Kondo quantum dot coupled to ferromagnetic leads: Numerical renormalization group study
M. Sindel,1L. Borda,1,2J. Martinek,3,4,5R. Bulla,6J. König,7G. Schön,5S. Maekawa,3and J. von Delft1
1Physics Department, Arnold Sommerfeld Center for Theoretical Physics, and Center for NanoScience, Ludwig-Maximilians-Universität München, 80333 München, Germany
2Research Group “Theory of Condensed Matter” of the Hungarian Academy of Sciences, TU Budapest, Budapest H-1521, Hungary
3Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan
4Institute of Molecular Physics, Polish Academy of Sciences, 60-179 Poznań, Poland
5Institut für Theoretische Festkörperphysik and DFG-Center for Functional Nanostructures (CFN), Universität Karlsruhe, D-76128 Karlsruhe, Germany
6Theoretische Physik III, Elektronische Korrelationen und Magnetismus, Universität Augsburg, D-86135 Augsburg, Germany
7Institut für Theoretische Physik III, Ruhr-Universität Bochum, 44780 Bochum, Germany 共Received 18 July 2006; revised manuscript received 3 November 2006; published 18 July 2007兲 We systematically study the influence of ferromagnetic leads on the Kondo resonance in a quantum dot tuned to the local moment regime. We employ Wilson’s numerical renormalization group method, extended to handle leads with a spin asymmetric density of states, to identify the effects of共i兲a finite spin polarization in the leads共at the Fermi surface兲,共ii兲a Stoner splitting in the bands共governed by the band edges兲, and共iii兲an arbitrary shape of the lead density of states. For a generic lead density of states, the quantum dot favors being occupied by a particular spin species due to exchange interaction with ferromagnetic leads, leading to sup- pression and splitting of the Kondo resonance. The application of a magnetic field can compensate this asymmetry, restoring the Kondo effect. We study both the gate voltage dependence共for a fixed band structure in the leads兲and the spin polarization dependence共for fixed gate voltage兲of this compensation field for various types of bands. Interestingly, we find that the full recovery of the Kondo resonance of a quantum dot in the presence of leads with an energy-dependent density of states is possible not only by an appropriately tuned external magnetic field but also via an appropriately tuned gate voltage. For flat bands, simple formulas for the splitting of the local level as a function of the spin polarization and gate voltage are given.
DOI:10.1103/PhysRevB.76.045321 PACS number共s兲: 72.15.Qm, 75.20.Hr, 72.25.⫺b, 73.23.Hk
I. INTRODUCTION
The interplay between different many-body phenomena, such as superconductivity, ferromagnetism, or the Kondo ef- fect, has recently attracted a lot of experimental and theoret- ical attention. A recent experiment of Buitelaaret al.1nicely demonstrated that Kondo correlations compete with super- conductivity in the leads. The interplay between Kondo cor- relations and itinerant electron ferromagnetism in the elec- trodes has theoretically been intensively studied within the last years,2–9initially leading to controversial conclusions.
For effectively single-level quantum dots共i.e., dots with a level spacing much bigger than the level broadening⌫兲, con- sensus was found that a finite spin asymmetry in the density of states in the leads results共in general兲in splitting and sup- pression of the Kondo resonance. This is due to the spin- dependent broadening and renormalization of the dot level position induced by spin-dependent quantum charge fluctua- tions. In terms of the Kondo spin model, it can be treated as an effective exchange interaction between a localized spin on the dot and ferromagnetic leads. Moreover, it was shown that a strong coupling Kondo fixed point with a reduced Kondo temperature can develop6,7even though the dot is coupled to ferromagnetic leads, given that an external magnetic field6or electric field7 共gate voltage兲 is tuned appropriately. Obvi- ously, in the limit of fully spin-polarized leads, when only one spin component is present for energies close to the Fermi surface 共half-metallic leads兲, the effective screening of the impuritycannot take place any more and the Kondo reso-
nance does not develop. A part of these theoretical predic- tions have recently been confirmed in an experiment by Pa- supathyet al.10 The presence of ferromagnetic leads could also nicely explain the experimental findings of Nygård et al.11
Since the interplay between ferromagnetism and strong correlation effects is one of the important issues in spintronic applications, there are currently many research activities go- ing on in this direction. The goal is to manipulate the mag- netization of a local quantum dot 共i.e., its local spin兲 by means of an external parameter, such as an external magnetic field or an electric field共a gate voltage兲, with high accuracy.
This would provide a possible method of writing information in a magnetic memory.12 Since it is extremely difficult to confine a magnetic field such that it only affects the quantum dot under study, it is of big importance to search for alterna- tive possibilities for such a manipulation共e.g., by means of a local gate voltage, as proposed by the authors13兲.
