Valley Kondo Effect in Silicon Quantum Dots Shiue-yuan Shiau, Sucismita Chutia, and Robert Joynt† Department of Physics, University of Wisconsin, 1150 Univ. Ave., 7 0 Madison, Wisconsin 53706, † and Department of Physics, 0 2 University of Hong Kong, Hong Kong, China n a Abstract J 5 Recent progress in the fabrication of quantum dots using silicon opens the prospect of observing 1 ] the Kondo effect associated with the valley degree of freedom. We compute the dot density of l e - states using an Anderson model with infinite Coulomb interaction U, whose structure mimics the r t s nonlinearconductancethroughadot. Thedensityofstatesisobtainedasafunctionoftemperature . t a and applied magnetic field in the Kondo regime using an equation-of-motion approach. We show m - that there is a very complex peak structure near the Fermi energy, with several signatures that d n distinguish this spin-valley Kondo effect from the usual spin Kondo effect seen in GaAs dots. We o c [ also show that the valley index is generally not conserved when electrons tunnel into a silicon 2 dot, though the extent of this non-conservation is expected to be sample-dependent. We identify v 2 featuresoftheconductancethatshouldenableexperimenterstounderstandtheinterplayofZeeman 2 7 1 splittingandvalleysplitting,aswellasthedependenceoftunnelingonthevalleydegreeoffreedom. 1 6 0 / t a m - d n o c : v i X r a 1 I. INTRODUCTION Experimentation on gated quantum dots (QDs) has generally used GaAs as the starting material, due to its relative ease of fabrication. However, the spin properties of dots are becoming increasingly important, largely because of possible applications to quantum in- formation and quantum computing. Since the spin relaxation times in GaAs are relatively short, Si dots, with much longer relaxation times [1, 2, 3, 4], are of great scientific and technological interest [5]. Recent work has shown that few-electron laterally gated dots can be made using Si [6][7], and single-electron QDs are certainly not far away. There is one important qualitative difference in the energy level structures of Si and GaAs: the conduction band minimum is two-fold degenerate in the strained Si used for QDs as compared to the non-degenerate minimum in GaAs. Thus each orbital level has a fourfold degeneracy including spin. This additional multiplicity is referred to as the valley degeneracy. For applications, this degeneracy poses challenges - it must be understood and controlled. From a pure scientific viewpoint, it provides opportunities - the breaking of the degeneracy is still poorly understood. Previous experimental work has shown that in zero applied magnetic field the splitting of the degeneracy is about 1.5±0.6 µeV in two-dimensional electron gases (2DEGs) [8] and that it increases when the electrons are further confined in the plane [8]. Surprisingly, the splitting increases linearly with applied field. Theoretical understanding of these results has historically been rather poor, with theoretical values much larger than experimental ones for the zero-field splittings [9][10] and theory also predicting a nonlinear field dependence [11]. Recentworkindicatesthatconsiderationofsurfaceroughnessmayresolvethesediscrepancies [13]. In Si QDs, a related issue is also of importance: what is the effect of valley degeneracy on the coupling of the leads to the dots? The spin index is usually assumed to be conserved in tunneling. Is the same true of the valley index? Recent measurements of the valley splitting in a quantum point contact show a valley splitting much larger than in the 2DEGs, about 1 meV in quantum point contacts [8]. A QD in a similar potential well is expected to produce a valley splitting of the same order of magnitude. The overall picture of the degeneracy is as shown in Fig. 1 for a single orbital level of a Si QD. Of particular interest is the fact that a level crossing must occur, and the rough value of the applied field at this point is B ≈ 2.5 T, given a reasonable zero-field cr 2 FIG. 1: Schematic diagram of the energy levels of a single orbital in a Si QD in a magnetic field. The quantum numbers are even (e), odd (o) valley