Ground State and Excitations of Quantum Dots with “Magnetic Impurities”
We consider an “impurity” with a spin degree of freedom coupled to a finite reservoir of non-interacting electrons, a system which may be realized by either a true impurity in a metallic nano-particle or a small quantum dot coupled to a large one. We show how the physics of such a spin impurity is revealed in the many-body spectrum of the entire finite-size system; in particular, the evolution of the spectrum with the strength of the impurity-reservoir coupling reflects the fundamental many-body correlations present. Explicit calculation in the strong and weak coupling limits shows that the spectrum and its evolution are sensitive to the nature of the impurity and the parity of electrons in the reservoir. The effect of the finite size spectrum on two experimental observables is considered. First, we propose an experimental setup in which the spectrum may be conveniently measured using tunneling spectroscopy. A rate equation calculation of the differential conductance suggests how the many-body spectral features may be observed. Second, the finite-temperature magnetic susceptibility is presented, both the impurity susceptibility and the local susceptibility. Extensive quantum Monte-Carlo calculations show that the local susceptibility deviates from its bulk scaling form. Nevertheless, for special assumptions about the reservoir – the “clean Kondo box” model – we demonstrate that finite-size scaling is recovered. Explicit numerical evaluations of these scaling functions are given, both for even and odd parity and for the canonical and grand-canonical ensembles.
pacs:73.23.Hk, 73.21.La, 72.10.Fk
The Kondo problem describes a single magnetic impurity interacting with a sea of electrons Hewson (1993). At temperatures of the order of or less than a characteristic scale, , the dynamics of the impurity and the sea of electrons become inextricably entangled, thus making Kondo physics one of the simplest realizations of a strongly correlated quantum system. In its original context, the impurity was typically an element of the 3d or 4f series of the periodic table, embedded in the bulk of a metal such as Cu with s conduction electrons. With the subsequent development of fabrication and control of micro- and nano-structures, it was pointed out Glazman and Raikh (1988); Ng and Lee (1988) that a small quantum dot with an odd number of electrons – small enough that its mean level spacing is much larger than the temperature – could be placed in a regime such that it behaves as a magnetic impurity.Glazman and Pustilnik (2005); Zaránd (2006); Grobis et al. (2007) The first experimental implementations of this idea were naturally made by connecting the “magnetic impurity” formed in this way to macroscopic leads Goldhaber-Gordon et al. (1998); Cronenwett et al. (1998); Pustilnik et al. (2001); Grobis et al. (2007). The flexibility provided by the patterning of two dimensional electron gas makes it possible, however, to design more exotic systems, by connecting the small magnetic impurity dot to larger dots playing the role of the electron reservoirs. Schemes to observe, for instance, two-channel SU(2) Oreg and Goldhaber-Gordon (2003); Potok et al. (2007) or SU(4)Borda et al. (2003); Le Hur and Simon (2003); Le Hur et al. (2004); Galpin et al. (2005); Le Hur et al. (2007); Choi et al. (2005); Makarovski et al. (2007a, b) Kondo have been implemented.
When the bulk electron reservoir of the original Kondo problem is replaced by a finite reservoir, two energy scales are introduced: the Thouless energy associated with the inverse of the time of flight across the structure, and the mean level spacing Kouwenhoven et al. (1997); Akkermans and Montambaux (2007); Argaman et al. (1993). A natural question which arises is therefore how these two new scales affect the Kondo physics under investigation.
Because a quantum impurity problem has point-like interactions, the local density of states completely characterizes the non-interacting sea of electrons ( and are the one-body eigenvalues and eigenfunctions of the reservoir). For , thermal smearing washes out the effects of both mesoscopic fluctuations and the discreteness of the reservoir spectrum. Indeed, in this regime, one may safely approximate by a constant ; the impurity behaves in much the same way as if it were in an infinite reservoir. In contrast, when , the impurity senses the finiteness of the reservoir through the structure of . The presence of these new energy scales (which are ubiquitous Kouwenhoven et al. (1997); Akkermans and Montambaux (2007) in reservoirs made from quantum dots) is hence an essential and interesting part of Kondo physics in nano-systems and deserves to be understood thoroughly.
The implications of a finite Thouless energy, and of the associated mesoscopic fluctuations taking place in the energy range , have been investigated mainly in the high temperature range , where a perturbative renormalization group approach is applicable Zaránd and Udvardi (1996); Kettemann (2004); Kaul et al. (2005); Yoo et al. (2005); Ullmo (2008) (see also related work Kettemann and Mucciolo (2006, 2007); Zhuravlev et al. (2007) in the context of weakly disordered system). Less is known about the implications of mesoscopic fluctuations in the temperature range .
There is on the other hand already a much larger body of work concerning the “clean Kondo box” problem Thimm et al. (1999), namely the situation where mesoscopic fluctuations are ignored (or absent as may be the case in some one dimensional models), and only the existence of a finite mean level spacing is taken into account. Simon and Affleck Simon and Affleck (2002) and Cornaglia and Balseiro Cornaglia and Balseiro (2003) have, for instance, considered how transport properties are modified if one dimensional wires of finite length are inserted between the macroscopic leads and the quantum impurity. Ring geometries Affleck and Simon (2001); Simon and Affleck (2001); Lewenkopf and Weidenmuller (2005); Simon et al. (2006), including the configuration corresponding to a two channel Kondo effect Simon et al. (2006), have also been investigated.
