Universal Efimov spectrum & interaction-induced zero mode transport in graphene

Reinhold Egger

Natal Workshop17-21 Aug 2015

Outline Brief introduction to graphene Electric dipole potential via adatoms

Universal Efimov scaling of bound state energies Prediction: observable in tunnel spectroscopy

De Martino, Klöpfer, Matrasulov & Egger, PRL 112, 186603 (2014)

Magnetic graphene waveguide: e-e interaction induced zero-mode conductance Flat E=0 band: no conductance without interactions Can interactions generate finite conductance for zero

modes? Yes they can! Characteristic filling dependence of conductance

Cohnitz, Häusler, Zazunov & Egger, PRB in press(arXiv:1506.05362)

Graphene: Tight binding description

Basis contains two atoms; nearest-neighbor hopping connects different sublattices:

Pseudospin

nmdda 14.0,3 ==

Wallace, Phys. Rev. 1947

Low energy limit

Two independent corner points K, K´ in 1st Brillouin zone

Valence and conduction bands touch K, K´ at E=0 „Dirac point“ = Fermi points in

undoped graphene Low energy:

Dirac light cone dispersion Emergent Lorentz invariance Smooth perturbations: no valley mixing!

( )

sec/10300 6 mcvKkq

qvqE

F

F

=≈

−=

±=

Relativistic quantum mechanicsLow energy continuum limit: only momenta close to K or K´ matter:relativistic Dirac Hamiltonian

Dirac spinorPauli matrices in pseudospin space:

Experimental confirmation for massless Dirac fermions in graphene monolayers:

cyclotron resonance and „half-integer“ quantum Hall effect

00 σσ VAceivH F +

+∇−⋅=

),(),( χφ=Ψ yx),( yx σσσ =

Novoselov et al., Nature 2005, Zhang et al., Nature 2005

Dirac fermion massFinite mass gap ∆ can (for example) be induced by strain effects (artificial) spin-orbit couplings substrate superlattice finite-size effects

first without magnetic field:

( ) ( ) 00 , σσσσ yxVivH zyyxxF +∆+∂+∂−=

potential e.g. due to adatoms

STM manipulation of charged impuritiesCrommie group, Nature Phys. 2012, Science 2013

Eva Andrei group, PRL 2014

charge on individual monomer is gate-tunable

Part I: Universal bound-state spectrum for gapped graphene in a dipole potential

Electric dipole in graphene

Dirac problem in electrostatic potential of two oppositely charged impurities ±Q at distance d

At long distance >> d: electric dipole potential, dipole moment p=Qd

Particle-hole symmetry: Energy spectrum comes in ± pairs, no zero mode!

De Martino, Klöpfer, Matrasulov & Egger, PRL 112, 186603 (2014)Gorbar, Gusynin & Sobol, arXiv:1506.08379

( )( ) ( )

2

2222

cos

2/2/,

rp

ydx

Q

ydx

QyxV

θ−→

+−−

++=

Qualitative pictureCoulomb impurities (e.g. Co atoms) arranged by STM tip

→ probe tunneling DoS by tunnel spectroscopy Wang et al. (Crommie group), Nature Phys. 2012 and Science 2013;

Luican-Mayer et al. (Eva Andrei group), PRL 2014

Near band edge Consider E=-∆+ε near (lower) band edge Dirac equation maps to effective Schrödinger equation

for „large“ (lower) spinor component

Solvable by separation ansatz Angular part: Mathieu functions Radial part: MacDonald functions

Dipole potential leads to universal hierarchy of infinitely many bound states

( ) 0,cos21

22 =

++∇

∆θχεθ r

rp

( ) ( ) ( )θθχ YrRr =,

Angular problem

With separation constant γ, angular part obeys energy-independent Mathieu equation

2π periodic solutions exist only at characteristic values

Quantum numbers: Parity κ=± & „angular momentum“ j=0,1,... (with j+κ>-1)

