L,Ri )⫺1⬎⌫_L,R⫺1 whereas the tunnel rate of the device is the sum of the tunnel rates of the N double barriers.

For a few representative cases we have explicitly worked out the approximate approach in the limit of weak coupling to both leads ⌫L,RⰆ␧i j,ti. For the case of N⫽2 dots we

have first calculated the two exact eigenstates of one electron in the array of coupled dots without the reservoirs. From these we obtained the tunnel rates to and from each of the delocalized states. Finally we summed the contributions of the independent channels to the total current: Eq. 共19兲 pre-cisely gives the result 共9兲 without the term (⌫R/2t1)2 whereas Eq.共21兲 gives the result 共16兲 without the term (⌫L

⫹⌫R) /(2t1)2 共the latter result was previously found in Ref. 4 for the case of a double quantum well兲. For the case of N dots with equal interdot couplings ti⫽t and aligned levels

␧i⫽␧j, a simple result is also possible because in each

de-localized state 兩i

⬘典

the electronic densities in the dots are

FIG. 4. Tunneling through N independent channels. 共a兲 Cou-lomb blockade case: an electron tunneling through one of the chan-nels blocks the remaining N⫺1 channels. The state must first decay to the vacuum before another electron can enter one of the channels.

共b兲 ‘‘Free’’-electron case: N parallel channels for tunneling are

spatially symmetric. Then the modification factors of the rates are equal for each channel (円

具

i

⬘

兩1

典

円2⫽円

具

i

⬘

兩N

典

円2) and they cancel in Eqs.共19兲 and 共21兲. Thus for weak coupling to both reservoirs we obtain

I_N,maxCB e ⫽ 1 1 ⌫L ⫹_⌫1 R N , 共22兲 I_N,maxF e ⫽ 1 1 ⌫L ⫹_⌫1 R , 共23兲

which is just Eqs. 共11兲 and 共15兲 without the terms (N⫺1) ⫻(⌫R/2t)2 and ⌫L⌫R/(2t)2, respectively. Coulomb

block-ade increases the effective time for tunneling out of the de-vice by a factor N with respect to the ‘‘free’’-electron case. The current increases linearly with each rate ⌫L,⌫RⰆt as

expected from the enhanced tunneling.

B. Strong coupling to the leads

The terms which were found missing above become im-portant when one of the tunnel rates to the reservoirs is com-parable to the coherent interdot coupling. The correlations between the conduction channels can no longer be disre-garded in this case: Eqs. 共7兲 and 共14兲 do not decouple into separate sets of equations for diagonal and nondiagonal ele-ments on the basis of delocalized states. However, we will now show that in the limit of strong coupling to one or both of the reservoirs, we can use the independent channel ap-proximation again. 共Note that the energy uncertainties are assumed to be smaller than the bias ⌫_L,RⰆ␮_L⫺␮_R.)

First we consider the case where only the last dot N is strongly coupled to the right lead: ⌫LⰆ␧i j,tiⰆ⌫R. The

electronic state of the dot N and reservoir R form a con-tinuum of states with a Lorentzian spectral density which is approximately constant over the energy range tN⫺1Ⰶ⌫R:

DN⫹R共␧兲⫽ 1 2␲ ⌫R 共␧⫺␧N兲2⫹共⌫R/2兲2 ⬇₂1_{␲ ⌫}4 R .

The array of the remaining N⫺1 dots is weakly coupled to this continuum of states with matrix element tN⫺1. Therefore

we can apply the independent channel approach: the tunnel rate from dot N⫺1 to the continuum on the right is found with the Golden Rule,

⌫˜R⫽2␲tN_⫺1 2 _D N⫹R共␧兲⫽ 4t_N2_⫺1 ⌫R . 共24兲

For the case of aligned levels␧i j⫽0 and equal couplings ti

⫽t, substitution of ⌫R→⌫˜R and N→N⫺1 in Eqs. 共22兲 and

共23兲, respectively, gives the maximum current in the limit of strong coupling to the right lead:

