Gate-controlled spin splitting in quantum dots with ferromagnetic leads in the Kondo regime

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Gate-controlled spin splitting in quantum dots with ferromagnetic leads in the Kondo regime

J. Martinek,^1,3,6M. Sindel,²L. Borda,^2,4J. Barnaś,^3,5R. Bulla,⁷J. König,⁸G. Schön,¹S. Maekawa,⁶and J. von Delft²

1Institut für Theoretische Festkörperphysik, Universität Karlsruhe, 76128 Karlsruhe, Germany

2Physics Department and Center for NanoScience, LMU München, 80333 München, Germany

3Institute of Molecular Physics, Polish Academy of Sciences, 60-179 Poznań, Poland

4Institute of Physics and Research Group of the Hungarian Academy of Sciences, TU Budapest, H-1521, Hungary

5Department of Physics, Adam Mickiewicz University, 61-614 Poznań, Poland

6Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan

7Theoretische Physik III, Elektronische Korrelationen und Magnetismus, Universität Augsburg, Augsburg, Germany

8Institut für Theoretische Physik III, Ruhr-Universität Bochum, 44780 Bochum, Germany 共Received 10 June 2005; published 14 September 2005兲

The effect of a gate voltage共V_g兲on the spin splitting of an electronic level in a quantum dot共QD兲attached to ferromagnetic leads is studied in the Kondo regime using a generalized numerical renormalization group technique. We find that theV_gdependence of the QD level spin splitting strongly depends on the shape of the density of states共DOS兲. For one class of DOS shapes there is nearly noV_gdependence; for another,V_gcan be used to control the magnitude and sign of the spin splitting, which can be interpreted as a local exchange magnetic field. We find that the spin splitting acquires a new type of logarithmic divergence. We give an analytical explanation for our numerical results and explain how they arise due to spin-dependent charge fluctuations.

DOI:10.1103/PhysRevB.72.121302 PACS number共s兲: 73.23.Hk, 72.15.Qm, 72.25.⫺b, 75.20.Hr

The manipulation of magnetization and spin is one of the fundamental processes in magnetoelectronics and spintronics, providing the possibility of writing information in a magnetic memory,¹ and also because of the possibility of classical or quantum computation using spin. In most situations this is realized by means of an externally applied, non- local magnetic field which is usually difficult to insert into an integrated circuit. Recently, it was proposed to control the magnetic properties, such as the Curie temperature of ferromagnetic semiconductors, by means of an electric field: In gated structures,² due to the modification of carrier-density- mediated magnetic interactions, such properties can be modi- fied by a gate voltage. In this communication we propose to control the amplitude and sign of the spin splitting of a quantum dot 共QD兲 induced by the presence of ferromagnetic leads, only by using a gate voltage without further assistance of a magnetic field. As a representative 共but not the only兲 example of this effect we investigate its influence on the Kondo effect and its spin splitting, which acts as a very sensitive probe of the spin state of the dot and the effective local magnetic field in the QD generated by exchange inter- action with the ferromagnetic leads.

Recently, the possibility of the Kondo effect in a QD attached to ferromagnetic electrodes was widely discussed,^3–9 and it was predicted that the Kondo resonance is split and suppressed in the presence of ferromagnetic leads.^7,8 This prediction has since been verified experimentally.¹⁰ It was shown that this splitting can be compensated by an appropri- ately tuned external magnetic field, and the Kondo effect is thereby restored.^7,8In all previous studies of QDs attached to ferromagnetic leads^3–9 an idealized, flat, spin-independent density of states共DOS兲 with spin-dependent tunneling am- plitudes was considered. However, since the spin splitting arises from renormalization effects, i.e., is a many-body effect, it depends on the full DOS structure of the involved

material, and not only on its value at the Fermi surface. In realistic ferromagnetic systems, the DOS shape is strongly asymmetric due to the Stoner splitting and the different hybridization between the electronic bands.¹

In this communication we demonstrate that the gate voltage dependence of the spin splitting of a QD level, and the resulting splitting and suppression of the Kondo resonance, are determined by the DOS structure and can lead to cru- cially different behaviors. We apply the numerical renormalization group共NRG兲technique extended to handle bands of arbitrary shape. For one class of DOS shapes, we find almost no Vg dependence of the spin splitting, while for another class the induced spin splitting, which can be interpreted as the effect of a local exchange field, can be controlled byVg. The spin splitting can be fully compensated and its direction can even be reversed within this class. We explain the physical mechanism that leads to this behavior, which is related to the compensation of the renormalization of the spin- dependent QD levels induced by electronlike and holelike quantum charge fluctuations. Moreover, we find that for the QD level close to the Fermi surface, the amplitude of the spin splitting has a logarithmic divergence, indicating the many-body character of this phenomenon.

