Kondo effect in a quantum dot coupled to ferromagnetic leads and side-coupled to a nonmagnetic reservoir I. Weymann1,2,∗ and J. Barna´s1,3 1Department of Physics, Adam Mickiewicz University, 61-614 Poznan´, Poland 2Physics Department, Arnold Sommerfeld Center for Theoretical Physics and Center for NanoScience, Ludwig-Maximilians-Universit¨at, Theresienstrasse 37, 80333 Munich, Germany 3Institute of Molecular Physics, Polish Academy of Sciences, 60-179 Poznan´, Poland (Dated: January 14, 2010) 0 1 Equilibrium transport properties of a single-level quantum dot tunnel-coupled to ferromagnetic 0 leadsandexchange-coupledtoasidenonmagneticreservoirareanalyzedtheoreticallyintheKondo 2 regime. The equilibrium spectral functions and conductance through the dot are calculated using n the numerical renormalization group (NRG) method. It is shown that in the antiparallel magnetic a configuration,thesystemundergoes aquantumphasetransition with increasing exchangecoupling J J, where the conductance drops from its maximum value to zero. In the parallel configuration, on 4 theotherhand,theconductanceisgenerally suppressedduetoaneffectivespinsplittingofthedot 1 levelcausedbythepresenceofferromagneticleads,irrespectiveofthestrengthofexchangeconstant. However, for J ranging from J = 0 up to the corresponding critical value, the Kondo effect and ] quantumcriticalbehaviorcanberestoredbyapplyingproperlytunedcompensatingmagneticfield. l l a PACSnumbers: 72.25.-b,73.63.Kv,73.23.-b,73.43.Nq,85.75.-d h - s e I. INTRODUCTION purity’s spin. If the coupling to one of them is larger m than to the other one, a usual single-channel Kondo state (spin singlet) is formed between the dot and more . Transport through a model single-level quantum dot at captures many interesting and important features of strongly coupled reservoir. This results in two compet- m ingKondogroundstatesofthesystem,dependingonthe transportphenomena in realquantum dots. One of such ratioofcouplingstrengthstothefirstandsecondconduc- - phenomena, which has been of great interest in the last d decade,is the Kondoeffect.1–3 When the dotisoccupied tion channels. Interestingly, these two Kondo states are n separated by a quantum critical point, where both cou- by a single electron, virtual transitions between the dot o plings are equal and an exotic two-channel Kondo state and electron reservoirs (external leads) cause spin fluc- c is formed, which cannot be described within the Lan- [ tuations in the dot. As a result, the dot’s spin becomes dau Fermi-liquid theory. Very recently, the two-channel screened by electrons of the reservoirs, which results in 1 Kondo effect has been explored experimentally in quan- the formation of a non-local spin singlet ground state of v tum dots.27 The experimental setup consisted of a small 5 the system. Furthermore, a resonance in the density of quantum dot coupled to external leads and to a large 7 states appears at the Fermi level, which gives rise to en- Coulomb-blockaded island. While the electrons could 4 hanced transmission through the dot. In experiments, 2 this leads to the well-known zero-bias anomaly, i.e. a tunnel between the dot and the leads, only virtual tun- 1. peak at zero bias in the differential conductance.1,2 neling processes between the dot and the island were al- lowed, resulting in an exchange coupling. By tuning the 0 When the reservoirs are ferromagnetic, the effective 0 exchange field generated by the electrodes may sup- exchangecoupling,itwaspossible to study the quantum 1 press the Kondo anomaly.4–9 More specifically, when phase transition between the two ground states of the : system and analyze transport behaviorin the non-Fermi v the dot described by an asymmetric Anderson model is liquid regime.27 Theoretically, such a two-channel setup i symmetrically coupled to ferromagnetic leads, then the X can be modelled for example by a quantum dot which Kondo effect becomes suppressedin the parallelconfigu- r istunnel-coupledtoexternalleadsandexchange-coupled ration, while in the antiparallel configurationthe Kondo a to another electron reservoir.28–30 anomaly survives. However, the Kondo effect in the