Magnetic Single-Electron Transistor as a Tunable Model System for Kondo-Destroying Quantum Criticality
aa r X i v : . [ c ond - m a t . s t r- e l ] A ug Magnetic Single-Electron Transistor as a Tunable Model Systemfor Kondo-Destroying Quantum Criticality
Stefan Kirchner a , ∗ , Qimiao Si a , a Department of Physics & Astronomy, Rice University, Houston, TX 77005, USA
Abstract
Single-electron transistors (SET) attached to ferromagnetic leads can undergo a continuous quantum phase transition as their gatevoltage is tuned. The corresponding quantum critical point separates a Fermi liquid phase from a non-Fermi liquid one. Here, weexpound on the physical idea proposed earlier. The key physics is the critical destruction of the Kondo effect, which underliesa new class of quantum criticality that has been argued to apply to heavy fermion metals. Its manifestation in the transportproperties is studied through an effective Bose-Fermi Kondo model (BFKM); the bosonic bath, corresponding to the spin wavesof the ferromagnetic leads, describes a particular type of sub-Ohmic dissipation. We also present results for general forms of sub-Ohmic dissipative bath, and consider in some detail the case with critical paramagons replacing spin waves. Finally, we discuss somedelicate aspects in the theoretical treatment of the effect of a local magnetic field, particularly in connection with the frequentlyemployed Non-Crossing Approximation (NCA).
Key words:
Single-electron transistor; Bose-Fermi Kondo model; quantum phase transitions; non-crossing approximation
PACS:
The term Kondo effect refers to the screening of a lo-calized moment in a metallic host, a process that is me-diated by particle-hole excitations of the host’s itinerantelectrons. The screened ground state is an entangled sin-glet between the local moment and the electrons, and theexcitation spectrum contains a Kondo resonance which hasthe same quantum numbers as a bare electron. The pos-sibility that quantum dots (QD), nanostructures with awell-defined local moment weakly interacting with nearbyelectrodes, could be used to model the Kondo effect wassuggested early on [1,2].Over the past decade, the Kondo effect has been real-ized in semiconductor heterostructures followed by nan-otubes and single-molecule devices [3,4,5,6,7]. These de-velopments have enhanced our understanding of the quan-tum impurity physics and lead to an increased interest inthe formation of Kondo correlations in various settings, e.g. far away from thermal equilibrium [8,9]. Over roughlythe same period, studies in the bulk correlated systems ofheavy fermion metals have focused attention on quantumcritical points (QCP). Historical work in heavy fermionsaddressed the heavy Fermi liquid on one hand [10], and ∗ Corresponding author. Tel: (713) 348-4291 fax: (713) 348-4150
Email address: [email protected] (Stefan Kirchner). the competition between Kondo and RKKY interactionson the other [11,12]. The recent studies have instead cen-tered around the critical destruction of the Kondo effecton the verge of an antiferromagnetic ordering at zero tem-perature [13,14,15,16]. It is natural to ask whether relatedeffects can be realized in nanostructures.We recently showed that a single-electron transistor(SET) attached to ferromagnetic leads [17] constitutes atunable quantum impurity model system for a Kondo-destroying QCP. The purpose of this article is, in additionto reviewing the basic physical idea and some salient resultson this setup, addressing two issues. First, we determinewhat happens when spin waves are replaced by criticalparamagnons; these results will be relevant when the fer-romagnetic leads are replaced by, for instance, palladium,which maybe better suited to form SET structures withcertain molecules [18]. Second, we discuss some method-ological issues that arise when considering the effect of alocal magnetic field.