Model and method.—The Anderson model 共AM兲 of a single level QD with energy⑀0and Coulomb interactionU, coupled to parallel-oriented ferromagnetic leads, is given by

H=

兺

_rk

␴⑀rk␴c_rk^†_␴crk␴+⑀0

兺

␴

nˆ_␴+Unˆ_↑nˆ_↓

+

兺

_rk_␴^共^V^rk^d^␴^†^c^rk^␴^{+ h . c}^兲⁻^BS^z^. ^共¹^兲

Herecrk␴andd_␴共nˆ_␴=d_␴^†d_␴兲are Fermi operators for electrons with momentumk and spin␴ in the leads 共r=L/R兲, and in PHYSICAL REVIEW B72, 121302共R兲共2005兲

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the QD, Vrk is the tunneling amplitude, Sz=共nˆ_↑−nˆ_↓兲/ 2, and the last term denotes the Zeeman energy of the dot. The energy⑀0is experimentally controllable by Vg共⑀0⯝Vg兲.

To discuss the gate voltage dependence of the QD level spin splitting, we consider a more realistic, both energy- and spin-dependent band structure 关␳r↑共␻兲⫽␳r↓共␻兲兴, violating p-h symmetry ␳r␴共␻兲⫽␳r␴共−␻兲. This leads to an energy- dependent hybridization function ⌫r␴共␻兲=␲兺k␦共␻⁻⑀k␴兲V_rk²

=␲␳r␴共␻兲V₀², where we take Vrk=Vr to be constant. We apply the NRG method^11,12extended to handle arbitrary DOS shapes and asymmetry. To this end, the standard logarithmic discretization of the conduction band is performed for each spin component separately, with the bandwidths, D_↑=D_↓=D₀.

Within each interval 关−␻n, −␻n+1兴 and 关␻n+1,␻n兴共with

␻n=D₀⌳⁻ⁿ兲 of the logarithmically discretized conduction band共CB兲the operators of the continuous CB are expressed in terms of a Fourier series. Even though we allow for a nonconstant conduction electron DOS, it is still possible to transform the Hamiltonian such that the impurity couples onlyto the zeroth-order component of the Fourier expansion of each interval.¹³Dropping the nonconstant Fourier compo- nents of each interval^11,12then results in a discretized version of the Anderson model with the continous spectrum in each interval replaced by a single fermionic degree of freedom 共independently for both spin directions兲. Since we allow for an arbitrary DOS foreachspin component␴ of the CB this mapping needs to be performed for each ␴separately. This leads to the Hamiltonian

H=

兺

_␴ ^⑀^␴^nˆ^␴⁺^Unˆ^↑^nˆ^↓⁺

^冑

^␰⁰^␴^/^␲

兺

␴ 关d_␴^†f₀_␴+f_0␴^† d_␴兴

+

兺

␴n=0

⬁

关␧n␴f_n^†_␴fn␴+tn␴共fn†␴

f_n+1␴+f_n+1␴^† fn␴兲兴, 共2兲 wherefn␴are fermionic operators at thenth site of the Wil- son chain,␰_0␴^{= 1 / 2}兰_−D^+D₀⁰⌫_␴共␻兲d␻关⌫_␴共␻兲=⌺r⌫r␴共␻兲兴,t_n_␴denotes the hopping matrix elements, and⑀_␴⁼⑀0−BSz. The absence of particle-hole symmetry leads to the appearance of nonzero on-site energies,␧n␴along the chain. In thisgeneral case no closed expression for the matrix elements tn␴ and

␧n␴, both depending on the particular structure of the DOS via ⌫_␴共␻兲, is known; they have to be determined recursively.¹⁴

We calculate the level occupation n_␴⬅具nˆ_␴典 and the

⑀0-dependent spin-resolved single-particle spectral density A_␴共␻兲= −共1 /␲兲ImG_␴^r共␻兲, where G_␴^r共␻兲 denotes a retarded Green’s function, via Eq. 共2兲. For symmetric coupling

⌫L␴共␻兲=⌫R␴共␻兲 the spin-resolved conductance takes the form G_␴=␲共e²/h兲兰_−⬁⁺^⬁d␻⌫_␴共␻兲A_␴共␻兲(−关⳵^f共␻兲/⳵␻兴), where f共␻兲 is the Fermi function.