parallel configuration can be restored, when an exter- As discussed above, both the Kondo effect in a quan- nalmagneticfield,whichcompensatesthe exchangefield tum dot coupled to ferromagnetic leads and the two- created by the ferromagnetic leads, is applied.10,11 This channel Kondo phenomenon were already extensively behavior was confirmed in a couple of recent experi- studied. However,the interplayofleads’ferromagnetism ments.12–17 and two-channel Kondo effect remains to a large extent The situation becomes more complex and physically unexplored. Therefore, in this paper we address the richerwhenthe dotis exchange-coupledtoanadditional two-channelKondoprobleminthe presenceofferromag- reservoir.18Suchamodelcapturestheessentialphysicsof netism. Inparticular,weconsideranAndersonquantum the so-called two-channel Kondo effect.19–26 In the two- dotcoupledtoferromagneticleadsandexchange-coupled channelKondoproblem,two separateelectronreservoirs toanonmagneticelectronreservoir. Usingthenumerical (channels) compete with each other to screen the im- renormalizationgroup(NRG)method,weanalyzethein- 2 G G Ls Rs seeFig.1. Itisassumedthattheexternalleadsaremade QD of the same ferromagnetic materialand their magnetiza- tions are collinear, so that the system can be either in the parallel or antiparallel magnetic configuration. The J total Hamiltonian is given by H =H +H +H +H +H . (1) FM NM QD tun exch Here, H describes the ferromagnetic leads, H = FM FM ε c† c , where c† is the electron creation rkσ rkσ rkσ rkσ rkσ FIG.1: (color online) Theschematic ofaquantumdot(QD) operatorwithwavenumberk,spinσintheleft(r =L)or tunnel-coupledtoexternalferromagneticleadsandexchange- rPight(r =R) lead, andε is the correspondingenergy. rkσ coupled to a nonmagnetic electron reservoir. The spin- The second part, H , corresponds to a nonmagnetic NM dependent coupling to the left (right) lead is described by electronreservoirandisgivenby,H = ε a† a , ΓLσ (ΓRσ), while J denotes the exchange coupling constant. witha† being therespectivecreatioNnMoperaktoσrakndkσεkσis The magnetizations of the leads can form either parallel or kσ P k antiparallel magnetic configuration, as indicated. the single-particleenergy. The quantumdotis described by the Anderson Hamiltonian, terplay between the effects due to ferromagnetismof the H = ε d†d +Ud†d d†d +BS , (2) QD d σ σ ↑ ↑ ↓ ↓ z leads and exchange coupling to the additional nonmag- σ X netic reservoir. Depending on the strength of the tunnel coupling t and exchange coupling J, the dot’s spin can where d†σ creates a spin-σ electron, εd denotes the en- bescreenedeitherbyelectronsintheferromagneticleads ergy of an electron in the dot, and U describes the or by electrons in the nonmagnetic reservoir. By analyz- Coulomb correlations between two electrons occupying ing the equilibrium spectral functions and the conduc- the dot. The last term correspondsto externalmagnetic tance through the dot, we show that in the antiparallel field B applied along the zth direction (gµB ≡ 1) and magnetic configuration, the system undergoes a quan- Sz = 21(d†↑d↑ −d†↓d↓). The tunnel Hamiltonian is given tum phase transition with increasing exchange coupling by J,wheretheconductancedropsfromthemaximumvalue to zero. For a certain criticalvalue of J, JAP, both elec- H = t d†c +c† d , (3) c tun rσ σ rkσ rkσ σ tron channels try to screen the dot’s spin and the con- Xrkσ (cid:16) (cid:17) ductanceapproachesahalfofthequantumconductance. Intheparallelconfiguration,onthe otherhand,the con- where trσ describes the spin-dependent hopping matrix ductance is generally suppressed, irrespective of the ex- elements between the dot and ferromagnetic leads. The change constant J, due to effective spin splitting of the coupling to magnetic leads can be described by Γrσ = dot level caused by the exchange field coming from fer- πρr|trσ|2, where ρr ≡ ρ is the density of states in the romagnetic leads.6,7 We show that the Kondo effect can lead r. We have thus shifted the whole spin-dependence be restored by applying a properly tuned external mag- into the coupling constants and assumed a flat band of neticfieldB forJ belowthecorrespondingcriticalpoint, width 