In this paper, we push forward our previous work by per- forming a systematic analysis on the dependence of physical quantities on different band structure properties. Starting from the simplest case, we add the ingredients of a realistic model one by one, allowing a deeper understanding of the interplay of the Kondo model and itinerant electron ferro- magnetism. In this paper, we extend our recent studies car- ried out in this direction6,7,13and illustrate the strength of the analytical methods by comparing the results predicted by them to the results obtained by the exact numerical renormal- ization group共NRG兲 method.14 While in Refs. 2–9 the dot was attached to ferromagnetic leads with an unrealistic, spin-
independent, and flat band—with a spin-dependent tunneling amplitude—we generalize this treatment here by allowing for arbitrary density of states共DOS兲shapes.13In particular, we carefully analyze the consequences of typical DOS shapes in the leads on the Kondo resonance. We explain the difference between these shapes and provide simple formulas 共based on perturbative scaling analysis15,16兲that explain the numerical results analytically.
We study both the effect of a finite lead spin polarization and the gate voltage dependence of a single-level quantum dot contacted to ferromagnetic leads with three relevant DOS classes:共i兲for flat bands without Stoner splitting,共ii兲for flat bands with Stoner splitting and,共iii兲for an energy-dependent DOS共also including Stoner splitting兲. For this sake, we em- ploy an extended version of the NRG method to handle ar- bitrary shaped bands.17
The article is organized as follows: In Sec. II, we define the model Hamiltonian of the quantum dot coupled to ferro- magnetic leads. In Sec. III, we explain details of the Wilson mapping on the semi-infinite chain in the case of the spin- dependent density of states with arbitrary energy depen- dence. Using the perturbative scaling analysis, we give pre- diction for a spin-splitting energy for various band shapes in Sec. IV. In Sec. V, the results for spin-dependent flat DOS are demonstrated, together with the Friedel sum rule analysis.
The effect of the Stoner splitting is discussed in Sec. VI, together with comparison to experimental results from Ref.
11and an arbitrary band structure in Sec. VII. We summarize our findings then in Sec. VIII.
II. MODEL: QUANTUM DOT COUPLED TO FERROMAGNETIC LEADS
We model the problem at hand by means of a single-level dot of energy⑀d共tunable via an external gate voltageVG兲and charging energyUthat is coupled to identical, noninteracting leads共in equilibrium兲with Fermi energy= 0. Accordingly, the system is described by the following Anderson impurity model:
Hˆ =Hˆ
ᐉ+Hˆ
ᐉd+Hˆ
d, Hˆ
d=⑀d
兺
nˆ+Unˆ↑nˆ↓−BSz, 共1兲
with the lead and the tunneling part of the Hamiltonian Hˆ
ᐉ=
兺
rk⑀rkcrk†crk, 共2兲Hˆ
ᐉd=
兺
rk共Vrkd†crk+ H.c.兲. 共3兲 Here, crk and d 共nˆ=d†d兲 are the Fermi operators for electrons with momentum k and spin in leadr 共r=L/R兲 and in the dot, respectively. The spin-dependent dispersion in lead r, parametrized by ⑀rk, reflects the spin-dependent DOS,r共兲=兺k␦共−⑀rk兲, in lead r;allinformation about energy and spin dependency in lead r is contained in the
dispersion function⑀rk.Vrklabels the tunneling matrix ele- ment between the impurity and lead r, Sz=共nˆ↑−nˆ↓兲/ 2, and the last term in Eq.共1兲 denotes the Zeeman energy due to external magnetic fieldB acting on the dot spin only. Here, we neglect the effect of an external magnetic field on the leads’ electronic structure, as well as a stray magnetic field from the ferromagnetic leads. The coupling between the dot level and electrons in leadrleads to a broadening and a shift of the level⑀d,⑀d→˜⑀d共where the tilde denotes the renormal- ized level兲. Theenergyandspindependency of the broaden- ing and the shift, determined by the coupling, ⌫r共兲
=r共兲兩Vr共兲兩2, plays the key role in the effects outlined in this paper. Henceforth, we assume Vrk to be real and k independent,Vrk=Vr, and lump all energy and spin depen- dence of⌫r共兲into the DOS in leadr,r共兲.18
Without loosing generality, we assume the coupling to be symmetric, VL=VR. Accordingly, by performing a unitary transformation,19 Hˆ
ᐉ simplifies to Hˆ
ᐉ=21兺k共⑀Lk
+⑀Rk兲␣sk†␣sk, where␣skdenotes the proper unitary com- bination of lead operators which couple to the quantum dot and we dropped the part of the lead Hamiltonian which is decoupled from the dot. With the help of the definitions V
⬅
冑
VL2+VR2, ␣k⬅␣sk, and ⑀k*⬅12共⑀Lk+⑀Rk兲, the full Hamiltonian can be cast into a compact form,Hˆ =
兺
k ⑀k*␣k†␣k+兺
k V共d†␣k+ H.c.兲+Hˆd, 共4兲withHˆ
d as given in Eq.共1兲.