Spectral function and conductance.—Here, we focus our attention onT= 0 properties. We have analyzed several types of DOS shapes and found three typical classes of the V_g dependence of the Kondo resonance splitting, which smoothly cross over into another. Since our method enables us to perform NRG calculations for arbitrary band shapes, we decide to choose an example which turns

out to encompass all three classes, namely ␳_␴共␻兲

=¹₂共3

冑

2 / 8兲D^−3/2共1 +␴Q兲

冑

_␻+D+␴⌬, where ␻苸关−D

−␴⌬,D−␴⌬兴,D₀=D+⌬,关␴⬅1共−1兲for↑共↓兲兴, a square-root shape DOS equivalent to a parabolic band共as for free elec- trons兲with Stoner splitting⌬共Ref. 15兲, and some additional spin andp-hasymmetryQ, which modifies the amplitude of the DOS关see Fig. 1共insets兲兴.

In Fig. 1 we present the weighted spectral function A˜共␻兲⬅␲^e²^/h⌺_␴⌫_␴共0兲A_␴共␻兲, normalized such that for ␻^{= 0} it corresponds to the linear conductanceG=A˜共0兲, as a function of energy␻^and⑀0. We focus on a narrow energy win- dow around the Fermi surface where the Kondo resonance appears; charge resonances are visible when⑀0 orU+⑀0ap- proach the Fermi surface, namely at energies⑀0/Uⲏ−0.1 or ⱗ−0.9. Although the NRG method is designed to calculate equilibrium transport, one can still roughly deduce, from the spin splitting of the Kondo resonance of the equilibrium FIG. 1.共Color online兲V_gdependence of the spin splitting: Nor- malized spectral function␲⌺␴⌫␴共0兲A_␴共␻兲as a function of energy␻ and gate voltage⑀0, for the three different parabolic DOS shapes 关insets: corresponding DOS for spin↑ 共red兲 and↓ 共blue兲兴charac- terized by a different Q, which modifies both the spin and p-h asymmetry:共a-c兲for magnetic fieldB= 0,共d兲B/U= 0.017,共e兲 and 共f兲B/U= 0.0083. The white dashed lines are obtained using Eq.共3兲. HereU= 0.12D₀,␲V₀²=UD/ 6,⌬= 0.15D, andT= 0.

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spectral functionA˜共␻兲, the splitting of the zero-bias anomaly

⌬V in the nonequilibrium conductance G共V兲, since e⌬V⬃2⌬⑀共Ref. 7兲共⌬⑀⬅^˜⑀_↑⁻^˜⑀_↓is the splitting of the renor- malized levels兲.

We present A˜共␻兲 for three DOS shapes depicted in the insets of Fig. 1: 共i兲 Q= 0 共a,d兲, 共ii兲 Q= 0.1 共b,e兲, and 共iii兲 Q= 0.3共c,f兲, with 2⌬= 0.3D共Ref. 16兲. Changing the param- eter Q, which tunes the spin and p-h asymmetry 关see the definition of␳␴共␻兲兴, causes the⑀0dependence of the Kondo resonance splitting to display three different classes of behavior, which smoothly cross over into another关and whose origin is explained after Eq. 共3兲 below兴. For 共i兲 we hardly find any⑀0dependence of the spin splitting; for共ii兲a strong

⑀0dependence without compensation of the spin splitting in the local-moment regime, and for 共iii兲 a strong ⑀0 dependence with a compensation共i.e., a crossing兲and a change of the direction of the QD magnetization. The compensation 共crossing兲 corresponds to the very peculiar situation where the Kondo effect共strong coupling fixed point兲can be recovered in the presence of ferromagnetic leads without any external magnetic field. A behavior as presented in Figs. 1共a兲 and 1共b兲 was recently observed experimentally,¹⁷ where in- deed a variation of the gate voltage results in two split conduction lines G共V,V_g兲 which are parallel for one case and converging for the other case, similar to our findings.

Effect of a magnetic field.—In Figs. 1共d兲–1共f兲 we show how a magnetic field B modifies the results of Figs.