2D,6,7 where D ≡ 1 is set as the energy unit, if J < J˜P. Furthermore, the quantum critical regime can not stated otherwise. c also be recovered, which however requires a fine-tuning By means of a unitary transformationin the left-right in the parameter space of J and B. basis,31,32 onecanmaptheproblemoftunnelingthrough The paper is organized as follows. In section II we quantum dot coupled to the left and right leads into a present the model as well as briefly describe the NRG problem where the dot is effectively coupled to a single method together with some details of calculations. In lead with a new coupling constant, Γσ = ΓLσ + ΓRσ. turn, in sectionIII we presentnumericalresults for sym- Thiscanbedonebyintroducingthefollowingsymmetric metricandasymmetricAndersonmodelsinbothparallel operators, αkσ = t˜LσcLkσ +t˜RσcRkσ, with dimensionless and antiparallel magnetic configurations of the system. coefficients t˜rσ = trσ/ t2Lσ+t2Rσ. Then, the tunneling Finally, we conclude in section IV. Hamiltonian can be written as p Γ H = σ d†α +α† d . (4) II. THEORETICAL DESCRIPTION tun sπρ σ kσ kσ σ Xkσ (cid:16) (cid:17) A. Model One can see that now the dot is tunnel-coupled to only one effective electron reservoir, H = ε α† α , FM kσ kσ kσ kσ The considered system consists of a single-level quan- with new spin-dependent coupling constant Γ . The σ P tum dot tunnel-coupled to left and right ferromagnetic other parts of the system Hamiltonian, Eq. (1), are not leads and exchange-coupled to a nonmagnetic reservoir, affected by this transformation. To parameterize the 3 spin-dependent couplings we also introduce the spin po- bothAbelianandnon-Abeliansymmetries.35,36 Incalcu- larizationofferromagneticleads,p=(Γ −Γ )/(Γ +Γ ). lationswehavethususedthe U(1)symmetryforthezth ↑ ↓ ↑ ↓ The couplingscanbe thenwrittenina compactformas, component of the total spin, the U(1) symmetry for the Γ = (1 ± p)Γ, where Γ = (Γ + Γ )/2. Assuming charge in the first channel, and the SU(2) symmetry for ↑(↓) ↑ ↓ symmetric coupling strength of the dot to the leads, the the chargeinthe secondchannel. Furthermore,incalcu- resultantcouplingin the antiparallelconfigurationis the lationswehavetakenthediscretizationparameterΛ=2 samefor the spin-upandspin-downelectrons,ΓAP =Γ. and kept 3000 states at each iteration step. ↑(↓) On the other hand, in the parallel configuration, the Using the NRG we can calculate the spectral function couplings are then different for the two spin directions, of the dot, A (ω) = −1ImGR (ω), where GR (ω) de- ΓP↑(↓) =(1±p)Γ, which effectively leads to spin splitting notes the Fourσier transfoπrm ofdtσhe dot retardeddσGreen’s ofthedotleveland,whenthisisthecase,theKondores- function, GR (t)=−iΘ(t)h{d (t),d†(0)}i. On the other dσ σ σ onance may become suppressed because of broken spin hand,thespectralfunctioncanbedirectlyrelatedtothe degeneracy.6,7 spin-resolvedlinearconductanceG bythefollowingfor- σ Finally, the exchange Hamiltonian describing the cou- mula pling between the dot and the second (nonmagnetic) reservoir is given by e2 4Γ Γ ∂f(ω) Lσ Rσ G = dωπA (ω) − , (6) σ σ Hexch = J2 S~a†kσ~σσσ′akσ′, (5) h ΓLσ+ΓRσ Z (cid:18) ∂ω (cid:19) σσ′ k XX wheref(ω)istheFermidistributionfunctionandtheto- whereS~ = 21 σσ′d†σ~σσσ′dσ′ isthe spininthe dot, J de- tal conductance is given by, G=G↑+G↓. At zero tem- notestheexchangecouplingconstantand~σ isavectorof perature, the spin dependent conductance for the paral- Pauli spin maPtrices. We note that in addition to the ex- lel configuration is given by, GP = e2(1±p)πΓAP , ↑(↓) h ↑(↓) changescatteringofelectrons[describedbyEq.