A. Ferromagnetic leads
For ferromagnetic materials, electron-electron interaction in the leads gives rise to magnetic order and spin-dependent DOS,r↑共兲⫽r↓共兲. Magnetic order of typical band ferro- magnets such as Fe, Co, and Ni is mainly related to electron correlation effects in the relatively narrow 3d subbands, which only weakly hybridize with 4sand 4pbands.20We can assume that due to a strong spatial confinement ofdelectron orbitals, the contribution of electrons from d subbands to transport across the tunnel barrier can be neglected.21In such a situation, the system can be modeled by noninteracting22s electrons, which are spin polarized due to the exchange in- teraction with uncompensated magnetic moments of the completely localizedd electrons. In mean-field approxima- tion, one can model this exchange interaction as an effective molecular field, which removes spin degeneracy in the sys- tem of noninteracting conducting electrons, leading to a spin- dependent DOS.
In experiments, very frequently, the electronic transport measurements are performed for two configurations of the leads’ magnetization direction:10the parallel and antiparallel alignments. By comparison of electric current for these two configurations, one can calculate the tunneling magnetoresis- tance, an important parameter for the application of the mag- netic tunnel junction.12
In this paper, we restrict the leads’ magnetization direc- tion to be either共i兲parallel, i.e., the left and right leads have the DOSL共兲andR共兲, respectively, so the total DOS
corresponding to the total dispersion⑀k* 共∀k苸关−D0;D0兴兲is given by 共兲=L共兲+R共兲, and 共ii兲 antiparallel, i.e., the magnetization direction of one of the leads 共let us con- sider the right one兲 is reverted so R共兲→R¯共兲, where
¯=↓共↑兲 if =↑共↓兲, so the total DOS is described then by
共兲=L共兲+R¯共兲. Here, D0 labels the full 共general- ized兲bandwidth of the conduction band共further details can be found below兲. Effects related to leads with noncollinear leads’ magnetization are not discussed here.23
For the special case of both leads made of the same ma- terial in the parallel alignment, L共兲=R共兲 and, in the antiparallel alignment, L共兲=R¯共兲. Therefore, for the antiparallel case, it gives the total DOS to be spin indepen- dent, ↑共兲=L↑共兲+R↓共兲=L↓共兲+R↑共兲=↓共兲. In a such situation, one can expect the usual Kondo effect as for normal共nonferromagnetic兲metallic leads; however, the con- ductance will be diminished due to mismatch of the density of states described by the prefactor of the integral in Eq.
共23兲.
B. Different band structures
In this paper, we will consider different types of total spin-dependent band structures 共兲 independently of the particular magnetization direction of leads. The particular magnetization configuration will affect only the linear con- ductanceGdue to the DOS mismatch for both leads, as will be discussed in detail later.
1. Flat band
The simplest situation where one can account spin asym- metry is a flat band with the energy-independent DOS
共兲=. Then, the spin asymmetry can be parametrized just by a single parameter, the spin imbalance in the DOS at the Fermi energy= 0. For flat bands, the knowledge of the spin polarizationP of the leads, defined as
P= ↑共0兲−↓共0兲
↑共0兲+↓共0兲, 共5兲 is sufficient to fully parametrize 共兲 共see Fig. 2兲. This particular DOS shape is special due to the fact that the particle-hole symmetry is conserved in the electrodes leading to a particular behavior for the symmetric Anderson model.
This type of model of the leads will be considered in Sec. V.
2. Stoner splitting
One can generalize this model and break the particle-hole symmetry by taking the Stoner splitting into account. A con- sequence of the conduction electron ferromagnetism is that the spin-dependent bands are shifted relative to each other 共see Fig. 1兲. For a finite value of that shift ⌬, the spin- band is in the range −D+⌬艋艋D+⌬ 共with the “origi- nal” bandwidthD兲. In the limit ⌬= 0, only energies within the interval关−D;D兴are considered, a scheme usually used in the NRG calculations.7 This relative shift between the two spin-dependent bands leads to the so-called Stoner splitting
⌬, defined as
⌬=⌬↓−⌬↑, 共6兲 at the band edges of the conduction band共see Fig.10兲. The flat band model with a Stoner splitting and consequences of the particle-hole symmetry breaking is considered in Sec. VI.
3. Arbitrary band structure
Since the spin splitting of the dot level is determined by the coupling to all occupied and unoccupied共hole兲electronic states in the leads, the shape of the whole band plays an important role. Therefore, it is reasonable to consider an ar- bitrary DOS shape, which cannot be parametrized by a par- ticular set of parameters as the imbalance between↑ and↓ electrons at the Fermi energy or a Stoner splitting共see Fig.
