Berry-Phase Induced Kondo Effect in Single-Molecule Magnets

Berry-Phase Induced Kondo Effect inSingle-Molecule MagnetsEduardo MuccioloDepartment of PhysicsUniversity of Central FloridaCollaborators: Michael Leuenberger (UCF) Gabriel Gonzalez (UCF)Tulane University (November 2008)

Overview- Single-molecule electronics- Single-molecule magnets:- Berry-phase blockade in single molecules- Kondo effect in SMMs

Single-molecule electronics:molecular transistorelectrode(source)V gatemoleculeelectrode(drain)1 - 10 nmV biasinsulated metal plate(gate)IPotential advantages:- fast operation- large energy scales (eV)- quantum effects at high temperatures ?!

Fabrication of the nano gaps:(Enrique del Barco, UCF)Electromigration and breaking of nanowires10Current (mA)8644K200.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4Voltage (V)

Multiwire chip: Trial and error approach6µm70 nm

Single-electron transistors: What is actually measured?μ N < μ S , μ D < μ N+1E c +ΔEμ N+1μ Dμ Sμ D < μ N+1 < μ Sμ N+1μ Sμ S = μ N+1 = μ Dμ Dμ N+1μ SDμ Nμ N-1Sμ Dμ N-1Dμ NSDμ Nμ N-1Sno current(Coulomb blockade)finite currentwith finite biasfinite current with(nearly) zero biasIdI/dVNN+1

Early experiments: Non magnetic moleculesNATURE | VOL 417 | 13 JUNE 2002 |www.nature.com/natureCoulomb blockade and the Kondoeffect in single-atom transistorsJiwoong Park*†‡, Abhay N. Pasupathy*‡, Jonas I. Goldsmith§,Connie Chang*, Yuval Yaish*, Jason R. Petta*, Marie Rinkoski*,James P. Sethna*, He´ctor D. Abrun˜a§, Paul L. McEuen* & Daniel C. Ralph** Laboratory of Atomic and Solid State Physics; and § Department of Chemistryand Chemical Biology, Cornell University, Ithaca, New York 14853, USA† Department of Physics, University of California, Berkeley, California 94720,USASeveral other groups around the world have also observed similar effects in molecular transistors

Our Motivation:How to increase the functionality of a molecular transistor ?Combine electronics with magnetism:(i) Use magnetic molecules(ii) Use magnetic contacts

How do QTM and Berry phase interferencemanifest themselves in electronic transportthrough a single SMM?- Spin-current blockade- Kondo effect

Recent experiments (Mn 12): Still not very conclusive...Herre van der Zant (Delft)Dan Ralph (Cornell)Enrique del Barco (UCF)No experiment has yetseen a unambiguousmanifestation of QTM,much less Berry phaseinterference...

Kondo effect: The case of quantum dots and molecules attached to leadsAt low temperatures, spin flip processes strongly renormalize the conductanceinitial statevirtual (intermediate) statefinal stateSharp resonance at the Fermi level appears at T T 3!E FEEF

Another way of probing QTM: Kondo effectno Coulomb blockade!The Kondo effect in a non-magnetic single-electron transistorhas already been observed by several groups...... but not yet for SMMs.

Unconventional Kondo effect in SMMs: How it happensM. Leuenberger and ERM [PRL 97, 126601 (2006)]S =mztS =m!12s = s =zztS =m!11U!1 +12z2z(!S = !1,z ! s z= +1)2~ t Uinelastic( E m ≠ E m-1 )(suppressed at zero bias)S =m S =mz!12tS =m!1U12s = s =zztzz!12(!S = 0,~ztU2! s = 0) zelasticJ zno spinflippingS =mzs =z!12tS z=m!121U!ES =!m+z1U12tS =!mz+1s z =2(!S = !2m, ! s = +1)~z2t !E2Uzelastic( E m = E -m )J ⊥spinflipping

Kondo effect in SMMs: detailed theoryeffective HamiltonianH = H SMM + H KondoH SMM = ∑ m ∗ E m ∗ |m ∗ 〉〈m ∗ |, E m ∗ = E −m ∗ [∆ m∗ ,−m ∗ (H ∗ x) = 0]H Kondo = ∑ k,α(ξ k + gµ BHxs ∗ )x2local exchange termψ † kα ψ kα + H exH ex = 1 2∑m∑ [j (m)+ Σ (m)k,k ′project it onto the {|m} subspace+ ψ † k↓ ψ k ′ ↑ + j (m)−Σ (m)−ψ † k↑ ψ k ′ ↓ + j z(m) Σ (m)z()]ψ † k↑ ψ k ′ ↑ − ψ † k↓ ψ k ′ ↓pseudo-spin flipping termspseudo-spin operatorsΣ (m)z(|m〉〈m| − | − m〉〈−m|)= 1 2Σ (m)± = | ± m〉〈∓m|anisotropic coupling constantsj (m)z= 2J z 〈m|S z |m〉j (m)± = J ± 〈±m|S ± | ∓ m〉

