interaction of dirac electrons with spins and point chargeseandrei/trieste18/lecture3... ·...

E.Y. Andrei

Vacancies in

graphene

Interaction of Dirac electrons with spins and point charges

Eva Y. Andrei

Rutgers University

August 27-28. 2018

1S

Vacancy Charge

and Tunable

artificial atom

Vacancy Magnetic

moment and

Kondo screening

Twisted bilayer

graphene

LECTURE NOTES POSTED AT:

http://www.physics.rutgers.edu/~eandrei/links.html#trieste18

E.Y. Andrei

Vacancies in

graphene

Interaction of Dirac electrons with spins and point charges

Eva Y. Andrei

Rutgers University

August 27-28. 2018

1S

Vacancy Charge

and Tunable

artificial atom

Vacancy Magnetic

moment and

Kondo screening

Twisted bilayer

graphene

Engineering electronic properties

• Density of states and Landau levels in graphene

• Scanning tunneling microscopy (STM) and spectroscopy (STS)

• Defects:

Atomic collapse and artificial atom

Kondo effect

• Substrate:

Twisted graphene

E.Y. Andrei

Density of states

VgSiO2

Si

r(E)

E EF electrons

Ef holes

Vg

+_

Density of states

1V a 7x1010 cm-2

Band structure

Ingredients:1. 2D2. Honeycomb structure 3. Identical atoms

sp2 Carbon Ultra-relativistic

Chiral quasiparticles

pvE F

222/33)(

Fv

EaE

r

skip

E.Y. Andrei

Landau levels in graphene from Dirac-Weyl equation

kx

ky

E

B0

D(E)

E

D(E)

E

-1

01

-2

1

2

3

4

-3

-4

Band structure Density of states

Finite B z Landau Levels

e/h;g4g:Degeneracy 00

,..1,0;22 NNNBevE cFN

meV B35/ BFc lv

0.

.0* p

pvH F

0).(

).(0* Aep

AepvH F

nm B25/ eB/lB

][105.2/ 214

0 TBmBegeneracyOrbital D g0

E.Y. Andrei

Graphene and conventional 2d electron systems

Conventional

semiconductor

Low energy excitations Density of states

D(E)

E

2

em2

hm

Graphene D(E)

E

pxpy

pvE

NBevE FN 2

px

py

E)2/1( N

m

eBEN

c

Landau levels

E.Y. Andrei

Landau levels: graphene and conventional 2D ES

//

,...1,0;22

eBvlv

NNNBevE

FBFc

cFN

px

py

E)2/1( N

m

eBEN

c

NBevE FN 2

EN=0 =0 Berry phase

Electron hole symmetric

E ~ N1/2 Level

E ~ B1/2

gi=4

Gap at E=0 Berry phase =0

Either electrons OR holes.

Equally spaced levels

E~B

gi=2

meB

NNE

c

cN

/

,...2,1,0)2/1(

Conventional 2DESGraphene

nm B25/ eB/lB

][105.8/g 214

00

TmBB

] /T[meV B33 1/2c GaAs /T]meV [4.1 Bc

Populating Landau levels

B

Flux line Electronic state

E1

E0

Landau levels

Wb104e

h 15

0

Flux Quantum

Flux Density 0/B

214

0o T 105.2/g mBegeneracyOrbital D

c

oo g4g igeneracyTotal Deg

B

n

ggg

nN s

ii

s 0

0

1

# of filled Landau levels

# of electrons per flux line

0g

nNg s

i

nm B25/ eB/lB

E.Y. Andrei

Summary

orbit cyclotron by enclosedflux 0

N

]m[Tesla cmgauss104.14 227-

0 e

h

flux ofunit Quantum

PhaseBerry

)2/2/1(2)( 2

NlkSBN

Onsager relation :k-space area of cyclotron orbit

)2/1( Nm

eBEN NBevE FN 2

Landau level energyNon-relativistic Ultra-relativistic (graphene)

degeneracy: gigo

0

4

B

0

2

B

E.Y. Andrei

Scanning tunneling microscopy and spectroscopy

Engineering electronic properties

• Density of states and Landau levels in graphene

• Scanning tunneling microscopy (STM) and spectroscopy (STS)

• Defects:

Atomic collapse and artificial atom

Kondo effect

• Substrate:

Twisted graphene

E.Y. Andrei

Scanning Tunneling Microscopy (STM

STM measures tunnel current across a vacuum barrier

r )(

0

exp),(),,( rz

eV

drVzrIb

Vb

zVb

E.Y. Andrei

r )(

0

exp),(),,( rz

eV

drVzrIb

Scanning Tunneling Microscopy (STM

STM measures tunnel current across a vacuum barrier

Vb

z Topography:

Imaging Atoms

Spectroscopy:

Density of states

I ,Vb constant z constant

),( bb VrdVdI r)(exp),( rzzrI

-400 -200 0 2000.0

0.2

0.4

0.6

0.8

1.0

dI/dV

(a.u

.)