I_N,maxCB e ⫽ 1 1 ⌫L ⫹N⫺1 ⌫˜R ⫽ 1 1 ⌫L⫹ ⌫R 4t2共N⫺1兲 , 共25兲 I_N,maxF e ⫽ 1 1 ⌫L ⫹ 1 ⌫˜R ⫽ 1 1 ⌫L⫹ ⌫R 4t2 , 共26兲

which is in agreement with our results共11兲 and 共15兲, respec-tively. For strong coupling to the right lead, the current

de-creases as ⌫_R⫺1, which is somewhat surprising because tun-neling is enhanced. The origin of this effect is the formation of linear combinations of the discrete state in dot N with reservoir states with energies roughly between ␧N⫾⌫R.

Be-cause the tunnel processes from reservoir states back into the discrete state destructively interfere,11the discrete state irre-versibly decays into the continuum. The resulting spectral density decreases with the energy uncertainty⌫_R. The even-tual decrease of the current with the tunnel rate⌫R can also

be interpreted as a manifestation of the quantum Zeno effect as discussed in Ref. 10. More generally we have the follow-ing relation: for ⌫LⰆ␧i j,tiⰆ⌫R,

IN CB_共␧ 1•••␧N,t1•••tN⫺1;⌫L,⌫R兲 ⫽IN_⫺1 CB

冉

_␧ 1•••␧N⫺1,t1•••tN⫺2;⌫L, 4t_N2_⫺1 ⌫R

冊

, I_NF共␧1•••␧N,t1•••tN⫺1;⌫L,⌫R兲 ⫽IN⫺1 F

冉

_␧ 1•••␧N⫺1,t1•••tN⫺2;⌫L, 4t_N2_⫺1 ⌫R

冊

.

This is clearly satisfied by Eqs. 共10兲 and 共17兲, respectively, and the general form共8兲 has this property.

Now consider the case where the array is coupled strongly to both leads, i.e., ␧i j,tiⰆ⌫L,R. In the ‘‘free’’-electron case

the reservoir L coupled to dot 1 gives a new continuum of states with a spectral density which is approximately con-stant over an energy range t1Ⰶ⌫L:

D1⫹L共␧兲⫽ 1 2␲ ⌫L 共␧⫺␧1兲2⫹共⌫L/2兲2 ⬇₂1_{␲ ⌫}4 L .

The tunnel rate from this continuum to dot 2 is

⌫˜L⫽2␲t1 2 D1⫹L共␧兲⫽ 4t₁2 ⌫L .

The remaining N⫺2 dots are weakly coupled to a continuum on the left with matrix element t1and to the right with tN⫺1

and we can apply the independent channel approach to the

N⫺2 conduction channels. For the case of aligned levels

␧i⫽␧j and equal couplings ti⫽t, substitution of both ⌫L

→⌫˜Land⌫R→⌫˜R in Eq.共23兲 gives the maximum current in

the limit of strong coupling to both leads,

I_maxF e ⫽ 1 1 ⌫˜L ⫹ 1 ⌫˜R ⫽ 1 ⌫L⫹⌫R 4t2 ,

in agreement with result 共15兲. More generally, we have the relation for ␧_i,t_iⰆ⌫_L,R:

I_NF共␧₁•••␧_N,t₁•••t_N⫺1;⌫_L,⌫_R兲 ⫽IN⫺2 F

冉

_␧ 2•••␧N⫺2,t2•••tN⫺2; 4t₁2 ⌫L ,4tN⫺1 2 ⌫R

冊

,

which is satisfied by Eq.共17兲. In the Coulomb blockade case, interdot repulsion prevents the discrete state in dot 1 from mixing with the reservoir: even if the energy uncertainty allows tunneling into dot 1 to occur on a small time scale ⌫L⫺1, the next electron will have to wait for the previous one

to tunnel out, which occurs on the much larger time scale max兵␧_ij⫺1,t⫺1,⌫_R⫺1其. In the limit of strong coupling to both leads, the current in this case is independent of ⌫_L and is correctly given by Eq.共25兲 in agreement with result 共11兲.