1兲. Therefore, in Sec. VII, we will consider a model with a more complex band structure and, in Sec. III, we will de- velop the NRG technique for an arbitrary band structure.
III. METHOD: NUMERICAL RENORMALIZATION GROUP FOR AN ARBITRARY SPIN-
DEPENDENT DENSITY OF STATES
In our analysis, we take the ferromagnetic nature of the noninteracting leads by means of a spin- and energy- dependent DOS 共兲 into account. A general example is given in Fig.1. To compute the properties of the model de- scribed above, we have extended the NRG technique calcu- lation to handle a spin-dependent density of states. In order to understand to what extent the method applied here is dif- ferent from the standard NRG, it is adequate to briefly re- view the general concepts of NRG.
The NRG technique was invented by Wilson in the 1970s to solve the Kondo problem;14 later, it was extended to handle other quantum impurity models as well.24–26,28In his original work, Wilson considered a spin-independent flat density of states for the conduction electrons. Closely fol-
∆
∆
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 11 1 1 11 1 1 11 1
−D
D D0
−Λ0
−Λ−1
−ΛΛ−n−n Λ−1 Λ0
11
σ(ω) ρ
1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 11 1 1 11 1 1 11 1
(a) (b)
−D0
V V
/D0 ω
∆
∆
µ=0
FIG. 1. 共Color online兲 共a兲 Example of an energy- and spin- dependent lead DOS共兲with an additional spin-dependent shift
⌬. To perform the logarithmic discretization, a generalized band- widthD0is defined. Since we allow for bands with energy and spin dependence, the discretization is performed for each spin- component separately, see panel共b兲. Since Hˆ
ᐉd does not include spin-flip processes, an impurity electron of spin关circles in panel 共b兲兴couples to lead electrons of spin, =↑共↓兲, only. Panel共b兲 also illustrates that impurity electrons couple to leads’ electron and hole states with arbitrary energy,兩兩艋D0.
lowing Refs.17and27, where the mapping for the case of energy-dependent DOS was given, we generalize that proce- dure for the case of the Hamiltonian given in Eq.共4兲which contains leads with an energy and spin-dependent DOS.
It is convenient to bring the Hamiltonian given by Eq.共4兲 into a continuous representation27 before the generalized mapping is started. The replacement of the discrete fermionic operators by continuous ones,␣k→␣, translates the lead and the tunneling part of the Hamiltonian into
Hˆ
ᐉ=
兺
冕
−1 1dg共兲␣† ␣, 共7兲
Hˆ
ᐉd=
兺
冕
−1 1dh共兲共d†␣+ H.c.兲. 共8兲
As shown in Ref.27, the hereby defined generalizeddisper- sion g共兲andhybridization h共兲functions have to satisfy the relation
g−1共兲
兵h关g
−1共兲兴其2=共兲关V共兲兴2, 共9兲
whereg−1共兲is the inverse ofg共兲, which ensures that the action on the impurity site is identical both in the discrete and the continuous representation. Note that there are many possibilities to satisfy Eq.共9兲.
The key idea of Wilson’s NRG14is a logarithmic discreti- zation of the conduction band, by introducing a discretization parameter ⌳, which defines energy intervals 兴−D0⌳−n;
−D0⌳−n−1兴 and 关D0⌳−n−1;D0⌳−n关 in the conduction band 共n苸N0兲. Within thenth interval of widthdn=⌳−n共1 −⌳−1兲, a Fourier expansion of the lead operators⌿np
±共兲 with funda- mental frequency⍀n= 2/dnis defined as
⌿np
±共兲=
冦
0冑
1dne±i⍀np ifotherwise.⌳−共n+1兲艋 ±⬍ ⌳−n冧
Here, the subscriptsn and p 共苸Z兲 label the corresponding interval and the harmonic index, respectively, while the su- perscript marks positive 共⫹兲 or negative 共⫺兲 intervals, re- spectively.
The above defined Fourier series now allows one to re- place the continuous fermionic conduction band operators a by discrete ones anp 共bnp兲 of harmonic index p and spinacting on thenth positive共negative兲interval only,
␣=
再 兺
np关anp⌿np+共兲+bnp⌿np−共兲兴冎
. 共10兲Impurity electrons coupleonlyto thep= 0 mode of the lead operators, given that the energy-dependent generalized hy- bridization h共兲 is replaced by a constant hybridization, h共兲→hn+for⬎0共orhn− for⬍0, respectively兲. Obvi- ously, the particular choice of constant hybridizationhn± de- mands the generalized dispersion g共兲 to be adjusted ac- cordingly, such that Eq. 共9兲 remains valid. Details of this procedure can be found in Appendix A. Since we adopt this strategy, the harmonic indexp 共the impurity couples only to
lead operators of harmonic indexp= 0兲will be dropped be- low.