The basic Kondo box configuration, namely a quantum impurity connected to an electron reservoir with a finite mean level spacing, turns out to be already a non-trivial problem and so has been investigated by various numerically intensive techniques such as the non-crossing approximation Thimm et al. (1999) or the numerical renormalization group Cornaglia and Balseiro (2002). In Refs. Kaul et al., 2006 and Simon et al., 2006 it was pointed out, however, that as only the regime is affected by the finiteness of , a lot of physical insight could be obtained by the analysis of the low energy many-body spectrum of the Kondo box system (i.e. the ground state and first few excited states). An analysis of this low energy many-body spectra and of an experimental setup in which it could be probed was given in Ref. Kaul et al., 2006.
In this article, we would like on the one hand to provide a more detailed account of some of the analysis sketched in Ref. Kaul et al., 2006, and furthermore to present an additional physical application, namely the low temperature magnetic response of the Kondo box system. See also Ref. Pereira et al., 2008 for an analysis of the addition energy of a Kondo box.
Since our focus is the consequences of a finite , we consider the simplest possible configuration: a double dot system with a small dot acting as the magnetic impurity and a larger one playing the role of the electron reservoir, as illustrated in Fig. 1. The Hamiltonian describing this double-dot system is
Here creates a state with spin and energy which is an exact one-body energy level in the bigger quantum dot R. These states include all the effects of static disorder and boundary scattering. is the number operator for electrons in the reservoir, the dimensionless gate voltage applied to the large dot R, and its charging energy. As the charging energy is the leading interaction for electrons in a finite system, we shall neglect all other interactions among the electrons on R. (See, e.g., Ref. Murthy, 2005; Rotter et al., 2008; Pereira et al., 2008 for work that includes interactions among electrons in R.) The last term in Eq. (1) contains the description of the small dot and the interaction between the dots.
We consider two models for the magnetic impurity quantum dot and its interaction with the reservoir R. For most of this paper, we use a “Kondo”-like model, which therefore includes charge fluctuations only implicitly. In this case, the smaller quantum dot is represented by a spin operator . The interaction with the screening reservoir R is given by the usual Kondo interaction,
describing the anti-ferromagnetic exchange interaction between the two dots, with the spin density in the large dot at the tunneling position and .
We also consider (see Sec. II) a multi-orbital “Anderson”-type model that explicitly includes the effect of charge fluctuations on the quantum dot S:
Here the quantum dot S is described by a set of spin-degenerate energy levels created by which couple to the state in R. Interactions are included through the usual charging term of strength , where and is the dimensionless gate voltage applied to the small dot. When takes only a single value, this reduces to the usual single-level Anderson model. The crucial feature of this model is that the R-S tunneling term (proportional to ) involves only one state in the reservoir.
For temperature much larger than not only the mean level spacing but also the corresponding Thouless energy of the reservoir dot, the discreetness of the spectrum as well as mesoscopic fluctuations in R can be ignored. Thus one expects to recover the traditional behavior of a spin-1/2 Kondo or Anderson model. If , however, significantly different behavior is expected. A simplifying feature of this limit is that many physical quantities can be derived simply from properties of the ground state and low-lying excited states.
To study the low temperature regime, we shall therefore in a first stage consider the low energy (many-body) spectrum of the Hamiltonian Eq. (1). Specifically, in Sec. II we extend (slightly) a theorem from Mattis Mattis (1967) that enables us to infer the ground state spin of the system. Using weak and strong coupling perturbation theory, we then construct in Sec. III the finite size spectrum of the Kondo problem in a box.
In a second stage, we consider a few observable quantities that are derived simply from the low energy spectra. We start in Sec. IV with tunneling spectroscopy, obtained by weakly connecting two leads to the reservoir dot (Fig. 1). Using a rate equation approach, we predict generic features in the non-linear - of our proposed device. We then address in Sec. V the low temperature magnetic response of the double dot system, and in particular discuss the difference between local and impurity susceptibilities which, although essentially identical for , differ drastically when . A further issue that we study is that the charging energy in R fixes the number of electrons rather than the chemical potential; thus, the canonical ensemble must be used rather than the grand-canonical. Use of the canonical ensemble accentuates some features in the susceptibility. Finally, we conclude in Sec. VI.
Ii Ground State Theorem
We now prove an exact ground state theorem for the models defined in Eqs. (1)-(3): the ground state spin of the system is fixed, and in particular cannot depend on the coupling between the small dot and the reservoir. We give the value of this ground state spin in a variety of cases.
The theorem is mainly an extension of a theorem due to Mattis Mattis (1967). It relies on the fact that in a specially chosen many-body basis, all the off-diagonal matrix elements of these Hamiltonians are non-positive. It is then possible to invoke a theorem due to Marshall Marshall (1955); Auerbach (1994) to infer the ground state spin – a proof of Marshall’s sign theorem is in Appendix A.
ii.1 “Kondo”-type models
The starting point of the proof is the tri-diagonalization of the one-body Hamiltonian of the reservoir, . Beginning with the state , one rewrites as a one dimensional chain with only nearest-neighbor hopping Wilson (1975); Hewson (1993). This transformation is illustrated in Fig. 2. In this one-body basis, the “Kondo”-type model Eqs. (1)-(2) can be rewritten as a sum of a diagonal and off-diagonal part:
Condition of the the Marshall theorem requires us to find a many-body basis in which all the off-diagonal matrix elements are non-positive. Consider the following basis:
with the quantum number of of the local spin and the site labels (positive integers) ordered according to and . Note that this basis is diagonal with respect to both the total number of electrons in R, , and the -component of the total magnetization, .