1. Quantum Criticality in a Ferromagnetic SET
The general setup of the magnetic SET is given inFig. 1a. The magnetic excitation spectrum of an itinerant
Preprint submitted to Elsevier 16 November 2018 µ V g DQ J LM g (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) Kondo QC V g (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) g c -4 -2 0 2 4!=T0K00.511.52 T ( ! ) D g = 0T0Kg = 4:17T0Kg = 5:83T0Kg = 6:35T0Kg = 6:50T0Kg = 8:33T0K (a) (c)(b) ε=2/3 Fig. 1. (a) Schematics of the ferromagnetic SET. The red arrowslabel the lead magnetization and µ i , i = L, R the chemical potentialin the L/R lead coupled to the quantum dot QD. (b) Phase diagramof the low-energy model of the ferromagnetic SET. Varying the gatevoltage V g tunes both, the Kondo ( J ) and the spin wave coupling( g ) along the dashed (blue) line. The dotted horizontal line is thepath across the transition used in Figs. 1c and 2. (c) Evolution ofthe Kondo resonance, for ǫ = 2 /
3, the case of critical paramagnons.The continuous (red) red curve is at the critical coupling, g = g c .The parameters adopted are: J = 0 . D , where D is the bandwidthassociated with E k , and correspondingly T K = 0 . D ; the cut-offenergy for the bosonic bath is Λ = 0 . D . ferromagnet consists of the Stoner continuum, i. e. tripletparticle-hole excitations, and spin waves. Given theZeeman-splitting of the bands, it might at first be surpris-ing that ferromagnetic leads can screen the local moment.The important point is that the local moment is coupled toall possible particle-hole combinations of both the sourceand drain leads. The resulting exchange coupling matrix issuch that the anti-symmetric combination of the two leadsdecouple from the local moment [1,2]: J ∼ V ∗ L V L V ∗ L V R V L V ∗ R V ∗ R V R = U V ∗ L V L + V ∗ R V R
00 0 U † , (1)where V i is the hybridization strength to the left/right ( i = L/R ) lead and the proportionality factor depends on thecharging energy of the dot and the chemical potential ofsource and drain. The local moment hence couples to thesum of the DOS of both leads. If the magnetization in thesource and drain are anti-aligned and the SET setup is oth-erwise symmetric w.r.t. the two leads, the local momentcouples to an effective band of unpolarized electrons andcomplete Kondo screening is recovered for arbitrary spinpolarization in the leads [19]. This was experimentally ver-ified by Pasupathy et al. [20]. Here, to illustrate the basicphysics, we will focus on such an anti-parallel case.The new observation we introduced in Ref. [17] is not thatStoner excitations can screen the local moment but thatthe spin waves in the ferromagnetic leads will also couple toit. The derivation of the effective low-energy model, givenin Ref. [17], confirms this symmetry argument. A gener-alized Schrieffer-Wolff transformation yields the following effective low-energy Hamiltonian [17]: H bfk = J X i S · s i + X k ,i,σ ˜ ǫ k σi c † k σi c k σi + h loc S z + g X β, q ,i S β ( φ β, q ,i + φ † β, q ,i ) + X β, q ,i ω q φ † β, q ,i φ β, q ,i . (2)where the local magnetic field h loc = g P i m i , with m i ,for i = L, R , being the ordered moment of the left/rightleads, ˜ ǫ k σi is the Zeeman-shifted conduction electron dis-persion, and φ β,i , with β = x, y , describes the magnon exci-tations. With the canonical transformation, Eq. (1), for thefermionic bath and a similar one for the bosonic bath, theeffective fermionic dispersion, labeled E k , becomes spin-independent; moreover, the antisymmetric combinations ofeach bath decouple. Hence, the low-energy properties of theferromagnetic SET are governed by a BFKM with an easy-plane anisotropy. For the anti-parallel alignment, m L = − m R , and h loc will vanish.Magnons are gapless bosons with a quadratic dispersion.The spectral density of the local dissipation they generateis sub-Ohmic, Z dq δ ( ω − ω q ) ∼ √ ω. (3)This feature turns out to be essential for the existence of aQCP [21]. Fig. 1b shows the corresponding phase diagramof the ferromagnetic SET. There are three renormalization-group fixed points: “Kondo” and “LM” refer to the Kondoscreened Fermi-liquid fixed point, and the critical local-moment fixed point, describing a quantum-critical phase.