(5)]there while for the antiparallelconfigurationone gets, GAP = couldbe alsopotentialscattering. However,inthis work ↑(↓) wearemainlyinterestedinthelowenergyphysics,where eh2(1−p2)πΓAA↑(P↓) = GAP/2, where APσ/AP is the zero- theKondoeffectemerges,sothepotentialscatteringmay temperature spectral function of the d-level operator in be neglected, as it does not lead to any Kondo-type cor- respective magnetic configuration, taken at ω =0. relations. B. Method III. NUMERICAL RESULTS To analyzethe equilibrium transportproperties of the considered system, we employ the numerical renormal- In the following we present numerical results on the izationgroupmethod33 –nonperturbative,verypowerful equilibrium spectral function and linear conductance, andessentiallyexactnumericalmethodto addressquan- when the quantum dot is in the Kondo regime. We will tum impurity problems.34 The NRG consists in a loga- distinguish between two different situations; symmetric rithmic discretization of the conduction band and map- (ε = −U/2) and asymmetric (ε 6= −U/2) Anderson d d ping of the system onto a semi-infinite chain with the models. The origin of such a distinction stems from the impurity (quantum dot) sitting at the end of the chain. wayinwhichferromagneticleadsactonthequantumdot. By diagonalizing the Hamiltonian at consecutive sites of Morespecifically,intheasymmetricAndersonmodelfer- the chain and storing the eigenvalues and eigenvectors romagnetism of the leads gives rise to a spin splitting of of the system, one can calculate the static and dynamic the Kondo resonance in the parallel configuration, while quantities of the system. In the case of model consid- in the symmetric model no such a splitting appears (as- ered in this paper, the Hamiltonian is mapped onto two sumingthatthedotiscoupledwiththesamestrengthto semi-infinite chains, where the first chain corresponds to theleftandrightleads).6,7,10 Inotherwords,aneffective ferromagnetic leads tunnel-coupled to the dot, while the exchange field, due to coupling to magnetic leads, acts second one to nonmagnetic reservoir exchange-coupled onthe dot in the formercase, while sucha field vanishes to the dot. Because such two-channel calculations are in the latter case. The effective field is directly related usually very demanding numerically, it is crucial to ex- to the difference inthe couplingstrengths ofthe dotand ploit as many symmetries of the system’s Hamiltonian ferromagnetic leads for the two spin orientations. Since as possible. Especially, using the SU(2) symmetry de- the coupling in the spin-up channel is larger than that creases the size of Hilbert space and thus increases con- in the spin-down one, energy of the spin-up (spin-down) siderably the accuracy of calculations. In particular, to electronin the dot decreases(increases)by ∆ε /2. Con- d efficientlyperformthe analysis,wehaveusedtheflexible sequently, the spin-dependent coupling acts as an effec- density-matrix numerical renormalization group (DM- tive magnetic field, leading to spin-splitting ∆ε of the d NRG) code, which can tackle with arbitrary number of dot level.6,7,10 4 0.5 a J=0 1.0 J=0.2 0.4 J=0.22 h] 0.8 J=0.226 / 2 J=0.228 e 2 A0 0.3 JJ==00..222288678 2p) 0.6 APA/σ 0.2 JJJ===000...22223392 −(1 0.4 J=0.24 [ 0.1 J=0.26 AP 0.2 G 0.0 0.0 J=0 0.0 0.1 0.2 0.3 0.4 b J=0.18 J 0.3 J=0.21 J=0.218 J=0.2204 FIG.3: Thelinearconductanceasafunctionofexchangecou- J=0.221 A0 0.2 J=0.22138 pling constant J for the antiparallel magnetic configuration. / J=0.2218 The conductance was determined from the spectral function PA↑ J=0.223 showninFig. 2(a). TheparametersarethesameasinFig. 2 J=0.228 and J is in unitsof D=1. J=0.24 0.1 J=0.26 Kondotemperaturedefinedasahalf-widthofthe d-level 0.0 spectral function for J =0 and p=0, T =2.5×10−4. J=0 K 0.8 c J=0.18 It can be seen that in the antiparallel configuration the J=0.21 spectral function is independent of the spin orientation J=0.218 0.6 J=0.2204 [Fig. 2(a)], AAP = AAP, while it depends on electron ↑ ↓ J=0.221 A0 J=0.22138 spin in the parallel magnetic configuration, AP↑ 6= AP↓, P/↓ 0.4 JJ==00..2222138 see Fig. 2(b) and (c). Note, that for symmetric Ander- A J=0.228 sonmodelthespectralfunctionissymmetricwithrespect J=0.24 to ω =0, therefore here it is shown only for positive en- 0.2 J=0.26 ergies, i.e. for energies above the Fermi level. Moreover, note also that the spectral functions are normalized to 0.0 that for the corresponding paramagnetic limit (p = 0 10-4 10-3 10-2 10-1 100 101 102 103 and J = 0), so A (ω = 0)+ A (ω = 0) 6= A in the ↑ ↓ 0 ω/TK parallel configuration. Let us consider first the situation with vanishing ex- FIG. 2: (color online) The spectral function of the d-level changecoupling