The off-diagonal matrix elements come from two terms, the spin-flip term and the fermion hopping. With regard to the fermion hopping term, first, since the fermions have been written as a one-dimensional chain, there is no sign from the fermionic commutation relation. Additionally, one can use the freedom to choose the phase that defines the one-body states to make the hopping integrals negative. Since the number of phases is the same as the number of hopping integrals , all the can be made negative, as in Eq. (II.1). This ensures that all off-diagonal matrix elements of the fermion hopping term in the many-body basis, Eq. (7), are non-positive. With regard to the spin-flip term, note that its sign in can be fixed by rotating the spin by an angle about the -axis. In order to ensure that the off-diagonal elements due to the spin-flip term are negative, we have to include the additional phase factor appearing in the definition of the basis states in Eq. (7).
Since the basis Eq. (7) is diagonal in and , we will work in a fixed sector. Condition (ii) of Marshall’s theorem – connectivity of the basis states by repeated application of – is easily seen to be satisfied for the “Kondo” model for all , in a given sector. However, when , condition 2 is violated: the impurity spin cannot flip and hence some basis states in a sector cannot be connected to each other by repeated applications of .
We have thus shown that the Kondo model satisfies the two conditions of Marshall’s theorem in a given sector. Now note that given and , the competing spin multiplets for the ground state spin () can either be integer spin multiplets or half-integer spin multiplets. Suppose for instance they are integer multiplets (this is true, e.g., when is odd and ). Marshall’s theorem guarantees that in the sector the lowest eigenvalue can never have a degeneracy; this ensures that in a parametric evolution there can never be a crossing in the sector. Since each competing multiplet has a representative state in the sector, we infer that the ground state spin does not change as the coupling is tuned. This is true as long as we do not cross the point , because this point (as explained above) violates condition 2 in the proof of the theorem. Hence the ground state spin can be different for ferromagnetic and anti-ferromagnetic , but does not change with the magnitude of the coupling: the ground state spin for all may hence be inferred by lowest order perturbation theory in . The ground state spin for a few representative cases is displayed in Table 1.
ii.2 “Anderson”-type models
We can prove a similar theorem for the model defined by Eqs. (1) and (3). We begin by tri-diagonalizing the electrons in the reservoir R, as for the Kondo case. In addition we have to tri-diagonalize the electrons in the quantum dot S, a process which begins with the state where . An organization of into diagonal and off-diagonal parts then yields
We note again that the sign of all the hopping integrals can be fixed as displayed above by an appropriate selection of the arbitrary phase that enters the definition of the and the . The appropriate basis that has only non-positive off-diagonal matrix elements is, then, simply
The total number of particles is now , and the -component of spin is .
Iii Finite Size Spectrum
In this section, we outline the main features of the low-energy finite-size spectrum for the Kondo problem, Eqs. (1)-(2). The basic idea is to use perturbation theory around its two fixed points: at the weak coupling fixed point () expand in , and at the strong coupling fixed point expand in the leading irrelevant operators (Nozières’ Fermi-liquid theory). We begin by analyzing the classic case of with anti-ferromagnetic coupling, and then turn to the under-screened Kondo problem realized by anti-ferromagnetic coupling and .
iii.1 : Screened Kondo problem
Weak-Coupling Regime: In the weak coupling regime defined by , given a realization of the reservoir R, we can always make small enough so that the spectrum can be constructed through lowest order perturbation theory.
The unperturbed system for odd is shown schematically in Fig. 3(a). At weak coupling the eigenstates follow from using degenerate perturbation theory in all the multiplets of the unperturbed system. The ground state and the first excited state are obtained by considering the coupling
where is the spin of the topmost (singly occupied) level of the large dot. The ground state is therefore a singlet () and the first excited state is a triplet with excitation energy
The next excited states are obtained by creating an electron-hole excitation in the reservoir [shown as a dashed arrow in Fig. 3(a)]. Combining the spin of the reservoir with that of the small dot, one obtains a singlet of energy separated from a triplet by a splitting , where we define with the mean local density of states.
In the even case depicted in Fig. 3(b), the ground state is trivially a doublet. The first excited eigenstate of the unperturbed system is an -fold degenerate multiplet obtained by promoting one of the bath electrons to the lowest available empty state [shown as a dashed arrow in Fig. 3(b)]. As is turned on, this multiplet gets split into two doublets and one quadruplet. In general the two doublets have lower (though unequal) energy than the quadruplet.