Sample Bias (mV)

E.Y. Andrei

Rutgers Low Temperature High field STM

Rutgers STM

– Temperature T=4 (2K)

– Magnetic field B=13 (15T)

– Scan range 10-10 -10-3 m

Topography a structure

Spectroscopy a Density of states B=0

Spectroscopy a Density of states B>0

Liquid Helium

Low temperature STM

Air Springs

E.Y. Andrei

Graphene on Graphite : the Perfect substrate

topography B=0 spectroscopy B>0 spectroscopy

skip-200 -100 0 100 200

0.0

0.5

1.0

dI/

dV

(a.u

.)

E (meV)

H = 4 T T=4.36 K

NBe2vE FN

G. Li , E.Y. A - Nature Physics, (2007)

G. Li, A. Luican, E. Y. A., Phys. Rev. Lett (2009)

Local doping

Electron-phonon coupling Local Fermi velocity

Quasiparticle lifetime

Coupling to substrate

-200 -100 0 100 2000.0

0.2

0.4

0.6

0.8

Sample bias (mV)

0.0 T

dI/dV

(a.u

.)

EF DP

E.Y. Andrei

Vacancies: Inducing local magnetic moments and charge

Engineering electronic properties

• Density of states and Landau levels in graphene

• Scanning tunneling microscopy (STM) and spectroscopy (STS)

• Defects:

Atomic collapse and artificial atom

Kondo effect

• Substrate:

Twisted graphene

E.Y. Andrei

Perfect Graphene

Carbon • No d, f electrons • But partially filled p shell

Magnetism: Spin of localized electrons

in partially filled inner d or f shell .

Graphite, graphene, …, Carbon allotropes

? Non-Magnetic ?

sp2 Carbon

E.Y. Andrei

Magnetism and Perfect Graphene

...repulsive Hubbard model + bipartite lattice + half-filled band:

spin of ground state with NA, NB populated sites:

S= 1/2(NA- NB)

Perfect graphene

Non-Magnetic!

Vacancies z NA>NB a magnetic moment

A.H. Castro Neto et al. Solid State Commun. (2009).

T. Wehling,. Phys. Lett. (2009)

O. Yazyev, et al Rep. Prog. (2010).

M. Vojta et.al, EPL, 90 (2010) 27006

T. O. Wehling, Phys. Rev. B 81, 115427(2010)

J. O. Sofo, et al Phys. Rev. B 85, 115405 (2012)

Pristine graphene (graphite) NA=NB z S=0

E.Y. Andrei

Imperfect Graphene – Vacancy Magnetic Moment

E

DOS

Dirac

bands

Dangling bond a localized state a 1B

pz a quasi-localized state on other sublattice a ~0.5-0.7B

Zero mode peak at ~Dirac Point

Zero mode peak

Yazyev & Helm (2007),

Popovi’c, Nanda, Satpathy (2012)

DOS

Broken

AB

symmetry

Energ

y

remove

Carbon atom

E.Y. Andrei

Vacancy Properties

Y Liu et al (2015)

Padmanabhan & Nanda (2016)Yazyev & Helm (2007)

Interaction of

ultra-relativistic electron

with Point charge ?

Charge ~ +1|e|

Interaction of

ultra-relativistic electron

with magnetic moment?

~ 1.7B

E.Y. Andrei

Vacancy Charge and Tunable artificial atom

Engineering electronic properties

• Density of states and Landau levels in graphene

• Scanning tunneling microscopy (STM) and spectroscopy (STS)

• Defects:

Atomic collapse and artificial atom

Kondo effect

• Substrate:

Twisted graphene

E.Y. Andrei

Pristine graphene

Katsnelson, Novoselov, Geim (2006)

Klein Tunneling

Pseudospin + chirality a Klein tunneling

a no backscattering

a large mobility

Katsnelson, Novoselov, Geim (2006)

No electrostatic confinement

• No quantum dots

• No switching

• No guiding

Can one use a point charge to control the carriers?