of the dotto the nonmagnetic reservoir, operator in the antiparallel (a) and parallel (b,c) magnetic J = 0. For ω < T , a Kondo peak develops in the dot K configurationsforthesymmetricAndersonmodelandfordif- spectral function due to screening of the dot’s spin by ferent values of the exchange coupling J. The parameters conduction electrons of ferromagnetic leads, which leads are: εd = −0.05, U = 0.1, Γ = 0.0077, p = 0.4, and T = 0. to the formation of a non-local spin singlet. The height TheKondotemperatureisdefinedasahalf-widthofthespec- tral function for J = 0 and p = 0, TK = 2.5×10−4, while of the Kondo peak is independent of spin in the antipar- A0 =PσAσ(ω=0)forJ =0andp=0. Alltheparameters allel configuration and depends on spin in the parallel are given in the unitsof D≡1. one. Apart from this, a Hubbard peak corresponding to ε +U is visible in the spectralfunction showninFig. 2. d This behaviorofthe Kondophenomenonin the presence of ferromagnetic leads is in agreement with that found A. Symmetric Anderson model by other methods, for instance by the equation of mo- tion for the Green functions37 and also by the real-time Forthe symmetricAndersonmodelweassumethe fol- diagrammatic technique.38 lowing parameters (in the units of D), ε = −0.05 and The situation changes when the electron in the dot is d U = 0.1. The zero-temperature spin-dependent spectral additionally exchange coupled to the nonmagnetic reser- functionA ,normalizedtoA ,withA = A (ω =0) voir. When the coupling is antiferromagnetic and the σ 0 0 σ σ takenforJ =0andp=0,isshowninFig.2forbothan- coupling parameter J increases, the width of the Kondo P tiparallel (a) and parallel (b,c) magnetic configurations peak becomes gradually narrower and narrower. The (notethelogarithmicenergyscale),andforindicatedval- hight of the peak, however, remains unchanged, as can ues of the exchange coupling parameter J. The spectral be clearly seen in Fig. 2 for some small values of the ex- functionisplottedasafunctionofω/T ,whereT isthe change coupling constant. In order to see this behavior K K 5 also for larger J, but still smaller than a critical value, 0.26 J = JP(AP), one should plot the spectral function for c 0.24 lowerenergies. ForJ <JP(AP),the systemisinthe spin c 0.22 singletgroundstateformedbythequantumdotspinand electrons in the ferromagnetic leads, which gives rise to 0.20 the Kondo resonance in the spectral function. However, PJc0.18 when J > JcP(AP), the coupling to nonmagnetic reser- 0.16 voir becomes larger than the coupling to ferromagnetic 0.14 G= 0.01 leadsandthedot’sspinbecomesscreenedbyelectronsof G= 0.0077 the nonmagnetic reservoir. Now the Kondo peak in the 0.12 G= 0.005 spectral function disappears for both magnetic configu- 0.10 0.0 0.2 0.4 0.6 0.8 1.0 rationsofthesystem,seeFig.2. Whenthetwocouplings p are equal, i.e. for J = JP(AP), the system is in an ex- c otic state where the two channels try to screen the dot’s FIG.4: (coloronline)Thedependenceofthecriticalexchange spin. The spectral function at ω = 0 is then equal to couplingJcP(inunitsofD=1)onthespinpolarizationofthe a half of its value corresponding to J = 0, 1AP/AP| , leadspinthecaseofsymmetricAndersonmodelandparallel 2 σ J=0 magneticconfigurationforthreedifferentvaluesofthetunnel for both magnetic configurations, see Fig. 2. This be- coupling Γ, as indicated in the figure. The other parameters havior reveals a quantum phase transition with increas- are thesame as in Fig. 2. ing strength of the exchange coupling. The origin of the phase transition follows from the interplay of the tunnel coupling to the ferromagnetic electrodes and exchange JAP. On the other hand, at the quantum critical point couplingtothe nonmagneticreservoir. Morespecifically, Jc= JAP, the linear conductance is equal to half of its thequantumphasetransitionoccursattheboundarybe- value fcor J =0, i.e. GAP =(1−p2)e2/h. Consequently, tween two different singlet ground states, involving the the dependence ofthe conductanceonthe exchangecou- dot’s spin and conduction electrons of the leads or side- pling J can be expressed as GAP = Θ(∆)(1−p2)2e2/h, coupledreservoir. Thebehavioroftransportcharacteris- where ∆ =JAP−J and Θ(x) is the step function. The tics aroundthis criticalpointin the case ofnonmagnetic c dependence of G on J for the parallel configuration is system was discussed in Ref. [28]. It was shownthat the qualitativelysimilartothatintheantiparallelconfigura- zero-temperature conductance depends step-like on the tion, therefore it is not shown here. difference∆betweenthe tunnelandexchangecouplings, In Fig. 4 the dependence of the critical exchange cou- and becomes equal to a half of its maximum value at plingJPonthespinpolarizationoftheleadspinthecase the critical point, i.e. when ∆ = 0. The discontinu- c ofsymmetricAndersonmodelandparallelmagneticcon- ity of the linear conductance with respect to ∆ reflects figurationisshownforthreedifferentvaluesofthetunnel the quantum phase transition in the parameter space of couplingΓ. Firstofall,thecriticalcouplingJP decreases tunnel coupling t and exchange coupling J. Since the with decreasing the coupling strength Γ. Mocreover, JP conductanceisdeterminedbythecorrespondingspectral c alsodecreaseswithincreasingthespinpolarizationofthe functions at ω = 0, quantum critical behavior is also leads. For p →1, JP tends to zero, as only spins of one reflected in the J-dependence of the spectral function. c orientationarecoupledtotheleadsandtheKondoeffect We also note that at finite temperature the transition is becomessuppressed. Thisbehaviorofthecriticalparam- smeared as T/T and turns rather into a crossover.28 K eter JP is consistent with the dependence of the Kondo c Using thepSchrieffer-Wolff transformation,39 one could temperature in a quantum dot coupled to ferromagnetic trytoestimatethecriticalvalueofJ. Forthesymmetric leads on the coupling strength Γ and spin polarization Anderson model and for antiparallel configuration one p.4–7,11 gets,3JS−W =Γ(πρ)−1U|ε |−1(ε +U)−1 ≈0.196. From c d d the numerical data, however, one finds JAP ≈ 0.22878 c for the antiparallel and JP ≈ 0.22138 for the parallel B. Asymmetric Anderson model c configurations, see Fig. 2. The difference between the value obtained using the Schrieffer-Wolff transformation Let us now consider the case of asymmetric Anderson and the numerical value may result for example from model, |ε | 6= U/2. For numerical calculations we as- d thefactthatthetransformationisbasedonperturbation sume ε = −0.05 and U = 0.2. The spectral function d expansion, and takes into account only the second-order in the antiparallel magnetic configuration as a function tunneling processes. of ω/T , where T = 3.4×10−5, is shown in Fig. 5 for K K The quantum critical behavior can be also seen in the indicated values of the exchange coupling parameter J. dependence of the linear conductance on the coupling The insetshowsthe behaviorofthe spectralfunctionas- constant J, which is shown in Fig. 3 for the antiparallel sociated with the Kondo peak. The general features of magnetic configuration. For J < JAP, the conductance the spectral function are similar to those of the corre- c is GAP = (1 − p2)2e2/h and drops to zero when J > sponding spectral function in the case of symmetric An- 6 0.5 0.12 a J=0 0.4 0.4 0.10 J=0.12 J=0 J=0.16 0.2 J=0.16 J=0.2 J=0.18 0.08 J=0.24 A0 0.3 0.0-0.1 0.0 0.1 JJ==00..1188451 /A0 0.06 JJ==00..2382 APA/σ 0.2 ω/TK JJJ ===000...11188855668 PA↑ 0.04 JJ==00..346 J=0.1865 J=0.1875 0.02 0.1 J=0.19 J=0.2 0.00 0.0 0.05 b 10-4 10-3 10-2 10-1 100 101 102 103 104 ω/TK 0.04 FIG. 5: (color online) The spectral function of the d-level A0 0.03 operator in the antiparallel magnetic configuration for the P/↓ asymmetric Anderson model, εd = −0.05, U = 0.2, and for A 0.02 indicatedvaluesoftheexchangecouplingJ. TheKondotem- perature for assumed parameters (and for J = 0 and p = 0) 0.01 isTK =3.4×10−5. Theotherparametersarethesameasin Fig. 2. 