Strong-Coupling Regime: For , on the other hand, the impurity spin is screened by the conduction electrons, and we can use Nozières’ “Fermi-liquid” theory.Nozières (1974, 1978) In the very strong coupling limit, one electron is pulled out of the Fermi sea to bind with the impurity; this picture essentially holds throughout the strong coupling-regime Glazman and Pustilnik (2005); Nozières (1974, 1978). For odd (even) one ends up effectively with an even (odd) number of quasi-particles that interact with each other only at the impurity site through a repulsive effective interaction
which is weak (). The quasi-particles have the same mean level spacing as the original electrons, but the spacing between two quasi-particle levels is not simply related to the spacing of the original levels in the chaotic quantum dot. This case is illustrated in Fig. 4.
For odd, the ground state is thus a singlet (as expected from our theorem), and the excitations start at energy since a quasi-particle must be excited in the reservoir. The first two excitations consist of a spin and a . Because the residual quasi-particle interaction is repulsive, the orbital antisymmetry of the triplet state produces a lower energy; the splitting is about .
In the even case at strong coupling, there are an odd number of quasi-particles in the reservoir, and so the ground state is a doublet. The first excited multiplet must involve a quasi-particle-hole excitation in the reservoir. There are two such excitations that involve promotion by one mean level spacing on average (either promoting the electron in the top level up one, or promoting an electron in the second level to the top level). Thus, the first two excitations are doublets.
Crossover between Weak- and Strong-Coupling: Remarkably, the ordering of the quantum numbers of the ground state and two lowest excitations is the same in both the and limits. It is therefore natural to assume that the order and quantum numbers are independent of . Thus we arrive at the schematic illustration in Fig. 5.
iii.2 : Under-screened Kondo problem
The theorem and perturbation theory analysis presented above has an interesting generalization to the under-screened Kondo effect, in which . We will consider for concreteness the case . Note that the under-screened Kondo effect has been realized experimentally in quantum dots.van der Wiel et al. (2002)
For even and , we find that the ground state for all has . At weak coupling, this follows directly from perturbation theory – the reservoir has spin zero and is too small to promote an electron so the spin of the ground state is just that of the small dot. The theorem then implies that for all . The first excited multiplet is at energy of order . It splits into a singlet, two triplets, and a quintuplet; as increases, the singlet has the lowest energy because the coupling is anti-ferromagnetic.
In the opposite limit of strong coupling, as , one electron from R binds to the impurity spin forming a spin-1/2 object. For , this spin does not interact with the quasi-particles in R; however, when , the flow to strong coupling generates other irrelevant operators that connect the spin to the quasi-particles. It is known from studies of the under-screened Kondo problem that the leading irrelevant operator is a ferromagnetic Kondo coupling Nozières and Blandin (1980) (the sign of the coupling follows heuristically from perturbation theory in ). However, since one of the electrons is bound to the spin, there is an odd number of quasi-particles in the effective low energy ferromagnetic Kondo description – the level filling is as in Fig. 4(b). Since the ferromagnetic Kondo problem flows naturally to weak coupling Hewson (1993), we are again justified in doing perturbation theory in the coupling, and so recover that the ground state has . From the small ferromagnetic coupling, we conclude that the first excited state is a singlet separated from the ground state by an asymptotically small energy (as ). The next excited state involves promotion of a quasi-particle to the next level within R and so has energy of order . It is a triplet because of the ferromagnetic coupling, with a nearby singlet in the strong coupling limit. Note that there are two possible quasi-particle excitations with energy of order (as discussed in the case), and so two singlet-triplet pairs.
The proposed crossover from weak to strong coupling for even is shown in Fig. 6(a). Note that in this case, level crossings of excited states must occur: the two singlets at energy of order at strong coupling come from energies greater than at weak coupling, and so cross the state. The two singlet-triplet pairs at strong coupling are shown to be slightly different because each involves a different level spacing; thus, there is an additional level crossing as one of the singlets comes below a triplet.
In the odd case, weak anti-ferromagnetic coupling implies that the ground state spin is . The first excited state is the other multiplet involving no excitations in the reservoir, . The next excited states are the and states that involve promoting one electron by one level. In the strong coupling limit, we repeat the mapping to a ferromagnetically coupled impurity, yielding this time an even number of quasi-particles in the reservoir. Now the first excited state involves promoting a quasi-particle in the reservoir by one level; the ferromagnetic coupling implies that the state has the lowest energy among the possible multiplets. Making again the reasonable assumption that the two limits are connected to each other in the simplest manner possible, we arrive at the schematic illustration in Fig. 6(b). In contrast to the even case, no level crossings are definitely required.
Iv Non-linear I-V characteristics of the R-S system
We now turn to the question of how to observe the features of the finite size spectrum delineated in the previous section. Any physical observable depends, of course, on the spectrum of the system and so could be used as a probe. We choose to concentrate on two: (1) In the next section, we discuss the magnetic susceptibility of the R-S system, a classic quantity in Kondo physics. (2) In this section we discuss the conductance across the device shown in Fig. 1. The advantage of this physical quantity is that the finite-size spectrum can be observed directly in the proposed experiment. The emphasis here is on transfer of electrons entirely by real transitions; cotunneling processes, which involve virtual states, are briefly discussed at the end of the section.