1共a兲–1共c兲: in 共i兲 the spin splitting can be compensated at a particular magnetic field B_comp 共Bcomp/U= 0.017兲 and the Kondo effect is visible in a wide range of ⑀0; for 共ii兲, at B/U= 0.0083, the Kondo effect is recovered only at one particular ⑀0 value, which depends on the applied magnetic field; case 共iii兲 shows that the crossing point shifts with B.

Since B_comp can be viewed as a measure of the zero-field splitting,⌬⑀共B= 0 ,⑀0兲⯝−B_comp共⑀0兲, the⑀0dependence of⌬⑀ can be measured by studying that of B_comp, for which one needs to measure the linear conductanceG共⑀0,B兲as a function of bothBand⑀0. In Figs. 2共a兲–2共c兲we plotG共⑀0,B兲for the three bands of Fig. 1. The two horizontal ridges 共reso- nances兲 in Figs. 2共a兲–2共c兲 correspond to quantum charge fluctuations共broadened QD level兲of width ⬃⌫. The bright lines with finite slope in Figs. 2共a兲–2共c兲 reflect the restored Kondo resonance and hence map out the ⑀0 dependence of B_comp共⑀0兲⯝−⌬⑀共⑀0兲 when the magnetic field compensates the spin splitting. Interestingly, the spin splitting and the cor- respondingB_comptend to diverge共兩⌬⑀兩→⬁兲when approach- ing the charging resonance, as is best visible in Fig. 2共c兲.

From Fig. 2共a兲–2共c兲, it is clear that even forB= 0, compensation can always be achieved共the bright lines always cross B= 0兲; the main difference between the three classes is whether this occurs in the local-moment regime共c兲or in the mixed-valence regime共a,b兲.

Such a finite slope inG共⑀0,B兲was observed for a singlet- triplet transition Kondo effect in a two level QD关Fig. 2共d兲in Ref. 18兴. The corresponding transition leads to a characteristic maximum in the valley between two charging resonances 关Fig. 3共c兲 in Ref. 18兴, similarly as in our Fig. 2共e兲. In that system the effective spin asymmetry共assumed by our model兲 is realized by the asymmetry in the coupling of two QD levels.¹⁹

In Fig. 2共d兲we show how the occupationn_␴and the magnetic moment共spin兲of the QDm=n_↑−n_↓= 2具Sz典change as a function of ⑀0 for the situation of Fig. 1共c兲. One finds that even though B= 0, it is possible to control the level spin splitting of the QD and thereby change the average spin direction of the QD from the parallel to antiparallel alignment with respect to the lead’s magnetization. This opens the pos- sibility共also forTⰇT_K, andTⱗ⌫_␴兲of controlling the QD’s spin splitting by a gate voltage without further need of an external magnetic field.

Perturbative analysis.—One can understand the behavior presented in Figs. 1共a兲–1共c兲 by using Haldane’s scaling method,²⁰ where charge fluctuations are integrated out. This leads to a spin-dependent renormalization of the QD’s level position^˜⑀_␴ and a level broadening ⌫_␴. In contrast to Ref. 7 we consider here the case of finite Coulomb interactions U⬍⬁, which means that also the doubly occupied state 兩2典 is of importance. The spin splitting is then given by

⌬⑀⬅␦⑀↑−␦⑀↓+B, where²¹

␦⑀␴⯝− 1

␲

冕

^d^␻

再

^⌫^␴^共^␻^␻^{兲关1 −}⁻^⑀^␴^f^共^␻^兲兴⁺^⑀^⌫^−␴^−␴^共⁺^␻^U^兲f⁻^共^␻^␻^兲

冎

^. ^共3兲

The first term in the brackets corresponds to electronlike processes, namely charge fluctuations between a single occupied state兩␴典and the empty兩0典one, and the second term to hole- FIG. 2. 共Color online兲 The QD’s linear conductance G as a function of gate voltage⑀0and external magnetic field B for the DOS shapes 共a兲共c兲 as for Figs. 1共a兲 and 1共c兲, respectively. 共d兲 Spin-dependent occupancyn_␴of the dot level as a function of gate voltage⑀0for the DOS shape as in Fig. 1共c兲 andB= 0.共d兲The⑀0

dependence of the total occupancy of the dotnand magnetizationm for the situation from Fig. 1共c兲. 共e兲 The conductance G for the situations from Figs. 1共c兲共dashed兲; 1共d兲共solid兲; and 1共f兲共long dashed兲. ParametersU,⌫, andTas in Fig. 1.