states and up and down spin projections for spin 1/2 states. A level crossing occurs at the magnetic field B . cr valley splitting of 0.5 meV, and the valley-spin slope of magnetic field for the ”o ↑ ” state to be 0.1 meV/T, as it is in Hall bars. The aim of this paper is to study the Kondo effect in the context of transport through a Si dot. The Kondo effect was originally discovered in dilute magnetic alloys. At low temper- atures, the electron in a single impurity forms a spin singlet with electrons in the conduction band, thus causing an increasing resistance as the temperature is reduced to zero. Since the first observations of the Kondo effect in GaAs QDs, there has been considerable experimen- tal and theoretical work done, mainly because QDs provide an excellent playground where one can tune physical parameters such as the difference between energy levels in QDs and the Fermi level, the coupling to the leads, and the applied voltage difference between the leads. The level of scientific understanding of this purely spin Kondo effect in QDs is on the whole quite satisfactory. The basic phenomena are as follows. At temperature T ≤ T , K where the Kondo effect appears, a zero-bias peak in the dot conductance is observed. An applied magnetic field splits the peak into two peaks separated by twice the Zeeman energy 2gµ B. These linear peak energy dependencies on the applied magnetic field have been B observed in GaAs QDs [14]. The Kondo effect only occurs when the occupation of the dot is odd. Clearly, Si dots will have a much richer phenomenology. There are multiple field dependencies as seen in Fig. 1, and the additional degeneracy can give rise to several Kondo peaks even in zero field. Furthermore, the Kondo temperature is enhanced when the degen- 3 eracy is increased. We build on previous work on the orbital Kondo effect. For example, an enhanced Kondo effect has been observed due to extra orbital degree of freedom in carbon nanotubes (CNTs) [15][16] and in a vertical QD, with magnetic field-induced orbital degen- eracy for an odd number of electrons and spin 1/2 [17]. More complex peak structure in their differential conductance suggests entangled interplay between spin and orbital degrees of freedom. We also note that a zero-bias peak has been observed in Si MOSFET structures [18]. Our goalwill be to elucidate the characteristic structures in the conductance, particularly their physical origin, and their temperature and field dependencies. Once this is done, one can hope to use the Kondo effect to understand some of the interesting physics of Si QDs, particularly the dependence of tunneling matrix elements on the valley degree of freedom. This paper is organized as follows. In Sec.II, we determine whether valley index conser- vation is to expected; in Sec. III, we introduce our formalism; in Sec. IV, we discuss the effect of valley index non-conservation; in Sec. V, we present our main results; in Sec. VI we summarize the implications for experiments. II. TUNNELING WITH VALLEY DEGENERACY The basic issue at hand is the conductance of a dot with valley degeneracy, and the influence of the Kondo effect on this process. The dots we have in mind are separated electrostatically from 2DEG leads on either side. These 2DEGs have the same degeneracy structure. Thus before we tackle the issue of the conductance we need the answer to a preliminary question: is the valley index conserved during the tunneling process from leads to dots? Non-conservation of the valley index will introduce additional terms in the many-body Hamiltonian that describes the dots, and, as we shall see below, it changes the results for the conductance. In this section, we investigate the question of valley index conservation in a microscopic model of the leads and dots. We do this in the context of a single-particle problem that should be sufficient to understand the parameters of the many-body Hamiltonian that will be introduced in Sec. III. If we have a system consisting of leads and a QD in a 2DEG of strained Si, then both the leads and the dot will have even and odd valley states if the interfaces defining the 2DEG