Finally, we consider the case where only dot 1 is strongly coupled to the left lead, i.e., ⌫_RⰆ␧_{i j},t_iⰆ⌫_L. In the Cou-lomb blockade case, the weak-coupling result still applies as explained above. In the ‘‘free’’-electron case the discussion is completely analogous to the case of strong coupling of dot

N to the right lead and the result is obtained by simply

inter-changing L↔R and 1↔N.

C. Intermediate coupling to the leads: Maximum current The competition between enhanced tunneling for weak coupling to the leads and destructive interference in the op-posite limit implies that the current reaches a maximum value when the rate for tunneling into a dot is comparable to the coherent coupling to the neighboring dot. We can find the precise location and value of this maximum with the results obtained in Secs. III and IV, which also hold in this interme-diate case.

First we consider the resonant current peak共11兲 and 共15兲 as a function of the transparency⌫R of the right tunnel

bar-rier as plotted in Fig. 5. Starting from zero, the current ini-tially increases linearly as expected from the enhanced tun-neling to the right lead. Then a maximum is reached:

I_maxCB e ⫽ 1 1 ⌫L⫹

冑

N共N⫺1兲 1 t , ⌫R⫽

冑

N N⫺12t, 共27a兲 I_maxF e ⫽ 1 1 ⌫L⫹ ⌫L 4t2⫹ 1 t , ⌫R⫽2t. 共27b兲

Increasing the transparency further will reduce the current as explained above. The maximum occurs when there is an op-timal balance of the coherent tunneling into dot N and inco-herent tunneling from this dot to reservoir R. At this point the effective time for tunneling out of the device is the same for weak 关Eqs. 共22兲 and 共23兲, respectively兴, and strong cou-pling to the right lead关Eqs. 共25兲 and 共26兲, respectively兴, i.e., at these values of ⌫R we have

1 ⌫R N⫽ 1 ⌫˜R 共N⫺1兲 共CB兲, 1 ⌫R⫽ 1 ⌫˜R 共F兲,

where⌫˜R is given by Eq.共24兲. For the case of ‘‘free’’

elec-trons the effective time for tunneling out and therefore the position of the maximum is independent of the number of dots N. 共For this case the ⌫_R dependence of the resonant current peak has been discussed for a double dot system in Ref. 11.兲 In the Coulomb blockade case the nonmonotonic variation of the current with⌫_Rpersists for all N. The maxi-mum occurs at a slightly higher value of ⌫R than for the

‘‘free’’-electron case: at⌫R⫽2t⫽⌫˜Rthe effective tunneling

time for weak coupling is still larger than for strong coupling because the fraction of channels excluded by Coulomb repul-sion is larger for N dots than for N⫺1 dots. The tunnel rate must be increased by a factor

冑

N/(N⫺1) to exactly balance

the effective tunneling times. For large N this difference be-comes negligible and the maximum occurs at the same posi-tion as for the ‘‘free’’-electron case but with a much smaller amplitude共Fig. 5兲.

Next we consider the resonant current peaks共11兲 and 共15兲 as a function of the transparency⌫Lof the left tunnel barrier.

In the ‘‘free’’-electron case the maximum current 共15兲 re-mains unchanged when we interchange ⌫L and⌫R.

There-fore, the resonant current peak共15兲 also displays a maximum as a function of⌫Lat⌫L⫽2t. As a function of both rates the

maximum current is

I_N,maxF

e ⫽

t

2, ⌫L⫽⌫R⫽2t.

As discussed in Sec. V B, there is no such effect in the Cou-lomb blockade case: due to interdot repulsion, the resonant current peak 共27a兲 will increase with ⌫L and saturate at a

maximal value when ⌫_LⰇt,

I_N,maxCB e ⫽ t

冑

N共N⫺1兲, ⌫LⰇ⌫R⫽

冑

N N⫺12t.

For N⫽2, the maximal current as a function of the rates is larger in the Coulomb blockade case whereas for N⬎2 it is larger in the ‘‘free’’-electron case.