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like processes, namely charge fluctuations between the states 兩␴典and兩2典. The amplitude of the charge fluctuations is pro- portional to⌫, which for⌫ⰇT determines the width of QD levels. Equation共3兲shows that ⌬⑀depends on the shape of

⌫_␴共␻兲 for all␻, not only on its value at the Fermi surface.

The same is true for the slope of the bright lines in Figs.

2共a兲–2共c兲, ⳵^Bcomp/⳵⑀0. The dashed lines in Figs. 1共a兲–1共c兲 show ±⌬⑀as a function of⑀0关from Eq.共3兲兴for the same set of parameters as in the NRG calculation, and are in good agreement with the position of the共split兲Kondo resonances observed in the latter. Equation共3兲shows that the dramatic changes observed in Fig. 1 upon changingQare due to the modification of both thep-h and spin asymmetry.

Equation 共3兲 predicts that even for systems with spin- asymmetric bands ⌫_↑共␻兲⫽⌫_↓共␻兲, the integral can give

⌬⑀= 0, which corresponds to a situation where the renormalization of⑀_␴ due to electronlike processes are compensated by holelike processes. An example is a system consisting of p-h symmetric bands,⌫_␴共␻兲=⌫_␴共−␻兲, where no splitting of the Kondo resonance 共⌬⑀^{= 0}兲 for the symmetric point,

⑀0= −U/ 2, appears. For real systems p-h symmetric bands cannot be assumed, however, the compensation⌬⑀= 0 is still possible, as shown in Fig. 1共c兲. Equation共3兲also shows that the characteristic energy scale of the spin splitting is given by⌫ rather than by the Stoner splitting⌬共⌬Ⰷ⌫兲, since the states far from the Fermi surface enter Eq. 共3兲 only with a logarithmic weight. However, the Stoner splitting introduces a strongp-h asymmetry, so it can influence the character of gate voltage dependence significantly.

For a flatband⌫_␴共␻兲=⌫_␴, Eq.共3兲can be integrated ana- lytically. For D₀ⰇU,兩␧0兩 one finds: ⌬⑀⯝共P⌫/␲兲

Re关␾共⑀0兲−␾共U+⑀0兲兴, where P⬅共⌫_↑−⌫_↓兲/⌫, ␾共x兲⬅⌿

关

¹₂

+i共x/ 2␲T兲

兴

, and ⌿共x兲 denotes the digamma function. For T= 0, the spin splitting is given by

⌬⑀⯝ 共P ⌫/␲兲ln共兩⑀0兩/兩U+⑀0兩兲, 共4兲 showing a logarithmic divergence for ⑀0→0 or U+⑀0→0.

Since any sufficiently smooth DOS can be linearized around the Fermi surface, this logarithmic divergence occurs quite universally, as can be observed in log-linear versions 共not shown兲 of Figs. 2共a兲 and 2共c兲. For finite temperature 共T⬎0兲 the logarithmic divergence for⑀0→0 or⑀0→−U is cut off, ⌬⑀⯝−共1 /␲兲P⌫

关

⌿

共

¹₂

兲

+ ln共2␲^T^/U兲

兴

, which is also important for temperaturesTⰆT_K.

In conclusion, we used an extended NRG technique for general band shapes to demonstrate the possibility of controlling the local exchange field, and thereby the spin splitting, of a QD attached to ferromagnetic leads by means of the gate voltage. This can be tested experimentally by mea- suring the linear and nonlinear conductance as a function of gate voltage and magnetic field.

We thank T. Costi, L. Glazman, W. Hofstetter, B. Jones, C. Marcus, J. Nygård, A. Pasupathy, D. Ralph, A. Rosch, M.

Vojta, and Y. Utsumi for discussions. This work was sup- ported by the DFG under the CFN and the SFB 484, the RT Network “Spintronics” of the EC RTN2-2001-00440, project PBZ/KBN/044/P03/2001, the EC Contract No. G5MA-CT- 2002–04049, the “Kompetenznetz Funktionelle Nanostruk- turen” of the Landesstiftung BW, and Project OTKA D048665. L.B. is a grantee of János Bolyai Scholarship.

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