Defining a fermionic operator f0⬅ 1
冑
0兺
n 共an␥n++bn␥n−兲, 共11兲with 0=兺n关共␥n+兲2+共␥n−兲2兴=兰−11 ⌫共兲d and the coeffi- cients␥n±
as given in Appendix A,29 reveals that the impu- rity effectively couples to asinglefermionic degree of free- dom only, the zeroth site of the Wilson chain 关for further details, see Eq.共A2兲兴. Therefore, the tunneling part ofHˆ can be written in a compact form as
Hˆ
ᐉd=
兺
冋 冑
0共d†f0+ H.c.兲册
. 共12兲The final step in the NRG procedure is the transformation of the conduction bandHˆ
ᐉinto the form of a linear chain. This goal is achieved via the tridiagonalization procedure devel- oped by Lánczos,30
Hˆ
ᐉ=
兺
n=0
⬁
关nfn†fn+tn共fn†fn+1+fn+1† fn兲兴. 共13兲
In general, the on-site energiesn and hopping matrix ele- mentstn along the Wilson chain need to be determined nu- merically. Besides the matrix elements ⑀n and tn, coeffi- cients unm and vnm, which define the fermionic operators fn,
fn⬅
兺
m=0
⬁
共unmam+vnmbm兲, 共14兲 already used in Eq.共13兲, need to be determined. One imme- diately anticipates the following from Eq.共11兲:
u0m=␥m+/
冑
0, v0m=␥m−/冑
0. 共15兲 Equations which determine the matrix elements⑀n andtn and the coefficientsunmandvnmare given in Appendix B.Note that the on-site energiesn vanish in the presence of particle-hole symmetry in the leads.
To summarize, Hamiltonians as the one given in Eq.共4兲 can be cast into the form of a linear chainHˆ
LC, Hˆ
LC=Hˆ
d+
冑
0/兺
共d†f0+f0† d兲 +
兺
n=0
⬁
关nfn†fn+tn共fn†
fn+1+fn+1† fn兲兴,
共16兲 even though one is dealing with energy- and spin-dependent leads. In general, however, this involves numerical determi- nation of the matrix elements n and tn,17 in contrast to Ref.24共where flat bands were considered兲, where no closed analytical expression for those matrix elements is known.
Equation 共16兲 nicely illustrates the strength of the NRG procedure. As a consequence of the energy separation guar- anteed by the logarithmic discretization, the hopping rate
along the chain decreases astn⬃⌳−n/2 共the on-site energies decays even faster兲, which allows us to diagonalize the chain Hamiltonian iteratively and, in every iteration, to keep the states with the lowest lying energy eigenvalues as the most relevant ones. This very fact underlines that this method does not rely on any assumptions concerning leading order divergences.31
IV. PERTURBATIVE SCALING ANALYSIS
We can understand the spin splitting of the spectral func- tion using Haldane’s scaling approach,16 where quantum charge fluctuations are integrated out. The behavior dis- cussed in this paper can be explained as an effect of spin- dependent quantum charge fluctuations, which lead to a spin- dependent renormalization of the dot’s level position˜⑀dand a spin-dependent level broadening⌫, which, in turn, induce spin splitting of the dot level and the Kondo resonance.
Within this approach, a spin splitting of the local dot level,
⌬⑀d⬅␦⑀d↑−␦⑀d↓+B, which depends on the full band struc- ture of the leads, is obtained13where
␦⑀d⯝− 1
冕
d再
⌫共兲关1 −−⑀df共兲兴+⑀⌫d¯¯共+U兲f共−兲冎
.共17兲 Note that the splitting isnotonly determined by the lead spin polarizationP, i.e., the splitting is not only a property of the Fermi surface. Equation共17兲is the key equation to explain the physics of the共spin-dependent兲splitting of the local level
⑀d. This equation explains the spin-dependent occupation and, consequently, the splitting of the spectral function of a dot that is contacted to leads with a particular band structure.
The first term in the curly brackets corresponds to electron- like processes, namely, charge fluctuations between a single occupied state 兩典 and the empty 兩0典 one, and the second term to holelike processes, namely, charge fluctuations be- tween the states 兩典 and 兩2典. The amplitude of the charge fluctuations is proportional to ⌫, which, for ⌫ⰇT, deter- mines the width of dot levels observed in transport.
The exchange field given by Eq.共17兲 giving rise to pre- cession of an accumulated spin on the quantum dot attached to leads with noncollinear leads’ magnetization is not dis- cussed here.23
A. Flat band
Equation 共17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 renormaliza- tion of⑀d due to electronlike processes is compensated by holelike processes. An example is a system consisting of particle-hole symmetric bands, 共兲=共−兲, where no splitting of the Kondo resonance共⌬⑀d= 0兲for the symmetric point,⑀d= −U/ 2, appears.