FIG. 5. Maximum resonant current as a function of the trans-parency of the right barrier: Coulomb blockade case for N

⫽1,2,3,4,5,6 共solid lines downwards兲 and ‘‘free’’-electron case 共dot-dashed line for any N⬎1). The dotted lines show the position

VI. CONCLUSIONS

We have extended the density-matrix approach to reso-nant tunneling to the case of a linear array of quantum dots with strong, long-range electron-electron interaction. We have found exact analytical expressions for the stationary current in the array for an arbitrary set of parameters共within the applicability of our model兲 characterizing the array. Cou-lomb repulsion was found to reduce the resonant current by a factor of the order of the number of dots. The formation of a localized state in one of the dots when the energy level is displaced results in an exponential decay of the current with increasing size of the array. Our approach takes into account correlations between conduction channels in the array due to

the coupling to the electron reservoirs. This makes our re-sults also valid for relatively strong tunnel coupling to the reservoirs where the independent channel approximation does not work. These correlations manifest themselves in the eventual decrease of the resonant current when the rate for tunneling into the reservoir is increased.

ACKNOWLEDGMENTS

The authors acknowledge valuable discussions with T. H. Stoof, G. E. W. Bauer, and especially S. A. Gurvitz. This work was supported by the Dutch Foundation for Fundamen-tal Research on Matter 共FOM兲 and by the NEDO joint re-search program共NTDP-98兲.

1_{L. P. Kouwenhoven, F. W. J. Hekking, B. J. van Wees, C. J. P.}

M. Harmans, C. E. Timmering, and C. T. Foxon, Phys. Rev. Lett. 65, 361 共1990兲; L. P. Kouwenhoven, Science 268, 1440

共1995兲.

2_{N. C. Van der Vaart, S. F. Godijn, Yu. V. Nazarov, C. J. P. M.}

Harmans, J. E. Mooij, L. W. Molenkamp, and C. T. Foxon, Phys. Rev. Lett. 74, 4702 共1995兲; T. H. Oosterkamp, S. F. Godijn, M. J. Uilenreef, Yu. V. Nazarov, N. C. van der Vaart, and L. P. Kouwenhoven, ibid. 80, 4951共1998兲.

3_{E. E. Mendez and G. Bastard, Phys. Today 46„7…, 33 共1993兲.} 4_{A. N. Korotkov, D. V. Averin, and K. K. Likharev, Phys. Rev. B}

49, 7548共1994兲.

5_{A. M. Frishman and S. A. Gurvitz, Phys. Rev. B 47, 16 348} 共1993兲.

6_{G. Chen, G. Klimeck, S. Datta, G. Chen, and W. A. Goddard III,}

Phys. Rev. B 50, 8035共1994兲.

7_{Yu. V. Nazarov, Physica B 189, 57共1993兲.}

8_{When a very large infinite on-site Coulomb repulsion is assumed,}

the electron spin can be included by taking into account the spin-degeneracy of the single dot levels. In a high magnetic field where all electron spins are aligned, this degeneracy is absent and our results apply directly.

9_{S. A. Gurvitz and Ya. S. Prager, Phys. Rev. B 53, 15 932共1996兲.}

In this paper the modified Liouville equation is derived for the case of a single and double quantum dot by ‘‘tracing out’’ the reservoir states from the Schro¨dinger equation for the device and the leads. However, the generalization to an arbitrary device

关result Eq. 共6.1兲 in Ref. 10兴 is not the most general equation; one

should instead use the form given in later publications, e.g., Eq.

共2.1兲 in Ref. 11. S. A. Gurvitz 共private communication兲. 10_{S. A. Gurvitz, Phys. Rev. B 57, 6602共1998兲.}

11_{E. Merzbacher, Quantum Mechanics, 2nd ed. 共John Wiley &}

Sons, New York, 1961兲, p. 479.

12_{T. H. Stoof and Yu. V. Nazarov, Phys. Rev. B 55, 1050共1996兲.} 13_{S. A. Gurvitz, Phys. Rev. B 44, 11 924共1991兲.}