“QC” refers to the quantum-critical fixed point, character-izing the critical Kondo screening on the entire separatrix(red line, corresponding to the critical coupling g c as a func-tion of J ). Most dissipation channels will not lead to sub-Omic fluctuation spectra; coupling to phonons, photons, orantiferromagnetic magnons will not lead to critical Kondoscreening.The generalized Schrieffer-Wolff transformation relatesthe coupling constants J (Kondo coupling) and g (magnoncoupling) of Eq. (2) to the coupling constants of the originalmodel: J ∼ Γ / ( ρ ∆) and g ∼ Γ / ( ρ ∆) , where Γ = πρV isthe hybridization width, and ρ is the lead density of statesat the Fermi energy. ∆ is the charge fluctuation energy andis linearly dependent on the gate voltage V g of the SET.The gate voltage is therefore able to tune the competitionbetween the Kondo coupling and the coupling to the fluc-tuating magnon field. Since the Kondo screening occurs onthe scale of the bare ( g = 0, no magnons) Kondo temper-ature T K = (1 /ρ ) exp ( − /ρJ ), the control parameter is g/T K . T K depends exponentially on J , whereas g ∼ J .This implies that g/T K is exponentially large deep in theKondo regime and becomes of order unity in the mixed va-lence regime. This situation is reminiscent of the so-calledDoniach-Varma picture for the Kondo lattice where theRKKY interaction ( ∼ J ) competes with the Kondo sin-glet formation ( ∼ T K ) [11]. This analogy is not accidental.2he quantum phase transition as g is tuned through g c is reflected in the narrowing of the Kondo resonance, asseen in Fig. 1c. The transport properties in the quantumcritical regime have been worked out in Ref. [17]. In theKondo phase the conductance has the well-known Fermi-liquid form, G ( T ) = a − bT , where a = 2 e /h followsfrom Friedel’s sum rule. In the critical local moment phase( g > g c ) at T = 0, the electrons are completely decoupledfrom the local moment and the conductance vanishes. Atfinite temperatures, we find G ( T ) = cT / . The conduc-tance versus temperature at the critical gate voltage showsfractional power-law behavior, G ( T ) = A + BT / , where A is smaller than a . The experimental feasibility of thesemeasurements has been extensively discussed in Ref. [17].We now make the connection between our results andthe physics of quantum critical heavy fermion systems.The BFKM has been put forth as an effective model fora Kondo-destroying QCP in heavy fermion systems [13].In this approach, the self-consistency relation between thelattice system and the effective impurity model gives riseto a sub-Ohmic spectrum. The inference about the de-struction of Kondo effect at the antiferromagnetic QCP ofheavy fermion systems have come from the collapse of alarge Fermi surface and the vanishing of multiple energyscales [15,16]. The ferromagnetic SET structure discussedhere provides a tunable model system to study the physicsof a critical destruction of Kondo effect.
2. The Case of Critical Paramagnons
If the leads contain critical paramagnons instead of spinwaves, the dynamical spin susceptibility of the leads willhave an over-damped form: χ leads ( q , ω ) ∼ q − iω/γq (4)where γ is a constant. The dissipative spectrum becomes Z dq Im χ leads ( q , ω ) ∼ | ω | / sgn( ω ) . (5)Since in this case the spin-rotational invariance in the leadsis not broken, the issue of anti-parallel alignment does notarise. Palladium, for instance, has a Stoner enhancementfactor of around 10; there will be a large frequency windowover which Eq. (5) applies. Furthermore, contact propertiesof palladium leads are well studied and seem to be charac-terized by a relatively small contact resistance [22]. It hasbeen argued [17] that temperature/frequency dependencesof the critical electronic properties of BFKM with easy-plane anisotropy are similar to those of the same model withSU(2) invariance. For the Kondo-destroying QCP and thecritical local-moment phase, it was further argued that theyare similar to those of a large-N limit of an SU(N) × SU(N/2)generalization of the BFKM: H BFK = (