are completely smooth. The valley state degeneracy is split by a small energy. If the valley 4 index is conserved during tunneling, then the even valley state in the lead will only tunnel into the even valley state in the dot. On the other hand, if the valley index is not conserved, then the even valley state in the lead will tunnel into both the even and odd valley states in the dot; similarly for the odd valley state in the lead. We can estimate the strength of the couplings between the various states in the leads and the dot by calculating the hopping matrix elements between them. Consider a system of a lead and a dot in a SiGe/Si/SiGe quantum well, separated by a barrier of height V . If V = ∞, there will be no tunneling between the lead and the dot b b and the eigenstates of the whole system can be divided into lead states |Ψ∞ i and dot L,(e,o) states |Ψ∞ i where e, o are the even and the odd valley states. Call the Hamiltonian for D,(e,o) this case H . When V is lowered to a finite value, there will be some amount of tunneling 0 b of the lead wavefunction into the dot. Call the Hamiltonian for this case H. This can be thought of as a perturbation problem where the nth perturbed eigenstate |ni is no longer the nth unperturbed eigenstate |n0i but acquires components along the other unperturbed eigenstates |k0i. The perturbationHamiltonian is H′ = H−H . H′ considered as a function 0 of position is not small, but the matrix elements of H′ are small. This means that in the 1st order perturbation theory we can write the perturbed nth state as V |ni = |n0i+ |k0i kn (1) E0 −E0 kX6=n n k where V = hk0|H′|n0i is the hopping or tunneling matrix term. kn If we further assume that the tunneling Hamiltonain conserves spin and that only one orbital state need be considered on the QD, we can expand the perturbed even valley lead wavefunction (with tunneling) as V V |Ψ i = |Ψ∞ i+ e,e |Ψ∞ i+ e,o |Ψ∞ i (2) L,e L,e |E0 −E0 | D,e |E0 −E0 | D,o Le De Le Do Usually, in a perturbation problem, the unperturbed wavefunctions and the hopping matrix elements are known and used to find the perturbed wavefunction. However in the present case we use a model to compute the perturbed and unpertubed wavefunctions and use these to determine the unknown hopping matrix elements V V . These will be needed in later e,e e,o sections. From the above equation, V = (|E0 −E0 |)hΨ∞ |Ψ i (3) e,e Le De De Le 5 V = (|E0 −E0 |)hΨ∞ |Ψ i (4) e,o Le Do Do Le The perturbed and the unperturbed wavefunctions and the corresponding energies are cal- culated by using an empirical 2D tight binding model with nearest (v ) and next nearest z neighbor interactions (u ) along the z (growth) direction and the nearest neighbor interac- z tions (v ) along the x direction (in the plane of the 2DEG). Potential barriers and edges of x the quantum well are modeled by adjusting the onsite parameter (ǫ) on the atoms. This is an extension of the 2-band 1D tight binding model outlined by Boykin et al. [9, 10], considering only the lowest conduction band of Si. This is a single particle calculation with the magnetic field B = 0. Note that the parameters from the Boykin et al. model are chosen precisely so as to reproduce the two-valley structure of strained Si. The single-particle Hamiltonian for the system is H = [ ǫ|φ(x,z)ihφ(x,z)|+v |φ(x,z)ihφ(x+1,z)|+v |φ(x,z)ihφ(x,z +1)| x z Xx + u |φ(x,z)ihφ(x,z +2)|+v |φ(x−1,z)ihφ(x,z)|+v |φ(x,z −1)ihφ(x,z)| z x z + u |φ(x,z −2)ihφ(x,z)| ] (5) z The parameters defining the system are as follows: the lead is 80 atoms long, the barrier between the lead and the dot is 16 atoms wide along x and 0.3 eV high, the QD is 38 atoms (≈ 10 nm) wide along x and 38 atomswide alongz, the barrier alongz defining the quantum well is 20 atoms wide on each side and its height is 0.3 eV. The nearest and next nearest neighbor interaction terms along z are v = 0.68264 eV, and u = 0.611705 eV, and that z z along x is v = −10.91 eV. The schematic of the system is shown in Fig. 2 x We consider two kinds of 2DEGs, one with no interface roughness at the interface and the other with a miscut of about 2◦. We find that the valley index is conserved