Transcript metals nonlocal - Baruch Rosenstein

Coulomb interactions make electrodynam
tansport of Dirac semi – metals nonloca
Baruch Rosenstein
Collaborators:
Theory: Hsien-Chong Kao (Nat. Taiwan Normal University)
Meir Lewkowicz (Ariel University, Israel)
Experiment: Wen – Bin Jian, Jian-Jhong Lai (Nat. Chiao Tung
University)
Tours, June, 13, 2016
Outline
1. Free relativistic massless fermions in graphene: finite
conductivity without either carriers or impurities.
2. Problems with calculation of the electron – electron
interaction effects and how they were resolved.
3. Nonlocal electrodynamics in graphene.
4. Experiment.
5. 3D Weyl - semi metals.
Carbon atoms in graphene are arranged in honeycomb 2D
crystal by the covalent bonds between nearest neighbours.
E. Andrei et al, Nature Nano 3,
491(08)
1. Finite conductivity and no screening in noninteracting graphene
Suspended samples are clearly undoped and exhibit the minimal
resistivity at zero temperature.
Andrei et al, Nature
Nano 3, 491(08)
On optical frequencies
conductivity is the same
 
 e2
 0   
 i0  O  
2 h
 
Nair et al, Science 102 10451 (08)
Tight binding model
Hˆ   
n
 k   hk ;hk   ei k



As  Bs
ˆ
 arn aˆrn   c.c.
1,2,3 s ,
Spectrum consists of two bands
with Fermi surface pinched right
between them: not an obvious
band insulator or a metal.
Fermi surface consists of two non - equivalent points of the
Brillouin zone with sufficient little group to support a two
dimensional representation
K
K


KK, K
K
Around K  the Hamiltonian is that of the left and right two
component Weyl spinors:  


H  vg σ  p
σ    x ,  y 

3 a
c
vg 

2
300
The Dirac point in band structure was pointed out very early
by Wallace: electrons in graphene should behave like a 2D
analogue of relativistic massless particles.
The L,R Weyl spinors can
be combined into two 4Dirac spinor described by
L    0 t   1 x   2  y  s
s
that possesses the
chiral
(sublattice) symmetry
 s   5 s ; 5  i 0 1 2 3
DOS of relativistic massless fermions
For the spectrum   vF k
D
E
n  k FD  F
vF
the number of electrons is
Wang et al, APL
101, 183110 (12)
dn E D 1
DOS 
 D
dE
v
At Dirac point the charge of carriers density flips sign: no “free”
carriers at the neutrality (Dirac) point.
Explanation: Schwinger’s electron – hole pair creation by
electric field
The basic picture of the pseudo –
diffusive resistivity in pure
graphene is the creation of the
electron – hole pairs by electric
field. The pairs carry current that
can be further increased by
reorientation of moving particles
i
E
 e2
1 e2
 ij     ij 0 ; 0 

2 h 4
j
Linear response for
current
Heuristic interpretation of the conductivity
The pair can be created with back to back momenta k   / vF
E
So that the current is proportional to electric field, number
of available states
J  k D 1
and to the life time of the virtual pair 1/ 
Therefore

k D 1


D 2
1 forD  2

 forD  3
Dynamical approach to linear response demonstrating the
absence of scale separation
Let’s try to understand qualitatively how massless fermions
react on electric field by just switching a homogeneous electric
field and observing the creation of electron - hole pairs and
the induced charge motion.
Lewkowicz, B.R., PRL102, 106802 (09)
One indeed observes that the current stabilizes at finite value. In
DC this requires the use of tight binding model rather than its
Dirac continuum theory.
J t 
 k2
 t  
 k
E
BZ  4 k
k  Im  hk*hk  /  k ;
k  Im  hk*hk ''  /  k
 e2

 2
k2
 2 k
sin 

2

BZ  k
hk   ei k


t

In addition to the finite term converging to
, there is also a
linear acceleration term that vanishes due to cancellation of
part near the Dirac points and far away. It can be presented as
a full derivative
J div
t

 
Sk  0
E
4 BZ k y
Kao, Lewkowicz, B.R.,
PRB81, 041416 (2010)
Therefore there is no perfect
scale separation when massless
fermions are involved. This is
also the source of the chiral
anomaly and doubling.
K
K
Accidental nature of the “pseudo-Ohmic” linear response
B.R, Lewkowicz, PRL 77, (09) ; B.R., Kao, Lewkowicz, PRB 81, R041416 (10)
2.5
J   0E
2.0
E0
E
2 6
E0
2 7
E0   / ea
pair creation rate
conductivity
2
E
E
2 8
E0
1.5
1.0
0.5
 1010V / m
0.0
0
20
40
60
80
100
120
time in units of t
Stays there till a crossover time tnl 
/ eEvg
to a linearly increasing Schwinger’s regime J  t  
2

2
1/ 2
g
ev
 eE 
 
 
3/ 2
t
At each k x one solves, using the WKB approximation, a
tunneling problem similar to that in the Landau – Zener
transition.
Gavrilov, Gitman, PRD53, 7162 (1986)
eEt /
4
The interband transition
n t  
2 k  0
probability at large times is
 2 
y

ky
1 eEt
2
  vF k x2 
kx exp   eE 
eE
1
1/ 2
3/ 2
 2  vF   eE  t
vF

Schwinger, PR82, 664 (1951)
k y  eEt
Hence
2 kx
J  t   2en  t  t
k y
k y  eEt
Schwinger, PR (1962)
Coulomb repulsion potential remains long range
The vacuum polarization
   0,q   q
0
0

Is too weak to exponentially screen the 3D
Coulomb with 2D Fourier transform
q
4
2
V q 
q
The relation between the polarization and conductivity obeys
the usual charge conservation relation that is sometimes
naively written as.