For a flat band, 共兲=, Eq. 共17兲 can be inte- grated analytically. For D0ⰇU, 兩⑀d兩, one finds
⌬⑀⯝共P⌫/兲Re关共⑀d兲−共U+⑀d兲兴, where 共x兲⬅⌿共12
+ix/ 2T兲 and ⌿共x兲 denotes the digamma function. For T
= 0, the spin splitting is given by
⌬⑀d⯝ P⌫
ln
冉
兩U兩⑀+d兩⑀d兩冊
, 共18兲showing a logarithmic divergence for⑀d→0 orU+⑀d→0.
B. Stoner splitting
For real systems, particle-hole共p-h兲symmetric bands can- not be assumed; however, the compensation⌬⑀= 0 using a proper tuning of the gate voltage⑀dis still possible. We can also analyze the effect of the Stoner splitting by considering a flat band structure in Fig.2 using the value of the Stoner splitting ⌬= 0.2
共
DD0兲
D0. Then, also from Eq. 共17兲, we can expect an additional spin splitting of the dot level induced by the presence of the Stoner field in the leads even for spin polarization P= 0 given by⌬⑀d共St兲⯝ ⌫
2ln
冋
共−⑀d共−+⑀Dd0+−D⌬兲共U0兲共U++⑀⑀dd++DD00兲−⌬兲册
,共19兲 which, for the symmetric level position,⑀d= −U/ 2, leads to
⌬⑀d共St兲⯝⌫
ln
冉
U/2 +U/2 +⑀d⑀+dD+0D−0⌬冊
, 共20兲Equation 共19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17兲 only with a logarithmic weight. The difference can be as large as 3 orders of magnitude, so, in a metallic ferromagnet, the Stoner splitting energy is of the order⌬⬃1 eV but the effective molecular field⌬⑀d
共St兲 gen-
FIG. 2. 共Color online兲 The lead density of states of a flat 关s共兲=兴but spin-dependent共↑⫽↓兲band for spin polarization P= 0.2. The dark region marks the filled states below the Fermi energy. Since⌬↑=⌬↓= 0, the generalized bandwidthD0=D.
erated by it is still a small fraction of⌫—of order of 1 meV, comparable with the Kondo energy scale for molecular single-electron transistors.10However, the Stoner splitting in- troduces a strong p-h asymmetry, so it can influence the char- acter of the gate voltage dependence significantly.
In the next section, we analyze the effect of different types of the band structure using numerical renormalization group technique and compare it to that obtained in this sec- tion by scaling procedure.
V. FLAT BANDS WITHOUT STONER SPLITTING We start our analysis by considering normalized flat bands without the Stoner splitting共i.e.,D0=D兲, as sketched in Fig.
2, with finite spin polarizationP⫽0 关defined via Eq.共5兲兴.32 In this particular case of spin 共but not energy兲 dependent coupling,⌫⬅V2, the coupling ⌫ can be parametrized via P,⌫↑共↓兲=12⌫共1 ±P兲 关here, ⫹ 共⫺兲 corresponds to spin ↑ 共↓兲兴, where ⌫=⌫↑+⌫↓. Leads with a DOS as the one ana- lyzed in this section have, for instance been studied in Refs.
7and8.
Since⌫⬅V2, the spin dependence of⌫ can be ab- sorbed by replacing V→V=V
冑
12共1 ±P兲 in Eq. 共4兲, i.e., a spin-dependent hopping matrix element, while treating the leads as unpolarized ones 共→兲. This procedure has the particular advantage that the “standard” NRG procedure24 can be applied, meaning that the on-site energies共tunneling matrix elements兲 关defined in Eq.共13兲兴along the Wilson chainnvanish, whiletnturn out to be spinindependent. There- fore, it does not involve the solution of the tedious equations given in Appendix B.
A. Spin and charge state
Consequently, we start our numerical analysis by comput- ing the spin-resolved dot occupation n⬅具¯n典, which is a static property. Figure 3 shows the spin resolved impurity occupation as a function of the spin polarization P of the leads. Figure 3共a兲 and3共c兲 correspond to a gate voltage of
⑀d= −U/ 3 共where the total occupation of the system n↑+n↓
⬍1兲 whereas the second line corresponds to ⑀d= −2U/ 3 共with n↑+n↓⬎1兲. The total occupation of the system, n↑ +n↓, decreases共increases兲 for ⑀d= −U/ 3 共⑀d= −2U/ 3兲 when the spin polarization of the leads is finite, P⫽0. Note that both situations, ⑀d= −U/ 3 and ⑀d= −2U/ 3, are symmetric with respect to changing particle into hole states and vice versa, which is possible only for the leads with particle-hole symmetry. For the gate voltage⑀d= −U/ 2, there is a particle- hole symmetry in the whole system leads with a DOS which does share this symmetry even in the presence of spin asym- metry. Note that whereas a finite spin polarization leads to a decrease inn↑+n↓for⑀d⬎−U/ 2关see Fig.3共a兲兴, it results in an increase in n↑+n↓ for ⑀d⬍−U/ 2 关see Fig. 3共b兲兴. Obvi- ously, for P= 0 and in the absence of an external magnetic field, the impurity does not have a preferred occupation,n↑
=n↓. Any finite value ofP violates this relation:n↑⬎n↓for P⬎0 and⑀d⬎−U/ 2共since␦⑀d↑⬃⌫↓⬍⌫↑⬃␦⑀d↓兲. For⑀d⬍
−U/ 2, on the other hand, the opposite behavior is found.
