Interference effects in interacting quantum dots

In this paper, we study the interplay between interference effects in quantum dots (manifested through the appearance of Fano resonances in the conductance), and interactions taken into account in the self-consistent Hartree–Fock approximation. In the non-interacting case we find that interference may lead to the observation of more than one conductance peak per dot level as a function of an applied gate voltage. This may explain recent experimental findings, which were thought to be caused by interaction effects. For the interacting case we find a wide variety of different interesting phenomena. These include both monotonous and non-monotonous filling of the dot levels as a function of an applied gate voltage, which may occur continuously or even discontinuously. In many cases a combination of the different effects can occur in the same sample. The behaviour of the population influences, in turn, the conductance lineshape, causing broadening and asymmetry of narrow peaks, and determines whether there will be a zero transmission point. We elucidate the essential role of the interference between the dot's levels in determining these outcomes. The effect of finite temperatures on the results is also examined.


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non-monotonous filling of each level as a function of an applied gate voltage. The dependence on the gate voltage may be discontinuous, even when all the dot's levels have finite widths. In many cases both continuous and discontinuous non-monotonicity can occur in the same case. We find how the behaviour of the population affects the conductance. Interference effects between the dot's levels play a very important role in determining which type of behaviour will occur in a given sample. We also discuss the effect of finite temperatures on the results, which is essential for analysing experimental data.
We conclude by reviewing our main findings and by discussing the limits of applicability of our Hartree-Fock method and the reliability of our results in section 5.

Model and methods of calculation
We consider the following model Hamiltonian, describing spinless electrons (experimentally realizable by applying a strong in-plane magnetic field) moving in a system composed of a (possibly interacting) dot and non-interacting leadŝ This Hamiltonian is composed of three parts: (i) The quantum-dot Hamiltonian Hereâ i andâ † i are creation and annihilation operators, respectively, of an electron in the dot's ith level; i,v = i − eV g is the corresponding single-particle energy, determined by the bare level energy i and modified by an applied gate voltage V g (e is the absolute value of the electronic charge); and U = e 2 /C is the strength of interaction between electrons in the dot, assumed to consist simply of a charging energy. (ii) The Hamiltonian of lead (= L or R for the left or right lead, respectively) whereĉ † k, ,ĉ k, are creation and annihilation operators, respectively, of an electron in the th leads kth mode, k, the corresponding single-particle energy. In the following we assume that each lead is a band of width 2D (much larger than any other energy scale in the system), and constant density of states. (iii) The tunnelling Hamiltonian where the tunnelling matrix elements t i k, are assumed to be real (i.e. there is no applied out-of-plane magnetic field), and independent of k. It is also assumed that t i L = t i R .

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For U = 0 (the non-interacting case), the Hamiltonian can be exactly solved. The matrix elements (in the dot states space) of the inverse retarded (advanced) Green function for the dot states is given by where i,j , the matrix elements of the widths of the dot's levels due to their coupling with lead , are given by ρ being the density of states in the th lead. Thus, ( i,j ) 2 = i,i j,j . In addition, by the above assumptions on the tunnelling matrix elements, L i,j = R i,j . Thus, one is free to choose the diagonal matrix elements of L , and the signs of the off-diagonal elements of both L and R . In the following we will denote the total width, L + R by , and its ith diagonal matrix elements by i .
Using the dot's Green functions, one can find the following averages, related to the average occupation of the dot's states at temperature T f( ) = 1/(exp(( − µ)/T) + 1) being the Fermi-Dirac distribution function with chemical potential µ and temperature T , using units where Boltzmann's constant equals unity. In particular, the average occupation of the dot's ith level is n i = â † iâ i . We can also find the linear conductance, following Meir and Wingreen [23] The interacting case (U = 0) is treated using the self-consistent Hartree-Fock approximation, which has been used by previous authors to study similar systems [14,22]. We will postpone the discussion of the range of validity of mean-field results to the problems studied until presenting our results (see section 5).
The Hartree-Fock approximation amounts to replacing the dot Hamiltonian (2) by an effective single-particle Hamiltonian, given bŷ The diagonal terms in this expression are the Hartree contribution, while Fock correction leads to the off-diagonal terms. Using equations (5) and (7), the problem is reduced to solving a set of self-consistent equations for the averages â † iâ j . One can show that the Fock terms vanish for any pair of levels i, j for which L i,j = − R i,j .