E.Y. Andrei

Electron in 1/r potential : Bohr atom

5 10 15

-4

-2

0

2

4

6

r

E

Ba

RE

e-

+Z-

22

2

mc1

E

a

R

cB

Length scale:

Bohr radius

Energy scale

(Rydberg)

137

1~

2

c

e

ZCoulomb coupling

constant

Fine structure

constant

c

c

r

mc

r,

Compton

wavelength

)2

1()(

2

2

r

rr mcE

r

ZePE

m

pKE

22

;2

rp

Non relativistic classical

ER

r

Dirac vacuum

mc2

1

21 mcER

QM ensures stability of the atom

Z>>1 Must Include relativistic effects

2222

)(2

mcpcm

p

Semiclassical approximation

Z

E.Y. Andrei

Electron in 1/r potential : Bohr atom

5 10 15

-4

-2

0

2

4

6

rE

BaNon relativistic massive particle

Z>>1 Must Include relativistic effects

2

1~,1~ mc

Z

22

2

mcE

a

R

cB

1

Bohr radius

Rydberg

RE

relativistic massive particlee-

+Z-

22

2

1

1

mc

c

)/2/1( 22 rr mcE

)/1/1( 22 rr mcE

eVZ

6.131

22

2mc

1ER

2222

)(2

mcpcm

p

E.Y. Andrei

Dirac 1928: Total energy of Electron near nucleus of charge Z :

Bohr atom: QM+ Relativity a Atomic Collapse

Zc ~1/ =137

1 cc Z

22 1 mcER

e-+ Positron

Z

Atomic Collapse

Spontaneous positron emission

Atomic Collapse

solutionNo1

Pomeranchuk and

Smorodinskii (1945))

c ~ 1.24 a Zc ~170

Regularize problem

finite size nucleus: r0~10-15 m

QM+RELATIVITY :

atom collapses

Zc~170

END OF PERIODIC TABLE

In fact

Periodic

table ends

earlier

E.Y. Andrei

Relativistic Electron in 1/r potential

relativistic massive Bohr atom

)/1/1( 22 rr mcE

5 10 15

0

1

2

r

E/m

c2

Ba

RE

+Z-

22

2

1

1

mcE

a

R

cB

Energy scale

Length scale

c=1 a Zc~137

relativistic massless Bohr atom

)2

1(

r

vE F

137

1~

2

c

e

ZCoulomb coupling

constant

Fine structure

constant

c

c

r

mc

r,

Compton

wavelength

2D, m=0, c=vF

E.Y. Andrei

Ultra-relativistic Electron in 1/r potential

No bound states

Klein tunneling

E

r

2/1E

r

Super-critical

E<0 Resonances

Artificial atom

Shytov, Katsnelson, Levitov 07

Pereira, Nilsson, Castro Neto 07

2/1

Z

e-

Z

holes1S

2S

1

23S

Sub-critical

)2

1(

r

vE F

Scale free

Graphene

Critical charge a Quantum Phase transition

2/1c Zc~1

O. Ovdat et al Nat. Comm. 8, 507 (2017)

+Z

gZ 2F

gv

c

Ultra-relativistic Bohr atom

Dielectric

constant

E.Y. Andrei

Ultra-relativistic Case

Shytov, Katsnelson, Levitov 07

Pereira, Nilsson, Castro Neto 07

Type of carriers is reversed as EF crosses DP

Klein tunneling couples electron-like states at small r to hole-like states at

large r

CE

Zer

2

*

momentum angularmM )2/1(

E.Y. Andrei

0.0 0.5 1.0 1.5

-0.20

-0.15

-0.10

-0.05

0.00

0.05

E-E

D (

eV

)

Schwinger (1950)

Zeldovich (1970)

What is the experimental signature of Atomic collapse

ATOMGRAPHENE + CHARGE

UNBOUND

Atoms

Dirac Point

“Collapse

States”

like artificial atom

UN

BO

UN

D

D. Moldovan, M.R. Masir, F. Peeters,

2S1S

Supercritical chargeAtomic CollapsePositron

production

170~cZ

E.Y. Andrei

0.0 0.5 1.0 1.5

-0.20

-0.15

-0.10

-0.05

0.00

0.05

E-E

D (

eV

)

Schwinger (1950)

Zeldovich (1970)

What is the experimental signature of Atomic collapse

GRAPHENE + CHARGE

UNBOUND

BOUND

COLLAPSE

STATES

Dirac Point

UNBOUND

2S

D. Moldovan, M.R. Masir, F. Peeters,

Wang et al Science 2013

+1S

Supercritical charge

“Collapse

States”

like artificial atom

E.Y. Andrei

Uncharged metastable state

Stabilized by strain or substrate

Can one host a stable charge in graphene?