in the case of smooth interfaces. We model the miscut interface as a series of regular steps of one atom thickness and width defined by the tilt, e.g., a 2◦ tilt corresponds to a step length of about 30 Si atoms. In this case, the valley index is no longer conserved. Strengths of the coupling between the e−e and e−o valley states is found to depend strongly on the relative distance between the edge of the closest step and the edge of the quantum dot. In Fig. 3, we plot the hopping matrix elements as a function of this relative distance. The terms show oscillations that have a period equal to the step size in this case. The coupling strengths are also found to depend on the tilt of the substrate on which the 2DEG is grown. We show in Fig. 4, the hopping matrix elements as a function of the 6 Qdot width z Vb Vb distance between x step and dot edge h td iw G tilt θ E D 2 Stepsize FIG. 2: The system of the lead and the dot in a miscut quantum well with the potential profile along the x-direction. The open circles represent Si atoms sandwiched between the SiGe atoms in the barrier region represented by the filled circles. stepsize (∝ 1/ tilt) and find that the hopping terms vary rapidly at small step sizes (i.e large tilts) but slowly at larger stepsizes (i.e., smaller tilts). Inaquantumwell, thephasesoftherapidoscillationsoftheelectronwavefunctions (along the z-direction) are locked to the interface and in a well with smooth interfaces, the phase remains the same for a particular valley throughout [12]. But in a miscut well, the phases are different for electrons localized at different steps even for the same valley eigenstate. The resultant valley splitting is due to interference of phase contributions from all the steps. Hence the valley splitting can be very different in the lead and the dot. The same argument explains the conservation of the valley index across the barrier in the lead-dot system. In the case of a smooth interface, the e eigenstate in the lead has same phase of the z-oscillations as the e eigenstate in the dot, and is 90◦ out of phase with the o eigenstate in the dot. Hence, there is a finite overlap between the e−e wavefunctions, while the e−o overlap cancels out due to interference. This preserves the valley index during the tunneling. In the case of a miscut well, we can no longer strictly label the eigenstates e or o, but let us continue to do so just for convenience. Here, because e in the lead does not really have 7 x 10−3 1.5 s m 1 er g t n pi p0.5 o H 0 0 5 10 15 20 25 30 35 40 45 50 40 s m er30 g t n ppi20 o h o of 10 ati R 0 0 5 10 15 20 25 30 35 40 45 50 distance between step edge and Qdot edge(atoms) FIG. 3: (a)Tunneling matrix elements (V ,V ) in units of eV and (b) Ratio of the tunneling e,e e,o matrix elements (V /V ) as a function of the relative distance between the step edge and the e,o e,e quantum dot edge for constant barrier height and width (for a miscut QW). The solid line with triangular markers indicate the V term and the dashed line with open circles indicate the V e,e e,o term. Here the stepsize is constant at 30 atoms (≈ 2o tilt) a single phase along z but is composed of different phases at each step and similarly for o in the dot, there is a fair chance that there will be a non-cancellation of the phases between the e, o wavefunctions depending up the the extent of tunneling. This will give rise to a finite hopping matrix term between all the dot and lead states and the valley index o, e can longer be said to be conserved. The results show that the conservation or non-conservation of the valley index depends sensitively onfine detailsof thedot morphologysuch the proximity of a stepedge tothe edge of the dot on atomic scales. It is not likely that good control at this level can be achieved, and thus some sample dependence must be expected. This means that it will normally be necessary to include hopping terms that do not conserve valley index into the Hamiltonian in order to understand the Kondo effect in realistic Si QDs, where the interfaces are rarely smooth. 