J/N ) X α S · s α + X k ,α,σ E k c † k ασ c k ασ G = G g =5:83T0Kg =6:17T0Kg =6:35T0Kg =6:5T0Kg =8:33T0K10(cid:0)7 10(cid:0)6 10(cid:0)5 10(cid:0)4 10(cid:0)3 10(cid:0)2 10(cid:0)1 1T=T0K10(cid:0)510(cid:0)410(cid:0)310(cid:0)210(cid:0)11 G = G g =0T0Kg =4:17T0K T ( ! ; T ) D T = 10(cid:0)5 T0KT = 10(cid:0)6 T0KT = 1:67 (cid:1) 10(cid:0)7T0K10(cid:0)8 10(cid:0)6 10(cid:0)4 10(cid:0)2 1 102 104 106 108!=T0:06250:1250:25 T ( ! ; T ) D T = 0:5 T0KT = 10(cid:0)1 T0KT = 10(cid:0)2 T0KT = 10(cid:0)3 T0KT = 10(cid:0)4 T0K ε=2/3ε=2/3 (a) (b)
Fig. 2. (a) DC conductance for different coupling strengths g , for ǫ = 2 /
3, the case of critical paramagnons. The zero temperaturevalue of the conductance in the Fermi liquid vase is fixed throughthe Friedel-Langreth sum rule. (b) ω/T -scaling at the QCP ( g = g c ).The universal scaling curve of the T-matrix can be probed via theAC conductance and Johnson noise measurements [17]. + ( g/ √ N ) S · Φ + X q ω q Φ † q · Φ q . (6)The large-N limit leads to a set of dynamical saddle-pointequations [25], which can be solved analytically at zero tem-perature and numerically at finite temperatures.Alternatively, the dynamical equations, exact in thelarge-N limit, can be used as an approximation for the N = 2 case. Ref. [17] considered the N = 2 version of theBose-Fermi Anderson model, H bfam = X k ,σ E k c † k σ c k σ + t X k ,σ (cid:18) c † k σ d σ + h.c. (cid:19) + ε d X σ d † σ d σ + U n d ↑ n d ↓ + g S d · Φ + X q ω q Φ † q · Φ q , (7)at U = ∞ (and, hence, particle-hole asymmetric). Thenumerical results presented in Ref. [17] are all for this N =2 case. At zero field, they have the same behavior as theexact results in the large-N limit of Eq. (6).We observe that the dissipative spectrum associated withthe critical paramagnons, Eq. (5), can be cast into the gen-eral form considered in Ref. [25], A Φ ( ω ) ∼ | ω | − ǫ sgn( ω ) , (8)with ǫ = 2 /
3. For general ǫ , the large-N results at zerotemperature [25] imply that, for the critical point ( g = g c ), T ′′ ( ω >
0) = const + const · ω ǫ/ . (9)Likewise, for the critical local-moment phase ( g > g c ), T ′′ ( ω >
0) = const · ω ǫ . (10)For the case appropriate to critical paramagnons, ǫ =2 /
3, we have carried out more detailed studies based onthe large-N limit of Eq. (6). Fig. 1c demonstrates the de-struction of Kondo resonance as the dissipative coupling g reaches g c and beyond. The DC conductance as a func-tion of temperature is given in Fig. 2a. The temperatureexponent at g = g c and g > g c are compatible to 1 / / T = 0 frequency dependence is consis-tent with ω/T scaling. The latter is further illustrated in3 .001 0.01 0.1 1h=T0K0.0010.010.11 M ( h ; T ) T (cid:25) 0:001T0KT (cid:25) 0:01T0Kf(x) = 0:43x(cid:0)1=3g(x) = 2:2x-8 -6 -4 -2 0 2 4 6 8 10 12!=T0K051015202530 T ( ! ) D h = 1:43T0Kh = 0 (a) (b) Fig. 3. (a) Kondo resonance in zero (dashed line) and finite local field(continuous line). The NCA, while capturing the Zeeman-split peaks,incorrectly produces a sharp resonance that is pinned to the Fermienergy ( ω = 0). This reflects its failure to capture the marginallyirrelevant character of the potential scattering term. (b) Local mag-netization at the critical coupling g c . The results are consistent withthe expectation based on hyperscaling. The parameters adopted are: ǫ d = − . D , U = ∞ , t = 0 . D , corresponding to T K = 4 . × − D ;the cut-off energy for the bosonic bath Λ = 0 . D . Fig. 2b, which demonstrates the ω/T scaling collapse of thedynamical T-matrix at g = g c . This ω/T scaling providesevidence for the interacting nature of the QCP. Because ǫ > /
2, the latter in turn is an indication for an uncon-ventional quantum criticality [25,26,27].
