Kondo effects in multilevel quantum dots (a renormalization group study) David Logan and Martin Galpin, Chris Wright Oxford University NATO ARW, Yalta, 20.9.07.
1. Introduction. Recent years have seen strong renewal of interest in Kondo physics , due in large part to the advent of quantum dot systems – from semiconducting QDs based on GaAs/n-doped GaAs heterostructures, through e.g. carbon nanotube dots, to molecular electronic devices. Semiconducting QDs in particular provide a tunable, controlled realisation of ‘mesoscopic atoms’ – cf the spin-1/2 Kondo effect in an odd-electron QD, manifest e.g. in the unitarity limit for the zero-bias conductance ......
Spin-1/2 Kondo effect ...... Odd electron valleys:– – spin-1/2 Kondo effect; – unitarity limit in zero-bias conductance for T ¿ T K : – associated in effect with a single dot level, singly occupied. – Kondo progressively killed by increasing T. V t l source lead V r V gl drain V bl van der Wiel et al, Science (2000).
Even electron valleys : – no Kondo; suppressed conductance Odd electron valleys : – spin-1/2 Kondo effect; van der Wiel et al, Science (2000).
This odd/even (or ‘Kondo/non-Kondo’) alternation is common in QDs, but not ubiquitous: – if the dot level spacing is sufficiently small, might expect instead to see S=1 Kondo physics in an even valley, i.e. S=1 (ferromag Hund’s coupling). even odd Such behaviour is indeed observed ....... e.g...... van der Wiel et al, Science (2000).
Kogan et al, PRB 67 , 113309 (2003) Triplet Kondo: Even valley: also strongly enhanced zero-bias conductance, indicative of (underscreened) spin-1 Kondo. Talk about aspects of the S=1 and S=1/2 Kondo regimes, and the transition between them …… Odd valley: spin -½ Kondo, strongly enhanced zero-bias conductance. V t l ¢ V g V bl
Some background refs: Theory: Pustilnik, Glazman, Kondo effect in real quantum dots, PRL 87 , 216601 (2001) Hofstetter, Schoeller, Quantum phase transition in a mulitlevel dot, PRL 88 , 016803 (2002) Vojta, Bulla, Hofstetter, Quantum phase transitions in models of coupled magnetic impurities, PRB 65 , 140405 (R) (2002) Posazhennikova, Coleman, Anomalous conductance of a spin-1 quantum dot, PRL 94 , 036802 (2005); Conductance of a spin-1 quantum dot: the two-stage Kondo effect, PRB 75 , 245329 (2007) Koller, Hewson, Meyer, Singular dynamics of underscreened magnetic impurity models, PRB 72 , 045117 (2005) Mehta, Andrei, Coleman, Borda, Zarand, Regular and singular Fermi-liquid fixed points in quantum impurity models, PRB 72 , 014430 (2005) Pustilnik, Borda, Phase transition, spin-charge separation and spin filtering in a quantum dot, PRB 73 , 201301 (R) (2006) Experiment: Kogan, Granger, Kastner, Goldhaber-Gordon, Shtrikman Singlet-triplet transition in a single-electron transistor at zero magnetic field , PRB 67 , 113309 (2003)
Outline:- – 2-level dot coupled to two conducting leads. 2. Model – some issues/questions. Results shown here obtained mainly from Wilson’s numerical renormalization group (NRG)* as method of choice – non-perturbative RG, providing essentially exact results on low-temperature/energy scales central to the physics. – static/thermodynamic properties. 3. Results – evolution of Kondo scale. – phase diagram(s). – single-particle dynamics, and transport. – experiment. [ * use both ‘standard’ NRG and full density matrix/complete Fock space NRG (Peters, Pruschke, Anders; Weichselbaum, von Delft …)]
MODEL: V 2L V 2R 2 2-level dot, coupled L to two conducting R leads. 1 V 1R V 1L n 2# ) + U 0 ^ H D = ² 1 ^ n 1 + ² 2 ^ n 2 + U (^ n 1" ^ n 1# + ^ n 2" ^ n 1 ^ n 2 ¡ J H ^ s 1 ¢^ s 2 (J H > 0 – ferromagnetic [Hund’s rule] coupling) X X ² k c y H 0 = k¸ ¾ c k¸ ¾ – two non-interacting, metallic leads ¸ = L ;R k;¾ X X X V i ¸ (c y k¸ ¾ d i ¾ + d y H V = i ¾ c k¸ ¾ ) – dot/leads tunnel coupling i = 1;2 ¸ = L ;R k;¾ Model both ‘general’ and rich. Look first at the dot states arising in the ‘atomic limit ’, where dot decouples from leads .......