Y Liu et al (2015)

Padmanabhan & Nanda (2016)

Charge ~ +1|e|

Metastable (+50meV)

50meV

UnchargedCharged

Ground state

E.Y. Andrei

Making Vacnacies in graphene

1nm

Pristine

20nm20nm

Irradiated

STM Topography

Single atom vacancy g triangular structure.

Sputtering with

100eV He+ ions

M. M. Ugeda, et al PRL 104, 096804 (2010).

E.Y. Andrei

Charged Vacancy Spectrum - Artificial Atom

0.0 0.5 1.0 1.5

-0.20

-0.15

-0.10

-0.05

0.00

0.05

E-E

D (

eV

)

1S 2S

gZ

Dirac Point

1S’

Zero mode measures vacancy charge

No zero mode in perfect graphene

D. Moldovan, M.R. Masir, F. Peeters,

Charged vacancy

J. Mao et al, Nature Physics 2016

Quasi bound staes

Artificial atom

Zero mode

Neutral vacancy

2/1c

E.Y. Andrei

Vacancy in Graphene/hBN

-0.2 0.0 0.20

1

2

dI/

dV

(a

.u.)

Vb(V)

0.0 0.5 1.0 1.5

-0.20

-0.15

-0.10

-0.05

0.00

0.05

E-E

D (

eV

)

1S 2S

Artificial atom

gZ

Dirac Point

1S’

J. Mao et al Nature Physics 2016

Negligibly small charge!

Zero mode measures vacancy charge

As prepared vacancy

Zero mode

DP

hBN

E.Y. Andrei

Charging a Vacancy

J. Mao et al, Nature Physics 2016

Pulse on vacancy

e-

+

Before

-0.2 0.0 0.2

0

10

20

30

dI/

dV

Bias (V)

2S

DP

Experiment

1S'1S

spectroscopy

E.Y. Andrei

Charge Buildup

-0.2 0.0 0.2

0

10

20

30

dI/

dV

Bias (V)

2S

DP

0

0.04

0.15

0.20

0.26

0.730.90

0.97

1.04

1.09

Experiment

1S'1S

0.0 0.5 1.0 1.5

-0.20

-0.15

-0.10

-0.05

0.00

0.05

E-E

D (

eV

)

1S 2S

Atomic collapse

1S’

spectroscopy Simulation

E.Y. Andrei

-0.2 0.0 0.2

0

10

20

30d

I/d

V

Bias (V)

2S

DP

0

0.04

0.15

0.20

0.26

0.730.90

0.97

1.04

1.09

Experiment

1S'1S

Charge Buildup

spectroscopy

10nm

Collapse peaks extended ~ 20nm

ZM peak tightly localized ~ 2nm

J. Mao et al, Nature Physics 2016

E.Y. Andrei

Spatial Dependence -Artificial Atom

B

1S1S’

10nm2S

DOS map

1S

1S’

2S

electrons

holes

Captured electron state

surrounded by halo of hole states

mechanism to confine electrons in graphene

uncharged

10nm

DOS

J. Mao et al, Nature Physics 2016

E.Y. Andrei

Asymmetric Screening

-3 -2 -1 0 1 2 3

0.9

1.0

1.1

Carrier density [1011

cm-2]

p doped n doped

NO ACS

Gate dependence

J. Mao et al Nature Physics 2016

Gate controlled switch for trapping and releasing carriers

E.Y. Andrei

1/2 Fgraphene ve

Summary of part III

• QED Fine structure constant measures strength of electromagnetic interaction

• IN QED interactions is weak a periodic table could survive to Z~137

Question we can ask:

Physics at large ?

22

2

42

)(2

1

2Zmc

meZRydberg

137

12

c

eQED

• In graphene fine structure constant is large a strong interactions

• “Atomic collapse” observable already for Z ~1