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In many cases there are several solutions to the Hartree-Fock equations, corresponding to different dot states being almost full or almost empty. In those cases one should average over the solutions, giving each a probability factor proportional to exp(− /T), where is the grand canonical free energy In particular, at zero temperature only the solution with the lowest free energy should be retained.
We mention in passing that the above integrals for the free energy, occupations and conductance can be expressed in terms of the logarithm of the gamma function of a complex argument and its first two derivatives (the digamma and trigamma functions) [24], respectively, to facilitate faster computation.

Interference effects in non-interacting dots
We first consider the non-interacting case. Here, for a two-level dot, there are two possibilities, depending on the relative signs of the matrix elements of the widths R,L between the two states. These will be termed the plus (minus) configuration for positive (negative) sign of R 1,2 / L 1,2 . If we make a transformation from the lead operatorsĉ k,R ,ĉ k,L to the combinationŝ c k,± = (ĉ k,R ±ĉ k,R )/ √ 2, we find that in the plus case the two dot states are connected to thê c k,+ states, whereas in the minus case one dot state is connected to theĉ k,+ states, while the other is connected with theĉ k,− states. Thus, in the plus case the two dot states are effectively coupled to a single lead, while in the minus case each dot state is connected to a different effective lead [5,14].
The local density of states of each level, ρ i , is given by (i is the index of the other level) [5] ρ As one can see, in the minus case, the density of states of each of the dot's levels is unaffected by the other level, because they are effectively decoupled, as was explained above. Thus, their populations follow the usual n i = 1/2 + tan −1 ((µ − i,v )/( i /2))/π law. However, in the plus case, the levels interfere. As a result, the local density of states of each level goes to zero at the position of the other level. Thus, as the two levels approach each other, the density of states of both levels develops a sharp peak, going from zero to its maximum in a gate voltage distance which goes as | 1 − 2 |, instead of the widths i . This causes the populations n i to vary fast for V g between the two level energies. When the level energies exactly coincide, this sharp feature becomes a delta function peak in the density of states, or a discontinuous jump in the level population as a function of the gate voltage. Indeed, in this latter case, due to the degeneracy of the dot levels, one can transform to a basis of the states where one level is totally decoupled from the leads [5,11,14,16].

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The conductance in each case is given by Since conductance occurs by transmission of electrons between the left and right leads, and not between theĉ k,± combinations, we find interference effects in the conductance in both cases. Thus, in the plus case, the conductance reaches its maximal possible value (of e 2 / h) when µ equals one of the dot's levels ( 1 or 2 ), and goes to zero for µ = + , i.e. between the conductance peaks. In the minus case, the conductance reaches its maximal value for µ in the vicinity of (but slightly different from) the dot's levels. It goes to zero at µ = − , which lies outside the peaks. When 1 and 2 are sufficiently different, the conductance reaches its maximal possible value of e 2 / h at the peaks. When the dot's levels are too close the two peaks near the level energies become smaller, and eventually merge into a single peak. Finally, for 1 = 2 and 1 = 2 , there is a complete destructive interference between the two levels, and g = 0 for all µ values. All the above results are exemplified in figure 1. Here, and in the following, we set µ = 0, and vary the gate voltage V g . One can see that the interference effects result in an asymmetric shape of the conductance peaks, usually referred to as 'Fano resonances' [2]- [4], [6]- [8].
We note that if the couplings to the left and right leads were different in magnitude, we would get the same qualitative picture, but in all the cases the conductance would have reached values lower than e 2 / h even at the peaks.
One can show the above results for the conductance obey the relation g ± = e 2 / h × sin 2 (π(n 1 ± n 2 )) at zero temperature; this relation is also obeyed in the interacting case, in our Hartree-Fock approximation (as we have explicitly checked). The validity of this equation is, however, much wider, since it is required by the relation between the zero temperature conductance and the scattering phase shifts, the latter being determined by the Friedel sum rule [12], [16]- [21], [25]. In general the phase shifts and the conductance are related to both the occupation of the dot and the change in the occupation of the leads due to the dot. However, in the wide band limit employed here, the population of the leads is not affected by the dot, so the conductance is determined by the dot population alone.
As one can see, in the minus case, we get three conductance peaks from only two dot levels. As we show in figure 2, this can be extended to the case of many narrow levels and a single wide level, where the latter's coupling to one of the leads has an opposite sign (in the sense discussed above) to that of the narrow levels. In this way we get two peaks in the conductance for each narrow level. The conductance curve is very similar to some of the experimental results of Johnson et al [8]. This shows that interference effects alone can qualitatively explain the experiments, without the need to resort to interaction effects, as was done by the above mentioned authors.