0.00 -104 -103 -102 -101 101 102 103 104 ω/T K dersonmodeldiscussedabove,seeFig.2. Thisisbecause intheantiparallelconfigurationtheresultantcouplingto FIG. 6: (color online) The spectral function of the d-level ferromagneticleadsdoesnotdependonspinandthesys- operator in theparallel magnetic configuration for theasym- temeffectivelybehavesasanonmagneticone. Asbefore, metric Anderson model. The other parameters are the same one observes a quantum phase transition at J = JAP, as in Fig. 5. c where now JAP ≈0.1858. The only difference is that for c |ε |6=U/2 the spectral function displays an asymmetric d behavior with respect to ω =0, see the inset in Fig. 5. respond to the dot level ε and its Coulomb counterpart d Thesituation,however,changessignificantlywhenthe ε +U. d magnetizations of the leads switch to the parallelconfig- When the coupling parameter J increases, the weak uration. The corresponding spectral function for spin-↑ Kondo resonancesin the spectralfunction gradually dis- andspin-↓isshowninFig.6. Note,thatnowthespectral appear for both spin orientations. The physics behind functionisshownforbothpositiveandnegativeenergies. this disappearance remains similar to that described As before, let us consider first the case of J =0. Due to above, i.e. screening of the dot’s spin by the nonmag- an effective exchange field originating from the presence netic reservoir exchange-coupled to the dot. Interest- of ferromagnetic electrodes, the spin degeneracy of the ingly,thereis noquantumphasetransitioninthe caseof dotlevelislifted. Atzerotemperature,themagnitudeof parallel magnetic configuration shown in Fig. 6. thesplittingduetoexchangefield,∆ε ,canbeestimated d Let us now assume that there is an external magnetic from the formula6,7,10 field B applied to the dot along the zth direction. In 2pΓ |ε | the case of antiparallel configuration, the magnetic field d ∆ε = . (7) d π |ε +U| destroys both the Kondo resonance and quantum phase d transitionwithchangingJ. However,whenthe leadsare For the assumed parameters one then finds, ∆ε ≈ alignedinparallel,theKondoeffectisalreadysuppressed d 2.15×10−3. The exchange field leads generally to the bytheeffectiveexchangefieldcomingfromferromagnetic suppression of the Kondo peak, however the reminiscent electrodes,andonemayconsiderthepossibilityofrestor- of the Kondo effect are still visible as relatively small ing the Kondo peak by applying an external magnetic peaks in the d-level spectral function. The position of fieldwhichcompensatestheeffectsduetoexchangefield. these peaks is shifted away from the Fermi level – to In Fig. 7 we show the spectral functions for the parallel positive energies for spin-↓ and to negative energies for magneticconfigurationinthe caseofanasymmetricAn- spin-↑. In fact, the peaks occur for energies comparable dersonmodel, calculated for three different values of the to the magnitude of the exchange field. As can be seen exchangeconstantJ inthepresenceofexternalmagnetic in Fig. 6, they develop at ω/T ≈ 102 for spin-↓ and fieldB. The insets display behaviorof the spectralfunc- K at ω/T ≈ −102 for spin-↑ components of the spectral tion associated with the Kondo peaks. When J =0, see K function. The other peaks in the spectral functions cor- Fig.7(a)and (b), the full Kondopeak atthe Fermi level 7 0.4 0.4 0.20 0.4 0.2 0.3 J =0.17875 a c e 0.3 00..12 0.3 0.2 0.15 0.1 PA/A0↑ 0.2 0.0-2 -1ω/0TK 1 2 0.2 0.-00.4 -0.2ω/0T.0K 0.2 0.4 0.10 0.-00.2 -0.1ω B/=0T.00K0.1 0.2 B=0.0019 0.1 0.1 J =0.17 0.05 B=0.001914 J =0 0.0 0.0 0.00 0.8 0.8 0.8 0.8 0.4 0.4 b 0.6 d f 0.3 0.4 0.4 0.2 0.6 0.2 0.6 0.3 0.1 P/A0↓ 0.4 0.0-2 -1ω/0TK 1 2 0.4 0.-00.4 -0.2ω/0T.0K 0.2 0.4 0.2 0.-00.2 -0.1ω/0T.0K0.1 0.2 A BB==00.002 BB==00..000022005645 B=0 B=0.001928 B=0.00191621 B=0.00204 B=0.00207 B=0.00192 B=0.001929 B=0.001918 0.2 B=0.00205 B=0.0021 0.2 B=0.001925 B=0.001931 0.1 B=0.00193 B=0.001927 B=0.00194 B=0.002 0.0 0.0 0.0 10-3 10-2 10-1 100 101 102 103 104 10-3 10-2 10-1 100 101 102 103 104 10-4 10-3 10-2 10-1 100 101 102 103 104 ω/TK ω/TK ω/TK FIG. 7: (color online) The spectral function of the d-level operator in the parallel magnetic configuration for the asymmetric Anderson model in the presence of external magnetic field B applied along the zth direction for different values of exchange coupling constant J =0 (a,b), J =0.17 (c,d) and J =0.17875 (e,f). The otherparameters are thesame as in Fig. 5. inthespectraldensitycanberestoredforbothspinorien- 0.00206 tations by properly tuned external magnetic field, which 0.00204 