A current through the R-S system clearly involves number fluctuations on it. For a general value of the gate voltage [ in Eq. (1)], however, the ground state will have a fixed number of electrons, and hence (Coulomb blockade). When is increased sufficiently, the Coulomb blockade is lifted, and has a sequence of peaks. It is possible to extract the excitation spectrum of the R-S system from the position of these peaksvon Delft and Ralph (2001). In principle, there is a peak in for every transition that involves a change in . As discussed in subsection IV.3 we shall however choose a particular setting such that only a limited number of these transitions play a role, making in this way simpler the reconstruction of the underlying low-energy many-body spectra.
In order to describe transport through the R-S system (realized through either a double dot or a metallic grain with a single magnetic impurity), we solve the appropriate rate equations for the real transitions Beenakker (1991); von Delft and Ralph (2001). The rate equations are a limit of the quantum master equation in which the off-diagonal elements of the density matrix are neglected. The dynamics of the quantum dot can then be described simply by the probability that the R-S system is in a given many-body state . In thermal equilibrium these are the Boltzmann weights. The electrons in the leads are assumed to always be in thermal equilibrium; hence, the probability that a given one-body state in the leads is occupied is given simply by the Fermi-Dirac function . Here, is the deviation from the electro-chemical potential , where and are the voltages on leads and respectively.
Steady state requires that the are independent of time. Hence, the various rates of transition from to , , must balance, leading to a linear system for the ,
In addition, the occupation probabilities should be normalized, .
There are four transitions that have to be taken into account: addition or removal of an electron from lead L1 or L2. We denote the rates for these four processes as , and the in Eq. (14) are sums of these four transition rates. Once we have the from (14), the current is simply
The conductance then follows by differentiating .
The rates can be calculated in second order perturbation theory in the reservoir-lead coupling term, using Fermi’s golden rule von Delft and Ralph (2001). For example, consider the addition of an electron to R-S from corresponding to a transition on R-S:
where . is the amplitude for the above process. Although, in general it will have some dependence on and as well as the coupling , we will ignore such dependence here. We will, however, retain the distinction between and allow these to be tuned by the gates that define the R-L1 and R-L2 junctions.
To summarize our approach, to find the conductance in the proposed tunneling experiment, we have solved the rate equations for transferring an electron from lead 1 to the reservoir and then to lead 2.Beenakker (1991); von Delft and Ralph (2001) We assume that (1) the coupling of the lead to each state in R is the same (mesoscopic fluctuations are neglected), (2) the Kondo correlations that develop in R-S do not affect the matrix element for coupling to the leads,fns () (3) there is a transition rate that provides direct thermal relaxation between the eigenstates of R-S with fixed , (4) the electrons in the lead are in thermal equilibrium, and (5) the temperature is larger than the widths of the R-S eigenstates due to L1 and L2.
iv.2 Magnetic Field
A Zeeman magnetic field can be used as an effective probe of the various degeneracies of the R-S system. We shall assume that the magnetic field does not couple to the orbital motion of the electrons:
This can be achieved in the semiconductor systems by applying the field parallel to the plane of motion of the electrons. The effect of an orbital magnetic field in ultra-small metallic grains is argued to be small in Ref. von Delft and Ralph, 2001 for moderate fields.
We may neglect the effect of on the lead electrons: The only characteristic of the lead electrons appearing in the rate equation calculations is the density of states at . All that does to the lead electrons is to make the modification . Since the band is flat and wide (on the scale of ) to an excellent approximation, this has no effect.
The effect of on the R-S system is complicated if the -factors for the and electrons are different, as would be the case for a magnetic impurity in a metallic nano-particle. If we assume, however, that the -factors for the electrons on the and systems are the same, as is relevant for the semiconductor quantum dot case illustrated in Fig. 1, then becomes simply . The energy of a given many-body level is then where is the corresponding eigenvalue of the many-body state.
iv.3 Application to R-S System
To identify characteristic features in the transport properties, let us analyze a situation in which only a limited number of transitions show up.von Delft and Ralph (2001) For a Kondo problem the most interesting features appear in the spectrum when there is an odd number of electrons in the reservoir. These states appear clearly when an electron is added to a even reservoir and the parameters are such that the excited states of the electron reservoir dominate.
We thus consider the following situation: For zero bias, assume that the R-S system is brought into a Coulomb blockade valley, not far from the transition. This could be done by adjusting and in the setup of Fig. 1 (with ), or more realistically with the the help of the additional gate voltage in Eq. (1). We take this setup as the origin of the bias potentials (). Upon applying a bias , electrons flow from lead 1 to lead 2.
We assume that the rates are sufficiently small that virtual processes (cotunneling) can be entirely neglected for this subsection; that is, all relevant transitions occur on shell and can be described by the Fermi golden rule expression Eq. (16). Furthermore, we take . Because , this means that it takes much longer to add an electron to the dot than to empty it. Thus, the dot tends to be occupied by electrons.
Several conditions are needed in order to restrict the discussion to just the lowest lying states of the system. First, we shall assume that so that in equilibrium only the ground state doublet, with energy , needs to be considered. In a non-equilibrium situation, however, higher excited states can also be populated: the excess energy of the electron supplied by the bias can be used to leave the dot in an excited state. Apart from the ground state doublet, we take all excited states to be higher in energy than .EFt () Then, if an excited state is populated, it will quickly relax to an energy below through a rapid exchange of particles back and forth between the dot and lead 2 (). Because off-shell processes are assumed negligible, this relaxation will stop as soon as an -electron state below is reached. We assume that the energy of the first excited doublet, , is large enough that . Then only the -electron ground state multiplet needs to be retained in the calculation.