Interaction effects
We now turn to the interactions. We will focus on two-level systems from now on. The simplest effect of the interactions is that a filled level pushes unfilled ones toward higher energies by the Hartree term. However, interactions can lead to much more interesting phenomena, such as non-monotonous population of the levels. In the two limiting cases in terms of the ratio between the widths of the two levels (i.e. one of the levels has zero width, or has the same width as the other level respectively), one finds either of the two most basic effects.
(i) As was first noted by Silvestrov and Imry [9], when one of the levels is completely decoupled from the leads (i.e. has zero width), it is either completely filled or completely empty at zero temperature. As the gate voltage is swept from low to high values, the wider level is first filled (if its energy is not too much higher than that of the narrow level), and pushes up the energy of the narrow level. Indeed, for low gate voltage values, filling the wider level gives lower total kinetic energy than filling the narrow one. However, at some point, for higher gate voltage values, the kinetic energy considerations make it advantageous to fill the narrow level instead, thus moving the wider one toward higher energy values, and reducing its population. Thus, at that point there is a sharp jump in the occupation of both levels, and in the conductance. Also, the two conductance peaks result from filling the same level twice, and thus have equal lineshapes. (ii) For levels of comparable widths, no discontinuity occurs. As the gate voltage is swept, the two levels start becoming populated. However, if one level has a lower bare energy, or is wider, this level gets populated faster, and repulses the electrons out of the other level, thus increasing even further its own population. When the gate voltage becomes higher, the process is reversed: the less-populated level starts to be occupied again, and in turn reduces the population of the more-populated level. This effect causes the conductance peaks to be asymmetric. This effect is most pronounced when the level widths are in fact equal [13,14].
We will begin by examining the case of levels in the minus configuration. This is the simpler case, since, as we have seen in the non-interacting case, the interference effects do not affect the dot's population. To start with, we give examples to the two limiting cases discussed above in figure 3. One can observe that, in contrast with the non-interacting case discussed in the previous section, the conductance does not go to zero in both cases-in the first, one level is completely decoupled from the rest of the system (except for the interaction), so no interference can occur even in the conductance. In the second case, the levels have equal widths, and −,v goes to infinity (see equation (12)).
In intermediate cases between those two limits, all kinds of combinations of these two phenomena can occur. Some typical cases are shown in figure 4. In all those cases the interaction is much stronger than the level widths, and the width of level 2 is much smaller than the width of level 1, but the distance between the two levels is different.
The most interesting feature of the results is that the phenomena of discontinuity of the dot's properties (level populations and conductance) as a function of V g is not restricted to the case of a level of zero width shown above, but can appear even when the widths of both levels are finite. As one can see in the figure, the discontinuity occurs when the energies of the levels are sufficiently close (cases (c)-(g)); otherwise, the variation is continuous. By varying the other parameters we have found that for any given value of level separation (including zero), this effect occurs if the interaction is strong enough, and the ratio between the width of the narrow level and the width of the wide level is small enough-the limiting values are less restrictive for close levels, and more demanding for far-away ones. In addition to this effect, the second, continuous effect of non-monotonous level filling can also be observed in cases (c)-(g). However, it is quite weak here, since the level widths are far from being equal.
Considering the conductance, we have seen in the non-interacting case that it goes to zero at a value of V g on the narrower peak side away from the wider peak. This effect is manifested only when there is no discontinuity, and results in an asymmetry of the narrower peak. In cases (c)-(g), the discontinuity skips over this point, but leaves the narrow peak asymmetric. In addition, since the wider level is being filled in both conductance peaks, it makes the narrow peak wider in comparison with the non-interacting case (although, in contrast with the case of figure 3(b), the narrow level has a finite width, and the two conductance peaks have different lineshapes).
The situation is, however, quite different in the plus configuration. Here, there is a strong interference effect on the populations of the two levels. This is due to both the off diagonal element of the total widths matrix , (which create imaginary off diagonal elements in the dot's inverse Green function), and the Fock term (which create real off diagonal elements in the dot's inverse Green function). Both these terms are absent in the minus configuration discussed above. Figure 5 shows the results in the plus configuration, where all parameters values are the same as in figure 4. Because of the interference effects on the populations of the two levels, there are no discontinuities here, except in the case of exactly degenerate levels (case (e)), where we observe the non-interacting discontinuity, which occurs even for free electrons (as we have seen in figure 1(d)). This point will be discussed more fully later on.
Instead of discontinuities we observe in figure 5 a continuous version of effect I, where the narrower level depopulates the wider one as V g is increased (like, e.g. case (d) for V g / 1 near zero), resulting in a broadening of the narrow conductance peak. We also observe effect II, where  the inverse occurs (such as, e.g. case (f) for V g / 1 near zero), contributing to the asymmetry of the wide peak.
We also note that here, in all the cases (except the exactly degenerate case (e)) there is a value of V g between the two conductance peaks where the conductance goes to zero, as for the non-interacting case.
Nevertheless, we can obtain discontinuities even in the plus configuration. This is exemplified in figure 6, in which the parameters are similar to figure 5, but the interaction is stronger and the ratio between the narrow level width and the wide level width is smaller. As in the minus case, there is no discontinuity when the levels are too far away (cases (a) and (i)). In contrast with the minus case, interference effects prevent the discontinuities when the levels are too close (cases (c)-(g), except the non-interacting discontinuity for the exactly degenerate case (e)). Only for the intermediate cases (in terms of energy levels distance) does discontinuity occurs (as can be seen in the insets to panels (b) and (h)).
In addition, continuous non-monotonicity of the level populations (and the resulting broadening and asymmetry of the narrow peak) occurs in cases (c), (d), (f) and (g), in a similar way to the corresponding cases in figure 5. Here too the conductance vanishes for some value of V g between the two conductance peaks in all cases (except (e)). This happens even in the discontinuous cases (in contrast with the situation in the minus configuration, figures 4(c)-(g)).
The reason is the smallness of the discontinuity even when it occurs. In section 3, we have seen that for a non-interacting dot in the plus configuration, when the two levels are almost degenerate, they can be treated as a linear combination of a wide and a very narrow effective levels, almost decoupled for each other. The presence of the narrow effective level causes a sharp variation in the population of the dot's original levels as a function of V g . In the limit of exactly degenerate levels, the narrow level width becomes zero (it becomes completely decoupled), so a discontinuity occurs [5,11,16].
The results shown so far seem to imply that in the presence of interaction this sharp variation disappears, except for the case of coinciding levels, where the non-interacting discontinuity is still observed. However, this is only true for strong interactions, which generate a strong Fock interference term between the levels. For weak interactions (too weak for discontinuities at intermediate level separations to occur), the situation is quite different. Here, because of the presence of the effective narrow level, the interactions can turn the sharp but continuous variation of the level populations in the non-interacting case into a discontinuous one. The jump actually occurs for the effective levels, increasing the population of the narrow effective level by almost one, and reducing the population of the wider effective level by a smaller amount (since the interaction is relatively weak).
A typical situation is exhibited in figure 7. Here the first dot level width, the level distance and interaction are fixed, but the second dot level width varies. When the second dot level is much narrower than the wide one (cases (a) and (b)), the picture is quite similar to the usual interacting discontinuity-at the discontinuity the population of the narrow dot level rises, while that of the wide dot level falls. This is because the narrow dot level is composed mainly of the narrow effective level, while the wide dot level is composed mainly of the wide effective level. When the narrow dot level width is comparable to that of the wider dot level (cases (d) and (e)), the results resemble the non-interacting discontinuity-the occupations of both the wider and the narrower dot levels rise at the discontinuity. This happens since the wider dot level now has a larger share in the narrow effective level, and, as was explained above, the increase in the population of the narrow effective level is larger than the (absolute value of the) decrease in the population of the wide effective level. There is an intermediate value of the narrow dot level width (case (c)) where only the narrower dot level population jumps, while the wider dot level population is continuous, since the effects of the wide and narrow effective levels on its population exactly cancel. (e) 2 / 1 = 0.7. In cases (a) and (b) n 1 decreases at the discontinuity, in case (c) it does not change, while in cases (d) and (e) it increases at the discontinuity. In all the cases, n 2 increases at the discontinuity.
Due to the large discontinuities in these cases, the conductance does not vanish between the two peaks in any of the cases considered. The result is two wide and overlapping conductance peaks, with discontinuous features in the conductance valleys between them.
To clarify these points, in figure 8 we compare the results with and without interaction for parameter values corresponding to figure 7(a). It can be clearly seen that although in the non-interacting case the variation of the physical parameters (especially the conductance) with V g is very fast, it is still continuous, and discontinuities can appear only with the addition of interactions.
Finally, we remark on the effects of finite temperatures. In figure 9, we examine one particular case (having parameter values intermediate between those of figures 4(c) and (d)). For temperatures lower than the narrow level width, the only effect of the temperature is to make the discontinuity smooth. Higher temperatures cause the smearing of the entire curves, and the lowering of the conductance peaks heights below the maximal value of e 2 / h. Thus, discontinuities may appear experimentally as fast gate voltage dependence of the physical parameters, which thus show temperature dependence for temperatures much lower than, e.g., the conductance peaks widths. Of course, one cannot differentiate experimentally between discontinuous features and continuous features having width smaller than the lowest accessible temperature.