The effect on the impurity of a finite lead polarization, namely, to prefer a certain spin species, can be compensated by a locally applied magnetic field, as shown in Figs. 3共c兲 and 3共d兲. For ⑀= −U/ 3 and B/⌫= −0.1 关see Fig. 3共c兲兴, the impurity isnotoccupied by a preferred spin species,n↑=n↓, for P⬃0.5. Due to particle-hole symmetry, the same mag- netic field absorbs a lead polarization of P⬃−0.5 for ⑀=
−2U/ 3关see Fig.3共d兲兴. The magnetic field which restores the conditionn↑=n↓, henceforth denoted as compensation field Bcomp共P兲, will be of particular interest below.
B. Single-particle spectral function
Using the NRG technique we can access the spin-resolved single-particle spectral density A共,T,B,P兲=
−1 ImGd,R 共兲 for arbitrary temperatureT, magnetic fieldB, and spin polarization P, where GR共兲 denotes a retarded Green function. We can relate the asymmetry in the occu- pancy,n↑⫽n↓, to the occurrence of charge fluctuations in the dot and broadening and shifts of the position of the energy levels 共for both spins up and down兲. For P⫽0, the charge fluctuations, and hence level shifts and level occupations, become spin dependent, causing the dot level to split6 and the dot magnetization n↑−n↓ to be finite. As a result, the Kondo resonance is also spin split and suppressed 关Fig.
4共b兲兴, similarly to the effect of an applied magnetic field.33 This means that Kondo correlations are reduced or even completely suppressed in the presence of ferromagnetic leads.
Note the asymmetry in the spectral function for P⫽0 which stems from the spin-dependent hybridization⌫. Fig- ure4共b兲, where the spin-resolved spectral function is plotted, reveals the origin of the asymmetry around the Fermi energy of the spectral function. The spectral function obtained for polarized leads has to be contrasted to that of a dot asserted
0.25 0.75
−0.5 0.5
P
−0.5 0.5
P 0.5
1
B /Γ=−0.1
a c
b n
↓ d
εd=−2U/3 εd=−U/3 B = 0
n↓ n↓
n↓
n↑ n↑
n↑ n↑
n↑+n↓
n↑+n↓
n↑+n↓
n↑+n↓
FIG. 3. 共Color online兲 关共a兲–共d兲兴 Spin-dependent occupationn of the local level as a function of the leads’ spin polarizationPfor
⑀d= −U/ 3关共a兲and共c兲兴and⑀d= −2U/ 3关共b兲and共d兲兴 共related by the particle-hole symmetry兲forB= 0共left column兲andB= −0.1⌫共right column兲. For finite spin-polarization, P⫽0, the condition n↑=n↓ can only be obtained by an appropriately tuned magnetic field, as shown in共c兲and共d兲. Parameters:U= 0.12D0and⌫=U/ 6.
to a local magnetic field, where a nearlysymmetric共perfect symmetry is only for the symmetric Anderson model兲 sup- pression and splitting of the Kondo resonance 共around the Fermi energy兲 appear.33 In Fig. 4共c兲, the gate voltage 共⑀d=
−2U/ 3兲 is chosen such that it is particle-hole symmetric to the case shown in Fig.4共a兲. Due to the opposite particle-hole symmetry, the obtained spectral function in Fig.4共c兲is noth- ing else but the spectral function shown in Fig.4共a兲mirrored around the Fermi energy.
Figures3共c兲and3共d兲showed that a finite magnetic fieldB can be used to recover the conditionn↑=n↓. Indeed, for any lead polarizationP, acompensation fieldBcomp共P兲exists at which the impurity is not preferably occupied by a particular spin species. To a good approximation, Bcomp共P兲 has to be chosen such that the induced spin splitting of the local level is compensated.36,37One consequently expects a linearPde- pendence of Bcomp from Eq. 共18兲. Figure5共a兲 shows the P dependence of Bcomp for various values of ⑀d obtained via NRG. The numerically found behavior can be explained through Eq.共18兲. It confirms that the slope ofBcompis nega- tive共positive兲for ⑀d⬎−U/ 2 共⑀d⬍−U/ 2兲 and that Bcomp共P兲
= 0 for⑀d= −U/ 2.