With regard to the electron states, we limit ourselves to a small enough bias such that only transitions to the three lowest excited multiplets need to be taken into account. In this way, only a small number of transitions will show up in the excitation spectrum, making it relatively simple to analyze.von Delft and Ralph (2001)
When a magnetic field is applied, note the following unusual behavior: Since there is no way to decay from the state to the state of the lowest doublet without involving virtual processes explicitly neglected here, the doublet will remain out of equilibrium: the state can be significantly populated even though .
iv.4 Results for
With the above assumptions, results for the differential conductance are shown in Fig. 7. We assumed that the system parameters (gate potential, , , and ) are such that the first three states of the electron system coincide with the ground state of the electron system for , , and , respectively, at . (See Fig 4. of Ref. Kaul et al., 2006 for in the case.) We are thus assuming that the excited triplet state lies midway between the two lowest singlet states (see Fig. 5), placing ourselves in the middle of the cross-over regime.
First, note that the ground state to ground state transition, , yields only one peak even at non-zero . This is because the state of the doublet cannot be populated before some current is flowing through the R-S system (). However, after the first transition, the state can decay into the state. Hence, we expect the higher transitions to split in a magnetic field, as in Fig. 7.
The next feature to understand is the two transitions. These two peaks occur because out of the six transitions between the multiplets, two are forbidden by spin conservation and the other four split into two degenerate sets.
How is one then to distinguish between a and state, since they both split into two as a function of ? One possible method is to observe the peak heights in keeping fixed. These are plotted in Fig. 7 for a variety of .fnG () A clear feature is that the two peaks are very asymmetric, while the are almost symmetric. This is for a robust physical reason: each peak gets contributions from both and initial states, while in the transition each peak gets a contribution from only one, the for the taller peak and for the shorter one. The associated probabilities, and , are shown in the lower panel of Fig. 7. Thus the peak heights in the transitions are insensitive to the difference between the probability of occupation of the two states in the doublet, while the peak heights in the transitions are sensitive to this difference.
iv.5 Energy Relaxation
In any real system, there are mechanisms of energy relaxation beyond the energy conserving exchange of electrons with the leads that is given by the rate equations. These mechanisms can involve, for instance, interactions with phonons or, more simply, higher order virtual processes between the R-S system and the leads that are neglected in the Fermi’s golden rule approach Eq. (16). These relaxation processes are particularly important for the second transition. If the system is in perfect thermal equilibrium this transition should yield a single peak, even in the presence of a . The second peak is suppressed even if only on-shell processes are taken into account, as discussed above, but explicit energy relaxation causes this suppression to be more pronounced.
To model energy relaxation, we include a transition rate between the states that satisfies detailed balance (i.e., with Boltzmann weights),
where is the energy eigenvalue of the state.
The effect of this term is shown in Fig. 8. Clearly as the relaxation rate is increased, the peak in the second transition coming from non-equilibrium effects is suppressed further. Note, however, that the heights of the transition are unaffected (in both relative and absolute magnitude).
iv.6 Cotunneling Spectroscopy
While the approach proposed above should be reasonably simple to implement, because the excitations of both the and electron systems may come into play, the resulting experimental conductance curves may in some circumstances be non-trivial to interpret. Therefore, we mention, without going into detail, an alternative way to extract the excitation spectra from the differential conductance. Though within the “Coulomb blockade diamond”, on-shell processes such as the ones considered above are forbidden by energy conservation constraints, a small current can nevertheless be measured, which is associated with virtual (cotunneling) processes.Aleiner et al. (2002)
At very low bias, these virtual processes are necessarily elastic as the electron transferred from one lead to the other does not have enough energy to leave the R-S system in an excited state. However, each time reaches a value corresponding to an excitation energy of the system with electrons, a new “inelastic” channel is open, as the electron has the option to leave the R-S system in an excited state as it leaves the structure. The opening of these new channels produce steps in the differential conductance within the Coulomb diamond. These steps are small, but clearly observable experimentally De Franceschi et al. (2001); Zumbühl et al. (2004); Makarovski et al. (2006).
Because of the smallness of the associated currents, observing this substructure within the Coulomb diamond is certainly more challenging experimentally than for observing the main peaks associated with on-shell processes. On the other hand, the time elapsed between the successive transfers of an electron across the structure is large enough that the initial state of the -particle system is always the ground state. If they can be measured accurately, the cotunneling steps within the Coulomb diamond may therefore lead more directly to the -particle excitation spectra.
Summarizing this section, we have shown in detail how measurements enable one to extract the finite size spectrum and spin quantum numbers of the R-S system, using the case when the ground state has an even number of electrons as an example. In particular, we argued that the relative peak height of the Zeeman split terms () reflects the spin quantum number of the excitation: asymmetric peak heights correspond to , whereas symmetric peak heights correspond to . The case when is odd is straightforward to analyze in a similar way. Transitions from to or can easily be distinguished: the former splits into two in a magnetic field while the latter splits into four.