Atomic limit: dot states n 2# ) + U 0 ^ H D = ² 1 ^ n 1 + ² 2 ^ n 2 + U (^ n 1" ^ n 1# + ^ n 2" ^ n 1 ^ n 2 ¡ J H ^ s 1 ¢^ s 2 ² 2 effectively a one-level problem; ‘filling level 1’. 1 2 1 2 2 1 1 2 1 2 1 2 1 2 1 2 1 2 ² 1
Atomic limit: dot states n 2# ) + U 0 ^ H D = ² 1 ^ n 1 + ² 2 ^ n 2 + U (^ n 1" ^ n 1# + ^ n 2" ^ n 1 ^ n 2 ¡ J H ^ s 1 ¢^ s 2 ² 2 effectively a one-level problem; ‘filling level 1’. 1 2 1 2 2 1 1 2 1 2 1 2 1 2 effectively a one-level problem; 1 2 1 2 ‘filling level 2’. ² 1
Atomic limit: dot states n 2# ) + U 0 ^ H D = ² 1 ^ n 1 + ² 2 ^ n 2 + U (^ n 1" ^ n 1# + ^ n 2" ^ n 1 ^ n 2 ¡ J H ^ s 1 ¢^ s 2 ² 2 States shown are spin S=1/2 or 0. 1 2 Finally, there’s the S=1 state. 1 2 2 1 1 2 1 2 1 2 1 2 1 2 1 2 ² 1
Atomic limit: dot states n 2# ) + U 0 ^ H D = ² 1 ^ n 1 + ² 2 ^ n 2 + U (^ n 1" ^ n 1# + ^ n 2" ^ n 1 ^ n 2 ¡ J H ^ s 1 ¢^ s 2 ² 2 Obvious question: what happens on coupling to the leads? .... – ‘deep’ in the S=1/2 regimes, low-energy model is usual spin-1/2 Kondo. 1 2 1 2 2 1 So ultimate stable fixed point (FP) is usual strong coupling FP: dot spin 1 2 quenched on scale T K ; strongly enhanced 1 2 G c (0) » 2e 2 =h: T=0 zero-bias conductance, 1 2 System a ‘regular’ Fermi liquid. 1 2 1 2 1 2 ² 1
Atomic limit: dot states n 2# ) + U 0 ^ H D = ² 1 ^ n 1 + ² 2 ^ n 2 + U (^ n 1" ^ n 1# + ^ n 2" ^ n 1 ^ n 2 ¡ J H ^ s 1 ¢^ s 2 ² 2 Obvious question: what happens on coupling to the leads? ... – ‘deep’ in the S=1 regime, low-energy model will be spin-1 Kondo. 1 2 1 2 Is it 1 - channel or 2-channel spin-1 Kondo? 2 1 Depends on coupling to leads….. 1 2 If 2-channel, S=1 wholly quenched on 1 2 coupling to leads (strong coupling FP); 1 2 and strong suppressed dc, G c (0) » 0 (Pustilnik/Glazman). 1 2 1 2 If 1-channel, S=1 only partially quenched 1 2 on coupling to leads (‘underscreened’ S=1 FP, Nozieres/Blandin);in this case, strongly G c (0) » 2e 2 =h: enhanced ² 1 System a ‘singular’ Fermi liquid (Mehta et al ).
Questions contd: – for S=1 regime, two -channel behaviour is generic. So apparent puzzle (Coleman G c (0) » 2e 2 =h? et al): why do exps see strongly enhanced Answer: low-energy model is generically 2-channel spin-1 Kondo with channel anisotropy : J ¡ J + : J + > J ¡ R 0 0 L 0 0 2-stage quenching of spin-1: R 0 0 : ! T K + » D exp(¡ 1=½ J + ) , quench spin S=1 S=1/2 by coupling to – on ‘high’ scale Here flow towards under screened S=1 fixed point. L 0 0 : ! – then on ‘low’ scale T K ¡ » D exp(¡ 1=½ J ¡ ); quench spin S=1/2 S=0 by coupling to (flowing then to fully screened, strong coupling FP) But the two scales may be vastly different in magnitude. T K ¡ ‘irrelevantly small’, in practice might as well be 0: And if – experiment will then ‘see’ the under screened spin-1 FP. We’ll take this for granted here; in practice, consider effective 1-channel set-up from beginning.
Atomic limit: dot states n 2# ) + U 0 ^ H D = ² 1 ^ n 1 + ² 2 ^ n 2 + U (^ n 1" ^ n 1# + ^ n 2" ^ n 1 ^ n 2 ¡ J H ^ s 1 ¢^ s 2 ² 2 Questions contd: – where does exp fit in to figure? 1 2 – so (Pustilnik/Borda) does there exist a 1 2 quantum phase transition (QPT) 2 1 between ‘regular’ spin-1/2 Kondo and spin-1 Kondo? 1 2 1 2 If so, what is the nature of the QPT? 1 2 – Can exp be explained? 1 2 1 2 1 2 ² 1
So consider explicitly:- Vsin µ Vcos µ 2 L R 1 Vsin µ Vcos µ V 2 0 R [two leads of course, but of 1-channel form: ] 1 V n 2# ) + U 0 ^ H D = ² 1 ^ n 1 + ² 2 ^ n 2 + U (^ n 1" ^ n 1# + ^ n 2" ^ n 1 ^ n 2 ¡ J H ^ s 1 ¢^ s 2 X X ² k c y H 0 = k¸ ¾ c k¸ ¾ – two non-interacting, metallic leads ¸ = L ;R k;¾ X X X V i ¸ (c y k¸ ¾ d i ¾ + d y H V = i ¾ c k¸ ¾ ) – dot/leads tunnel coupling ¸ = L ;R i = 1;2 k;¾
Recall discussion of ‘atomic limit’:- What of the phase boundaries on coupling to leads? ² 2 – all phases that were S=1/2 or 0 in the atomic limit are continuously connected (as we know..); here the stable fixed point is the usual strong coupling (or frozen impurity) FP. 1 2 1 2 – but the underscreened triplet state has a 1 2 distinct FP. So we expect a quantum phase transition; and hence the qualitative 1 2 1 2 behaviour shown. Indeed. 1 2 1 2 1 2 1 2 ² 1
For purposes of illustration here, will now fix ² 1 , and progressively decrease ² 2 from ‘high up’ in the spin-1/2 Kondo regime, down into the spin-1 Kondo regime, to ² 2 = ² 1 : ² 2 We’ll chose specifically ² 1 = ¡ U=2¡ U 0 (‘midpoint’); the point ² 2 = ² 1 is then particle-hole symmetric. 1 2 Will consider sequentially: – static/thermodynamic properties. 1 2 – evolution of the Kondo scale. – phase diagram(s). – single-particle dynamics. – transport/differential conductance. – and experiment. ² 1
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