3. Issues on NCA in a finite field
In the case of ferromagnetic leads, a local magnetic fieldwill arise if the ordered moments of the two leads are par-allel, or if the couplings to the leads are asymmetric in theanti-parallel configuration. This refers to h loc of Eq. (2),along the direction of magnetic ordering. The effect of thisfield goes beyond Eqs. (6,7). In the following, we briefly dis-cuss what would happen if we were to incorporate a localfield in Eqs. (6,7). This effect is relevant if an external lo-cal field is applied along any of the spin-wave directions inthe ferromagnetic case, or along any direction in the caseof critical paramagnons. We further restrict to the case ofEq. (7), where for g = 0 the large-N equations reduce tothe commonly applied NCA formalism. Our purpose is toillustrate some delicate aspects in the theoretical treatmentof such a local field, h .The Kondo effect ( g = 0) in the presence of a magneticfield is a well-studied subject [23]. The poor performanceof the NCA for this problem has, however, not been exten-sively discussed in the literature. It was shown in Ref. [24]that within the NCA the potential scattering term of theAnderson model incorrectly scales in the same manner asthe spin exchange coupling. In a magnetic field, the up anddown fermions will be Zeeman-split. This gives rise to thesplitting of the Kondo resonance which is reproduced bythe NCA, see Fig. 3a. The NCA does however overestimatethe asymmetry of the two peaks and, more significantly,it incorrectly predicts a sharp feature at the Fermi energy( ω = 0). This sharp resonance is due to the NCA’s incorrecttreatment of the potential scattering term. Since this termis not affected by the local field, the ’Kondo resonance’ dueto this term remains at ω = 0. At the QCP, on the other hand, the Kondo effect hasbeen destroyed. One might therefore expect that the NCAcan still be used to obtain universal properties at a finitelocal field. Following a hyperscaling analysis similar to thatgiven in Ref. [28], and using the fact that χ stat ∼ T ǫ − , wefind that, for ǫ = 1 / M ( h, T = 0) ∼ | h | ǫ/ (2 − ǫ ) = | h | / , (11)and we expect | h | /T (2 − ǫ ) / = | h | /T / -scaling. For h << T the magnetization should therefore behave as M ( h, T ) ∼| h | , whereas for h >> T it will be M ( h, T ) ∼ | h | / . (Wehave set gµ B = 1.) This behavior is correctly reproducedby the NCA, see Fig. 3b. We conclude that the NCA, gen-eralized to incorporate the coupling to the bosonic bath,correctly captures certain universal properties of the quan-tum critical BFKM in a finite local field.In conclusion, a SET with ferromagnetic electrodes con-stitutes a tunable spintronic system that allows to experi-mentally access a quantum critical Kondo state. Nonequi-librium properties of this boundary quantum phase transi-tion are readily obtained by having µ = µ [see Fig. 1a].The ferromagnetic SET therefore seems to be an ideal sys-tem to address out-of-equilibrium aspects of quantum crit-icality both theoretically and experimentally.This work was supported in part by NSF, the Robert A.Welch Foundation, the W. M. Keck Foundation, and theRice Computational Research Cluster funded by NSF, anda partnership between Rice University, AMD and Cray.References [1] L. I. Glazman and M. E. Raikh, JETP Lett. 47 (1988) 452.[2] T. K. Ng and P. A. Lee, Phys. Rev. Lett. 61 (1988) 1768.[3] D. Goldhaber-Gordon et al., Nature 391 (1998) 156.[4] J. Nygard et al., Nature 408 (2000) 342.[5] J. Park et al., Nature 417 (2002) 722.[6] W. Liang et al., Nature 417 (2002) 725.[7] L. H. Yu and D. Natelson, Nano Letters 4 (2004) 79.[8] J. Paaske et al., Nature Physics 2 (2006) 460.[9] B. Doyon and N. Andrei, Phys. Rev. B 73 (2006) 245326.[10] A. Hewson, The Kondo Problem to Heavy Fermions , CambridgeUniversity Press (1993).[11] S. Doniach, Physica B 91 (1977) 231.[12] C. M. Varma, Rev. Mod. Phys. 48 (1976) 219.[13] Q. Si et al., Nature 413 (2001) 804.[14] P. Coleman et al., J. Phys. Cond. Matt. 13 (2001) R723.[15] S. Paschen et al., Nature 432 (2004) 881.[16] P. Gegenwart et al., Science 315 (2007) 969.[17] S. Kirchner et al., Proc. Natl. Acad. Sci. USA 102 (2005) 18824.[18] D. Natelson, private communication.[19] J. Martinek et al., Phys. Rev. Lett. 91 (2003) 247202.[20] A. N. Pasupathy et al. Science 306 (2004) 86.[21] L. Zhu and Q. Si, Phys. Rev. B 66 (2002) 024426; G. Zarandand E. Demler, ibid. (2002) 024427.[22] B. Babic and C. Sch¨onenberger, Phys. Rev. B 70 (2004) 195408.[23] T. A. Costi, Phys. Rev. Lett. 85 (2000) 1504.[24] S. Kirchner and J. Kroha, J. Low Temp. Phys. 126 (2002) 1233.[25] L. Zhu et al., Phys. Rev. Lett. 93 (2004) 267201.[26] M. Vojta et al., Phys. Rev. Lett. 94 (2005) 070604.
27] M. Glossop and K. Ingersent, Phys. Rev. Lett. 95 (2005) 067202.[28] K. Ingersent and Q. Si, Phys. Rev. Lett. 89 (2002) 076403.27] M. Glossop and K. Ingersent, Phys. Rev. Lett. 95 (2005) 067202.[28] K. Ingersent and Q. Si, Phys. Rev. Lett. 89 (2002) 076403.