Discussion and conclusions
In this paper, we examined some of the various phenomena which may occur in a quantum dot where both interaction effects and inter-level interference effects are important. We now summarize our results.
We have first shown that the occurrence of more conductance peaks than energy levels in the dot can happen even in the non-interacting case, and may help explain recent experimental observations.
The interacting case was studied using the self-consistent Hartree-Fock approximation. We have found that in the minus configuration, where the interference between the two levels does not affect their populations, discontinuities occur if the levels are close enough, the interaction is strong enough, and the ratio of the level widths is much different from unity. In contrast, in the plus configuration interference effects are important. In the strong interaction regime discontinuities can only occur when the levels are not too far away but not too close; again, the widths ratio needs to be significantly different from one. In the weak interactions regime, the fast variation of the population in the non-interacting case for almost degenerate levels can lead to discontinuities in the interacting case even for comparable widths. These results agree with the 'phase diagram' picture of Golosov and Gefen [22] in the overlapping part of the parameter space.
In both configurations, even when no discontinuity occurs, there is usually a continuous version of mechanism I of non-monotonous filling, accompanied by a broadening of the narrow conductance peak. In addition, if the levels are not too far apart, the continuous or discontinuous type I non-monotonicity is accompanied by type II non-monotonous behaviour, causing asymmetry of the conductance peaks.
In either the plus or minus case, when no discontinuity occurs, or when the discontinuity is weak enough, there is a conductance zero in a location similar to the non-interacting case. Strong enough discontinuities can cause the conductance zero to be skipped.
Finite temperatures smear the discontinuities. The latter leave their mark as sharp features in the gate voltage dependence of the different physical properties, which show very strong temperature dependence relative to other parts of the gate voltage dependence curves.