Kondo effect in SMMs: microscopic derivation of coupling constants !1EE(!1)s,m(!1)a,menergy levels relevantto the Kondo effect 0[]j z = ± 2t 2 U + ∆ 0U + ∆ 0(U + ∆ 0 ) 2 − ∆ 2 +1 (U − ∆ 0 ) 2 − ∆ 2 1[]j ⊥ = 4t 2 ∆ 1∆ 1(U + ∆ 0 ) 2 − ∆ 2 +1 (U − ∆ 0 ) 2 − ∆ 2 1|j z | ≫ j ⊥U2EE(0)a,m(0)s,m≈ ± 4t2U≈ 8t2 ∆ 1UU2 1q=!1 q=0 q=1anisotropic exchange interaction!1!100(1)Ea,m1(1)E s,m1The sign depends onthe intermediate spinstate of the molecule!S 1 = S 0 − 1 2S 1 = S 0 + 1 2AFFM

Kondo effect in SMMs: poor man’s Renormalization Group!total Hamiltonian in projected subspaceH = ∑ m[E m(Σ (m)z) 2+ η (m) H ∗ xΣ (m)x]+ ∑ k,αξ k ψ † kα ψ k,α + H ex~DDEη (m) = 1 − νj(m) ⊥2Knight shiftmolecule’s pseudo-spin couples to transversal fieldRenormalization Group flow equations...ζ = ln( ˜D/D)dj zdζ = −2ν j +j −dj ±dζ = −2ν j ±j z... and solutions1.5dηdζ = ν22 (j + + j − ) j zj 2 z − j 2 ⊥ = C > 0(√ )1C2ν √ C arctanh j z= ln( ˜DT K)! (T / T K)10.50-0.5H ∗ x −→ H ∗ x/η (m)˜D≈T-10 2 4 6 8 10T / T K

Kondo effect in SMMs: ConductanceLinear conductance (T dependence)scattering amplitudeG(T ) = G 0∫dωA (m) ≈ j(m)ω≈ ˜D ⊥,ω≈ ˜D = √ C(− df ) π 2 ν 2dω 16[∑m e−E m/k B T |A (m) (ω)| 2∑m e−E m/k B T(ω/T K ) 2ν√ C(ω/T K ) 4ν√ C− 1]1st order perturbation theorysingularity for T

Kondo effect in SMMs: Berry phase oscillations!The tunnel splitting is an oscillating function ofthe transverse magnetic field due to the Berryphase interference.123Htwo-fold degeneracypoints (Kondo effect)H1< H 2 < H 3GConsequences:1) The Kondo peak splitting is a non-monotonicfunction of the transverse magnetic field.0eV2) The period of Berry oscillations is renormalized by the Kondo effect(strongly temperature dependent, with a universal function form).

Kondo effect in SMMs: Ni 4, the best candidate[Ni 4(ROH) 4L 4O 12] (R=Me,Et) Sieber et al. (2005)Spin tunneling splitting Δ (numerical simulations)=!E S,-S(H ) [K]10 010 -110 -210 -310 -4" = 34 #T / T K= 2T / T K= 3T / T K= 10!E S,-S(H ) [K]10 010 -110 -210 -310 -4" = 41 #0 1 2 3 4 5 6 7H [T]

Some estimates (Ni 4):M. Leuenberger and ERM [PRL 97, 126601 (2006)]T K ≈ D exp[ (√ )−arctanh C/jz /2ν √ ]Cν(J z − J ± ) ≈ 0.15∆E = D = ∣ ∣A [ S 2 − (S − 1) 2]∣ ∣ ≈ 9.3 KT K ≈ 1.2 K (m = ±4)crucial requirements:i) large spin tunnel splittingii) large coupling to states in the leadsissues under investigation:i) quantitative theory for transport (NRG, DMRG?)ii) spin/angular momentum relaxation in isolated moleculesSee also related work by the Aachen group (H. Schoeller).

Some Questions and Challenges for the STM group:(1) How does the SMM bind to metallic surfaces?(2) Where does the additional electron go in a SMM?chemistry/electronic structure(3) Can the SMM be manipulated by the SMT tip?(move it, flip it, and extract or modify ligands)technical challenge(4) Does the tip position change the electric responseof the SMM?(5) Can a SP-STM measure the magnetization curve ofa SMM (quantum tunneling, coherent oscillations,decoherence)?physicsSP!tipSMMmagnetic islandnonmagnetic substrateSMMs have intrinsically largemagnetization and stronganisotropy, so a magneticisland may not be necessary.

The End