8 x 10−3 2.5 2 s m er1.5 g t n pi 1 p o H 0.5 0 0 10 20 30 40 50 60 70 80 90 100 50 s m40 er ng t30 pi p ho20 of atio 10 R 0 0 10 20 30 40 50 60 70 80 90 100 step size(atoms) FIG. 4: (a)Tunneling matrix elements (V ,V ) in units of eV and (b) Ratio of the tunneling e,e e,o matrix elements (V /V ) as a function of the stepsize for constant barrier height and width (for e,o e,e a miscut QW). The solid line with triangular markers indicate the V term and the dashed line e,e with open circles indicate the V term. Here the step edge coincides with the quantum dot edge. e,o III. EQUATION-OF-MOTION APPROACH IN THE U → ∞ LIMIT Sec. II has given us some insight into the nature of the tunneling between the leads and the dot. We now introduce a Hamiltonian to describe the full many-body problem, and discuss the computational method we will use to solve it. This Hamiltonian must include the single-particle energy levels of the leads and the dot, the tunneling matrix elements that connect these levels, and the Coulomb interaction between electrons on the dot. We shall use the Anderson impurity model: H = ε c+ c + ε f+ f + V (c+ f +f+ c ) k ikmσ ikmσ mσ mσ mσ 0,ik ikmσ mσ mσ ikmσ iXkmσ Xmσ iXkmσ U + VX,ik(c+ikmσfm¯σ +fm+¯σcikmσ)+ 2 nm′σ′nmσ iXkmσ m′σX′6=mσ Here σ indicate two-fold spin 1/2 and m two-fold valley indices. m¯ means the opposite valley state of m. The operator c+ (c ) creates (annihilates) an electron with an energy ikmσ ikmσ ε in the i lead, i ∈ L,R, while the operator f+ (f ) creates (annihilates) an electron k mσ mσ with an energy ε in the QD, connected to the leads by Hamiltonian couplings V and mσ 0,ik 9 V which correspond to an electron tunneling between the same valley states and different X,ik valley states, respectively. We assume that the V do not depend on spin index σ. U 0(X),ik is the Coulomb interaction on the dot. It is assumed to be independent of m. Previous studies on the orbital Kondo effect have mostly assumed conservation of orbital indices [20]. Recent theoretical work has found that a transition occurs from SU(4) to SU(2) Kondo effect by allowing violation of conservation of orbital index in CNTs [16, 19] or in double QDs [21]. For Si QDs, we have shown from the previous section that tunnelings between different valley states have to be taken into account. Experiments measure the current I as a function of source-drain voltage V . The differ- sd ential conductance G = dI/dV is given by the generalized Landauer formula [22] sd e D 2Γ (w)Γ (w) L,mσ R,mσ I = dw(f (w)−f (w)) Im[G (w)] h Z L R Γ (w)+Γ (w) mσ Xmσ −D L,mσ R,mσ where D is the bandwidth and Γ (w) = 2π (V2 +V2 )δ(w−ε ). The f′s are Fermi i,mσ k 0,ik X,ik k P functions calculated with chemical potentials µ and µ with µ = µ +eV . In the fol- L R L R sd lowing argument, we assume a flat unperturbed density of states and V = V , V = V . 0,ik 0 X,ik X We then define Γ = Γ (w)+Γ (w) = πV2/D with V2 = V2 +V2. With these ap- L,mσ R,mσ 0 X proximations we maydifferentiate this equationandwe finddI/dV ∼−Im[G (eV )]/π = sd mσ sd density of states(DOS). Here G (w) is the retarded Green’s function: mσ ∞ G (w) =≪ f ,f+ ≫= −i eiαt < {f (t),f+ (0)} > dt. (6) mσ mσ mσ Z mσ mσ 0 where α = w+iδ. Here {,} and <,> denote the anti-commutator and statistical average of operators, respectively. Hence our task is to compute G (w). Although the approximation mσ of a flat density of states is not likely to be completely valid, we still expect that sharp structures in G (w) will be reflected in the voltage dependence of dI/dV . mσ sd Several approximate solution methods for this type of Hamiltonian have been used suc- cessfully: numerical renormalization group, non-crossing approximation(NCA), scaling the- ory, and equation-of-motion (EOM) approach. In this paper, we use the EOM approach to investigate the transport through a Si QD at low temperature, in the presence of magnetic field. The EOM approach has several merits. The most important for our purposes is that it can produce the Green’s function at finite temperature, and works both for infinite-U and finite-U. We basically follow the spirit of the paper by Czycholl [23] which gives a thorough analysis of EOM in the large-N limit(N is the number of energy level degeneracy) 10