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Finally, we turn to discuss the applicability of the self-consistent Hartree-Fock approach to the problems studied in this paper. It is well known that the mean-field approximation neglects correlation effects. For example, in the minus configuration, and for levels of equal widths, our model is equivalent to the Anderson model of magnetic impurities [26], where the role of the two spin states is replaced by the two orbital levels (since each of them is coupled to a different lead), and the level separation is equivalent to an applied magnetic field [14], [19]- [21]. In this case the Hartree-Fock approximation (used in Anderson's original work on the model bearing his name), wrongly predicts a ground state with broken spin rotation symmetry, and thus fails completely to account for the Kondo effect. This may raise some concern regarding the validity of our Hartree-Fock results (in particular-the discontinuous phenomena), especially in the cases when the levels are close to each other, and correlation effects become important [14].
However, the discontinuous behaviour found in this study occurs only when one level is about an order of magnitude narrower than the other one, and this asymmetry between the two levels suppresses Kondo correlations [19]- [21]. Moreover, in the plus case the two levels are coupled to the same effective lead, so Kondo physics cannot apply at all. Thus, although it is reasonable to assume that a more complete treatment will modify the Hartree-Fock results and may reduce the parameter range in which discontinuities occur, it is not implausible that some region of discontinuous behaviour will persist. This is supported by similar results obtained in a Bethe-ansatz study of related spinful models [12]. Even if using more accurate methods the discontinuities are eliminated altogether, our results imply that the behaviour of the system should be governed by some new small energy scale (not related to the Kondo temperature, at least in the plus configuration), whose properties are yet to be found. In addition, it can be expected that the various continuous phenomena found will remain correct, at least qualitatively, even in a more complete theory.