Mathieu functions

( ) 0cos22

2

=

∆−+ θθγ

θYp

dd

( )pj κγγ ,=

( ) ( )( ) ( )∆=

∆=

−

+

pYpY

jj

jj

4,2/se4,2/ce

2,

2,

θθ

θθ( ) ( )

( ) ( )∆=

∆=

−

+

pbp

pap

jj

jj

441

441

2,

2,

γ

γ

Radial problem

Radial equation

General solution (with decay at ∞) isMacDonald function Divergent at origin: fall-to-center problem Regularize by Dirichlet condition at r=r0=d/4

precise value from comparison to full solution of two-center problem

Eigenenergies follow from zeroes of MacDonald function: zeroes exist only when γ<0

( )pj κγγ ,=

( ) 02122

2

=

∆−−+ rR

rdrd

rdrd εγ

( )rK ∆εγ 2

Bound state energies

Bound state energies come in towers (radial quantum number n=1,2,...) at given (j,κ) Tower (j,κ) appears only above critical dipole

strength p>pj,κ where γj,κ(p) <0 lowest tower (j=0, κ=+) always realized

Universal Efimov scaling

Efimov scaling (within tower) involves universal numbers

accumulation of bound states near band edge infinitely many bound states in each tower

... reflects the long-distance behavior of the dipole potential, cf. three-boson problem

Efimov scaling follows also from Abramov-Komarov asympotic solution of full two-center problem

( )( )

0),0(,

,956.0,2

,, >

+=

∆−

∆=

jj

ppp

sj

j

κ

κκ

κπ

εε ,/21 js

n

n e−+ =

Density distribution

Mathieu functions: characteristic asymmetric angular features

MacDonald function explains radial density distribution

Probe by tunneling spectroscopy !

5=∆p

1=n 2=n

( )+,0

( )−,1

Away from band edge: Numerical diagonalization of Dirac equation

Numerics for dipole potential: graphene disk of radius 40d

No zero modes !

Part II: Interaction-induced zero mode conductance

Orbital magnetic field

Now consider inhomogeneous magnetic field & massless case

Single-particle problem: Dirac equation

relativistic Landau levels for homogeneous caseInhomogeneous fields: Confine electrons in magnetic dots, guide electrons in channels, magnetic barriers ...

De Martino, Dell‘Anna & Egger, PRL 2007

AeyxBB z

×∇== ),(

( ) Ψ=Ψ⋅

+∇− EyxA

ceivF σ

,

Inhomogeneous magnetic fields

Almost arbitrary magnetic field profiles can be experimentally created Deposition of lithographically defined

ferromagnetic layers on graphene covered by thin insulating layer Cerchez, Hugger, Heinzel, Schulz, PRB 2007

Here: disorder-free (clean) case Another possibility: strain-induced pseudo-

magnetic fields

Magnetic graphene waveguide (MGW) Magnetic field reversed within

central strip of width d Precise form of field profile

not important Classical snake orbits along

zero field lines: waveguide Quantum-mechanical

dispersion relation?

Magnetic length

Magnetic energy

eBclB =

BFB lvE /=

Single-particle problem

Landau gauge

Momentum (k) conserved along waveguide

Particle-hole symmetry:Inversion symmetry:

( )

2/2/||2/

,,,

)(

ˆ

dxdxdx

dxxdx

BxA

exAA y

><−<

−−+

⋅=

=

( ) ( )xeL

yx

EH

kniky

ykn

knknkn

,,

,,,0

1, ψ=Ψ

Ψ=Ψ ( ) ( )( )

=

xix

xkn

knkn

,

,, χ

φψ

1D states

real functions

knkn EE ,,0 −≠ −=

knkn EE ,, =−

Single-particle spectrum

Match wavefunctions in different regions with constant field (then expressed in terms of parabolic cylinder functions)

Ghosh, De Martino, Häusler, Dell‘Anna & Egger, PRB 2008

Probability maximum in x direction is at X ~ - k Approach to Landau

levels for |k|>>d Pair of snake states

(linear dispersion), electron- or hole-like

waveguide states: |k|<d Flat band (n=0) :

zero mode

Bld 2=

Zero mode band

Eigenstates for n=0:

All zero-mode current matrix elements vanishsince there is no upper spinor component

( )

( ) 2/2/

,

,1~

0

2222

/2/2/)/(/||

,0

,0,0

dxdx

ee

xi

B

BBBldx

kllxlxdk

kk

<>

⋅

=

−+−χ

χψ

0',0,0',0;,0 == ∫ +kykFkk dxvI ψσψ

Transport through finite-energy bands

If Fermi level intersects n=1 band: transport through quantum-mechanical snake orbits

Snake states spatially well separated for d>lB → Weak disorder / irregularities in magnetic field irrelevant:conductance quantization

independent of n=1 band filling (i.e. Fermi momentum)

Include Coulomb interactions: no change in G at T=0 → Luttinger liquid

heG /2=

Häusler, De Martino, Ghosh & Egger, PRB 2008

What if zero mode is partially filled?

All zero-mode current matrix elements vanish→ without interactions conductance vanishes Finite-energy bands have quantized conductance,

qualitatively different Intra-band n=0 Coulomb interaction matrix

elements cannot generate upper spinor component →

Finite zero-mode conductance requires inter-band transitions !

Coulomb interaction consider 3D Coulomb interaction potential,

neglect retardation effects include image potential due to parallel

metallic backgate at distance D Interaction strength: effective fine structure

constant

For density n, typical kinetic energy Typical Coulomb energy:

nEk ~

nreEC ~~

2

rFrk

C

ve

EE

εεα 2.22

≈==

not tunable through density!

Conductance for partially filled zero mode: Strategy

1. Hartree-Fock theory for intra-band interactions → break huge degeneracy of zero mode band

2. Then: Perturbation theory in inter-band Coulomb matrix elements → Compute T=0 conductance from Kubo formula vs filling factor

degeneracy (index theorem): ( )22

2

B

yxs l

LdLN

π−

=

sNN=ν

HF theory for zero mode bandAfter projection to n=0 band: self-consistent calculation of HF parameters with constraint

→ HF orbital energies

kkk ccn ,0,0+=

∑ ==k

sk NNn ν

( ) ''

)0(';',

)0(0;', k

k

nkkqkk

nqkkk nVV∑ =

−==

= −=ε

5.02

===

αBldD

HF „single-particle“ dispersion Dips for |k|<d, correspond to waveguide states Pauli holes at boundaries also follow from d=0 analytical

considerations, do not contribute to conductance Interaction-induced Fermi momentum and Fermi velocity

Diagrammatic expansionConductance computed by perturbative expansion in powers of W = residual inter-band interactions

(a)

(b)

(c)

(a) (b) (c)

(d) (e)

(f) (g)

First order diagrams: G=0 still holds

Second-order diagrams: G>0

Zero-mode conductance

0.1 0.15 0.2 0.250

0.2

0.4

0.6

0.8

1

G / G0 0 0.2 0.4 0.60

0.5

1G / G

0

Zero mode conductance

Pronounced dependence of G on filling factor Interplay of current matrix renormalization by virtual

inter-band transitions and electron-hole pair fluctuation effects

Both effects compete: minimum at finite filling No conductance quantization anymore! For finite-energy bands completely different behavior

Interaction-induced zero-mode conductor→ direct probing of electron-electron interactions

Cohnitz, Häusler, Zazunov & Egger, PRB in press (arXiv:1506.05362)

Summary

Introduction Electric dipole in graphene

Universal Efimov scaling for bound state energies Observable in tunnel spectroscopy

De Martino, Klöpfer, Matrasulov & Egger, PRL 112, 186603 (2014)

Magnetic graphene waveguide: interaction induced zero mode conductance Flat band: zero conductance without interactions! Can interactions generate finite conductance for zero

modes? Yes they can! Filling factor dependence of conductance

Cohnitz, Häusler, Zazunov & Egger, PRB in press (arXiv:1506.05362)

THANK YOU FOR YOUR ATTENTION!