V Magnetic response of the double dot system
We turn now to studying a second physical observable which probes the finite size spectrum of the system, namely the magnetic susceptibility defined by
where is the canonical or grand-canonical partition function depending on the ensemble considered. As in Section IV, we assume that the magnetic field is in plane so that only the Zeeman coupling needs to be considered, Eq. (17). We furthermore distinguish between the local susceptibility , corresponding to the case where couples only to the quantum impurity spin (), and the situation where couples to the total spin of the R-S system (). In the latter case, the impurity susceptibility is defined as the difference between the total magnetic response and that of R in the absence of the impurity dot.
For a wide ( and flat () band the local and impurity susceptibilities are essentially identical Clogston and Anderson (1961). Indeed the effect of the magnetic field on the reservoir electrons is just to shift the energies of the spin up electrons with respect to the spin down by a fixed amount. If the spectrum is featureless, this only affects in practice the edge of the band, which in the limit and with fixed will not affect the Kondo physics. More precisely, for small but finite (and again for a wide flat band) the impurity susceptibility, being associated with the correlator of a constant of the system, can be written as
where is a universal function of the ratio . On the other hand, since the spin of the impurity is not conserved, a multiplicative renormalization factor needs to be introduced for the local susceptibility so that . For we use a form motivated by two loop renormalization, for , with the coefficient of the quadratic term determined empirically, . (For a discussion of in the context of two loop renormalization, see e.g. Ref. Barzykin and Affleck, 1998.) We note here that in the universal regime , one also has and hence . In practical numerics, even though is small enough that the correction can be neglected, need not be as small; hence, it is necessary to include the prefactor correction to observe good scaling behavior.
In the regime that we consider here, however, the reservoir electron spectrum is not featureless near the Fermi energy, and the Zeeman splitting of the conduction electrons affects in practice the whole band, and not just the band edge. One therefore does not particularly expect any simple relation between the local and impurity susceptibilities. We now discuss the behavior of these quantities in this regime. We start with the canonical ensemble, for which the number of particles , and therefore the parity of , is fixed. We’ll consider in a second stage the grand canonical ensemble and so neglect charging effects in the reservoir [ in Eq. (1)]; in this case the spin degeneracy induces finite fluctuations of the particle number even in the zero temperature limit.
v.1 Canonical ensemble
Since is a good quantum number, the impurity susceptibility in the canonical ensemble follows immediately from the information contained in Fig. 5, i.e. from the knowledge of the total spin and excitation energy of the first few many-body states. Neglecting all the levels with an excitation energy of order (because ), we simply get for a spin 1/2 Curie law for even , and a spin 1 Curie law damped by for odd N. In this latter case, the magnetic response in the absence of the impurity is also a spin 1/2 Curie law; thus, for one finds
The local susceptibility on the other hand involves which is not a conserved quantity. Its computation therefore requires knowledge of the eigenstates, in addition to the eigenenergies and total spin quantum numbers contained in Fig. 5. We can follow the same approach used in Section III and analyze the two limiting regimes of coupling between the reservoir and the impurity quantum dot. We will then use a numerical Monte Carlo calculation in the intermediate regime and investigate how well it is described by a smooth interpolation between the two limiting regimes.
In the weak coupling regime, , we assume that even if some renormalization of the coupling constant takes place, the eigenstates are the ones obtained from first-order perturbation theory in this parameter. For even at , the impurity spin decouples from the (frozen) electron sea, and one obtains again a spin 1/2 Curie law. For odd at , the system formed by the impurity spin and the singly occupied orbital decouples from the set of doubly occupied levels. The magnetic response is the same as for two spin 1/2 particles interacting through Eq. (11). We thus obtain
valid for .
Turning now to the strong coupling regime, we follow Nozières’ Fermi liquid picture Nozières (1974, 1978), where low energy states (with ) are constructed from quasiparticles which interact locally according to Eq. (13). In a local magnetic field, the energy of a state is modified to
where the sum is over all the many-body excited states . The first term in this expression yields the effect of a change of the quasiparticle phase shift on the energy. It is important when is even: one of the quasiparticle states is singly occupied, and its energy is shifted by an amount . Thus the system acts like a spin-1/2 particle with an effective -factor given by . The result is a weak Curie susceptibility at low temperature.
The second term captures the effect of electron-hole quasiparticle excitations. It produces a non-zero contribution even when the discreteness of the spectrum is ignored, as may be seen as follows. First, the density of states of particle-hole excitations of energy in a Fermi liquid is proportional to . In a Kondo state, the density of single-particle states is increased by a factor of because of the Kondo resonance. Thus we may replace the sum in Eq. (23) by an integral using a density of states proportional to . The integral should be cutoff at an energy of order , where the Kondo resonance ends. Thus the second term in Eq. (23) gives a contribution to the energy, and a corresponding contribution to .
Note that this second contribution is independent of the finite-system parameter and so is the universal (bulk) part of the local susceptibility.Fin () It should behave smoothly as becomes smaller than one. In particular, if one considers a system without mesoscopic fluctuations (i.e. with constant spacings and wave function amplitudes at the impurity), we expect to recover the bulk behavior for . For even this relation holds only if the weak Curie behavior of the first term is not too large; more precisely, we expect to follow the universal behavior as long as .