happens forB =B =0.0020545,where B (in the units c c 0.00202 of D) denotes the compensating field. This is in agree- 0.00200 mentwiththeresultobtainedearlier.6,7 Similarbehavior alsoappearsforlargerpositiveJ,e.g. forJ =0.17shown Bc 0.00198 inFig.7(c)and(d). Now,theKondoresonancebecomes 0.00196 restored when the compensating field is B = 0.001928. c 0.00194 Note that the magnitude of magnetic field necessary for 0.00192 fullrestorationoftheKondoeffectslightlydecreasesasJ increases. Thequestionwhicharisesnowiswhethersuch 0.00190 0.00 0.04 0.08 0.12 0.16 a restoration by magnetic field is also possible for larger J values of J. By fine-tuning in the parameter space of J and B, we have found that this is the case for J below FIG. 8: The dependence of the compensating magnetic field a certain critical value J < J˜P = 0.17875. Here J˜P de- Bc on the exchange coupling constant J in the case of the c c notes the critical value of J in the parallel configuration parallel configuration and asymmetric Anderson model. The and in the presence of the compensating magnetic field. other parameters are the same as in Fig. 5. J and Bc are in unitsof D=1. Fromnumericalresults(notshownhere),followsthatfor J >J˜P, the magnetic field can only partially restore the c Kondo effect, leading to small side peaks in the spectral function, while the full Kondo peak at ω = 0 cannot be restored. OnemaynowexpectthatforJ =J˜P,themag- c netic field shouldalso restore the quantum critical state. Indeed, by fine-tuning in the parameter space we have found that the quantum critical state can be recovered can note that the compensating field B decreases with c for B = 0.00191621. This situation is shown explicitly increasing the exchange coupling J. This is explicitly c in Fig. 7(e) and (f). Thus, we have shown that in the shown in Fig. 8, where we have calculated the depen- parallel configuration the properly-tuned magnetic field dence of B on the exchange coupling J. For J < J˜P, c c canrestore both the full Kondo effect for J <J˜P as well the Kondo resonance can be fully restored by applying c as the quantum critical state for J =J˜cP. compensatingfieldBc. Ontheotherhand,whenJ >J˜cP, themagneticfieldcannotcompensatetheexchangefield, BycomparingnumericalcurvespresentedinFig.7,one sothenotionofcompensatingfieldbecomesmeaningless. 8 IV. CONCLUSIONS creasing the exchange constant J, these peaks become suppressed. In this paper we have considered spectral and trans- We have also considered the influence of an external port properties of a single-level quantum dot connected magneticfieldonthed-levelspectralfunctionandshown to external ferromagnetic leads and exchange-coupledto thatintheparallelconfigurationtheKondoeffectcanbe a nonmagnetic reservoir. Using the numerical renor- restored by applying appropriately tuned compensating malization group method we have calculated the zero- magnetic field for J <J˜P, where J˜P is the critical value c c temperature d-level spectral function and the conduc- of J in the compensating magnetic field. If, however, tance through the dot. We have shown that in the an- J > J˜P, the full Kondo effect cannot be restored by a c tiparallelconfiguration,dependingonthestrengthofthe magneticfield. Inaddition,wehavefoundthatthequan- exchange interaction J, the Kondo singlet ground state tum criticalbehavior,whichis suppressedin the parallel canform,inwhichthe conductionelectronseither inthe configuration,can also be recoveredby tuning the exter- ferromagnetic leads or in the nonmagnetic reservoir are nal magnetic field. involved. In the former case, the conductance is maxi- mum,whereasinthelattercasetheconductancebecomes fully suppressed. For a certain critical value of J, JAP, c Acknowledgments both electron channels try to screen the dot’s spin and the conductance is equal to a half ofits maximumvalue. The boundary between the two ground states is a quan- This work was supported by funds of the Polish Min- tum phase transition. istry of Science and Higher Education as a research In the parallel magnetic configuration, on the other projectforyears2006-2009. 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