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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