The fact that this occurs simultaneously with the disap- pearance of the Kondo resonance splitting suggests that the local spin is fully screened at Bcomp. The expectation of an unsplit Kondo resonance in the presence of the magnetic fieldBcompis nicely confirmed by our numerics in Fig. 5共b兲 for⑀d= −U/ 3. Moreover, the sharpening of the Kondo reso- nance in the spectral function upon increasingP共indicating a decrease in the Kondo temperatureTK兲can be extracted from this plot. The associated binding energy of the singlet 共the
Kondo temperatureTK兲is consequently reduced.
Having demonstrated that the Kondo resonance can be fully recovered for Bcomp even thoughP⫽0, we show that there is also the possibility to recover the unsplit Kondo resonance via an appropriately tuned gate voltage. In Fig.6, we plot the spin-resolved spectral function for various values of⑀d for P= 0.2 andB= 0. Note that one can easily identify whether the Kondo resonance is fully recovered from the positions ofA共兲 relative to each other. Since a dip in the total spectral function A共兲=兺A共兲 is not present for a modest shift ofA共兲with respect to each other, we identify an unsplit Kondo resonance with perfectly aligned spin- resolved spectral functions henceforth.
Clearly, one can identify from Fig. 6 that the spectral function is split for any value ⑀d⫽−U/ 2. This splitting changes its sign at ⑀d= −U/ 2. For the particular case ⑀d=
−U/ 2 关see Fig. 6共b兲兴, the spectral function reaches the uni-
−1 1
ω/ U 0
1
πΓA(ω)/4 P=0.0 P=0.2 P=0.4
P=0.6 −0.050 0 0.05
1
πΓA(ω)/4
−0.05 0 0.05 ω/ U 0
1
πΓAσ(ω)/2
−0.05 0 0.05
ω/ U 0
1
πΓA(ω)/4
a
εd=−U/3 b
c
P=0.4
εd=−2U/3 εd=−U/3
FIG. 4. 共Color online兲Total spectral functions A共兲=兺A共兲 for various values of the spin polarizationP.共a兲For⑀d= −U/ 3, an increase in P results in a splitting and suppression of the Kondo resonance共see inset兲. The effect of finitePon the Hubbard peaks is less significant. The spin-resolved spectral functionA共兲, shown in共b兲, reveals thatA↑共兲and A↓共兲 differ significantly from each other forP⫽0.共c兲Spectral function for the same values ofPas in 共a兲but for⑀d= −2 / 3U关due to particle-hole symmetry, the results are mirrored as compared to共a兲 but with inverted spins兴. Parameters:
U= 0.12D0,B= 0, and⌫=U/ 6.
−0.05 0 0.05
ω/U 0
0.2 0.4 0.6 0.8 1
πΓA(ω)/4
P=0.0 P=0.4 P=0.8
−1 0 1
P
−0.2 0 0.2
Bcomp/Γ
εd=−U/3 εd=−U/2 εd=−7U/12
a
εd=−U/3 b
FIG. 5.共Color online兲 共a兲Compensation fieldBcomp共P兲for dif- ferent values of⑀d. For flat bands,Bcompdepends linearly on the lead polarization. At the point where there is particle-hole symme- try, so for gate voltage共⑀d= −U/ 2兲,Bcomp共P兲= 0 for any value ofP, i.e., the spectral function is not split for any value ofPeven though B= 0. 共b兲 Spectral function for various values of P for B=Bcomp. Note the sharper resonance in the spectral function, i.e., a reduced Kondo temperatureTK.
0.5 1
πΓAσ(ω)/4
σ=↑+↓
σ=↑
σ=↓
−0.05 0 0.05
ω/U
−0.05 0 0.05
ω/U 0.5
1
πΓAσ(ω)/4
P=0.2
a
c
b
d
εd=−7U/12 εd=−U/3
εd=−5U/6 εd=−U/2
FIG. 6. 共Color online兲 Spin-dependent spectral functionA共兲 for various values of ⑀d, fixed P= 0.2, and B= 0 关共blue兲 dashed:
A↑共兲;共green兲long dashed:A↑共兲;共red兲dotted: A共兲=兺A共兲兴. The splitting betweenA↑andA↓changes its sign at⑀d= −U/ 2. The spectral function A共兲 is plotted for several values of P for ⑀d=
−U/ 3 in Fig.4共a兲. The splitting of the spectral functionA共兲 de- pends onPas well on⑀d. Parameters:U= 0.12D0and⌫=U/ 6.