With these arguments, we have thus arrived at a complete description of the magnetic susceptibility in both the weak and strong coupling limits for the canonical ensemble. Note in particular the difference in the conditions for having from those for having . Both limits hold in the regime no matter what the value of . However in addition, for any as long as .
v.2 Grand-canonical ensemble
Use of the grand canonical ensemble [which involves neglecting charging effects in the reservoir, in Eq. (1)] introduces additional complications compared to the canonical ensemble case above. To illustrate, recall first the behavior in the absence of the impurity, i.e. for a system of independent particles occupying doubly degenerate states . For in the grand canonical ensemble, the magnitude of the fluctuation of the number of particles will be significantly larger than one. Thus, even if the canonical ensemble result for even is quite different from that for odd [Eqs. (21)-(22)], such an odd-even effect would be completely washed out here: whatever the choice of the chemical potential , configurations with odd or even would be as probable.
In the low temperature regime on the other hand, as soon as (which is usually ), there is a fixed, even number of particles in the system. It is possible to make the average number of particles odd by choosing for some orbital . In that case, as , all orbitals are doubly occupied, all orbitals are empty, and independent of the orbital has probability to be empty, to be doubly occupied, and to be singly occupied. For quantities showing some odd-even effect in the canonical ensemble but no strong dependence on once the parity is fixed (such as the local susceptibility), the grand canonical ensemble produces a behavior which is the average of the the odd and even canonical response, even though the mean number of particles is odd.
Turning now to the full R-S system, the above non-interacting picture should certainly still hold in the weak coupling regime. If, either by adjusting or by making use of some symmetry of the one particle spectrum, is kept fixed with an even integer value as , one should recover the canonical magnetic response for even . In contrast, the magnetic response for odd should be the average of the canonical odd and even responses.
In the strong coupling regime (following again Nozières’ Fermi liquid description), one also has an essentially non-interacting picture, but with effectively one less particle since one reservoir electron is used to form the Kondo singlet. The role of “odd” and “even” are then exactly reversed from the weak coupling case: for the grand canonical response will be the average of the canonical odd and even response for even , and will be exactly the canonical response for odd .
v.3 Universality in a clean box
The previous discussion mainly addressed the two limiting behaviors – weak and strong coupling. To investigate the intermediate regime we now turn to numerical calculations. In particular, we use the efficient continuous-time quantum Monte Carlo algorithm introduced in Ref. Yoo et al., 2005, with in addition adaptations to compute quantities in the canonical ensemble can (). We study the behavior of the singlet-triplet gap and the local susceptibility ; the impurity susceptibility follows directly from .
To focus on the consequences of the discreetness of the one particle spectrum while avoiding having to explore an excessively large parameter space, we disregard the mesoscopic fluctuations of the spectrum and the wave-functions. That is, we consider the simplified “clean Kondo box” model Thimm et al. (1999); Simon and Affleck (2002); Cornaglia and Balseiro (2003) defined by and independent of . For initial results for the more realistic “mesoscopic Kondo model” see Refs. Kaul et al., 2005 and Yoo et al., 2005.
Under these conditions, the problem is described by only three dimensionless parameters: the coupling , and the two energy ratios and . For small and large (), and can be scaled away in the usual manner so that, except for renormalization prefactors such as which may still contain some explicit dependence on , physical quantities depend on and only through the Kondo temperature .
We therefore expect, again up to the factor , that both susceptibilities for the “clean Kondo box” model will be universal functions of the two parameters and . This function may, however, be different for the local and impurity susceptibilities, and will also depend on the parity and type of ensemble considered.
Before discussing how well the limiting behaviors discussed above describe the whole parameter range, we shall first check that our numerics confirm the expected universality. For the impurity susceptibility, since Eq. (21) is valid in the full range of coupling as long as , we only need to verify that for odd the singlet-triplet excitation energy is, as expected from the same argument, a universal function of : with the limiting behaviors
The strong coupling behavior follows directly from the discussion in Sec. III.1. The scaling behavior of at weak-coupling can, on the other hand, be obtained from a perturbative renormalization group argument: in the perturbative expression for Eq. (12), replace the coupling constant by its renormalized value at the scale . Within the one-loop approximation, yields
Substituting the one-loop expression for the Kondo temperature, , now yields the second line of (24).
In the crossover regime, we find through continuous time quantum Monte Carlo (QMC) calculations using a modification of the algorithm presented in Ref. Yoo et al., 2005 with updates which maintain the number of particles (canonical ensemble) can (). To extract , we measure the fraction of states with visited in the Monte-Carlo sampling at temperature . For a fixed and large , can be excellently fit to the form valid for a two-level singlet-triplet system. Repeating this procedure yields for a variety of and .
Fig. 9 shows the results of our calculations, plotted as a function of . We emphasize three features: (i) The inset shows that the fit of our QMC data to the simple two-level singlet-triplet form is indeed very good. (ii) The limiting behaviors of Eq. (24) are clearly seen. (iii) Data for a wide variety of bare parameters is shown; the excellent collapse onto a single curve in the main figure is a clear demonstration of the expected universality.
We now turn to the local susceptibility and explore the expected scaling ansatz
for variety, we use the grand canonical ensemble. By fixing the chemical potential in the middle of the spectrum of the reservoir, particle-hole symmetry ensures that even in the presence of the Kondo coupling the mean number of particles has a fixed parity: if is aligned with a level,