 , qx ,0    xx , qx ,0 
2
qx
2. Electron – electron interactions effect on
conductivity and screening
H int
e2

2
1
r ,r '   r  r  r '   r ' 
  r      r  rn  cˆnAs †cˆnAs    r  rn   3  cˆnBs †cˆnBs 
n
They break the (pseudo) relativistic invariance and are still 3D,
namely less long range in 2D than that of the 2+1 dimensional
massless QED.
How strong are the interactions?
Coulomb interactions naively are nonperturbatively strong due
to coupling constant of order 1
e2
c
300
g 
 QED 
vF vF
137
Just rescale the field in
2
e
H  vF      
r
2
†
1
†



r

 0  r ' 


r ,r ' 0
r  r'
†
Even if screened this would lead to a strongly coupled electronic
system that for example might exhibit the chiral (exciton)
condensate
This is very similar to the scalar
relativistic interaction that exhibits
chiral phase transition of non –
Wislon universality class
 s s  0
This would make quasiparticle
massive and the phase insulating.
Lattice simulation indeed
demonstrated formation of the
condensate above
  
g
Drut, Lande, PRB 79, 79167425 (09)
Ulyubishev et al, PRL 111, 056801
13).
…
However in the previous discussion, quite complete
understanding of transport was achieved ignoring the
interactions. The interaction effects have been observed, but are
small.
Siegel et al, PRL. 110, 146802 (13)
The first definitive measurement of the
coupling (on the BN substrate) gave the
value  g  . The actual expansion
parameter is  g   , small enough to use
perturbation theory.
Elias et al, Science 337, 1196 (12)
Conflicting perturbation theory results in continuum
limit (Dirac approximation)
The leading correction to conductivity would be of a value, and
can be directly calculated via Kubo formula by calculating
diagrams
x
x
x
x
First attempt in which sharp momentum cutoff was used gave
a disappointingly large correction indicating that interactions
are not weak
 xx 
 e2
2
1  C

h
g
 ... 
25 
C1 
  0.51
12 2
Herbut, Juricic, Vafek, PRL100 , 046403 (08)
However subsequent calculation of the polarization provided a
different value. The charge conservation leads to a general
relation between the charge and current transport functions:
qi ij , q  qi   , q 
For a regular (means local) isotropic time reversal invariant
(no magnetic field) system, one can expand
 ij , q    ij    O  q
 xx    lim
q0

q
2
2

q2 , q    , q 
 , q    0 1  C2 g  ... 
19 
C2 
  0.012
12 2
Mishchenko, PRL98, 216801)07); EPL100,
046403 (08)
0
0
0
0
This value was considered to be favorable phenomenologically.
However in order to force the two quantities, the polarization
and the “direct” conductivities to be the same, the interaction
was modified with the long range part modified to depend on
UV cutoff. It was claimed however that this way the Ward
identities of the charge conservation were violated by the cutoff
procedure
The polarization was recalculated with Ward identities obeyed by
using a variant of dimensional regularization with yet different
answer
22 
C1 
  0.25
12 2
Juricic,Vafek, Herbut, PRB82 , 235402 (10)
The new calculations indicated problems with the dimensional
regularization and reaffirmed the polarization value C2
MacDonald et al, PRB84, 045429 (11); Sodemann, Fogler, PRB86, 045429 (12)
Calculation on the lattice
Note that unlike in relativistic QFT, in graphene (or numerous
other “Dirac” systems in condensed matter the natural
regularization exists: a microscopic theory. To resolve the
ambiguity we have performed the calculation within the tight
binding model (lattice 3D with Coulomb interactions between
orbitals).
1
ˆ
H int   cˆns †cˆnsV  rn  rm  cˆms †cˆms
2 n,m
B.R., Maniv, Lewkowicz, PRL110, 066602 (13);
B.R., H.C. Kao, Lewkowicz, PRB 90, 045137 (2014)
The result for the conductivity is C1 , while the polarization
plus conservation of charge gives C2
How to reconcile this?
The way out is to reconsider the basic assumptions involved in
the derivation of the customary relations
 ij , q    ij    O  q 2 
q2 , q    , q 
Undoped graphene is not a “regular” system due to lack of
separation of scales and an additional nonanalytic term appears
 ij  , q    ij n   
qi q j

   ij  2
q

qi q j
q2
 nl    O  q 2 
qi q j

2






O
q






 T
L
2
q

3. Local vs. nonlocal electrodynamics
When and why the Ohm’s low (locality) is obeyed
If a conducting system has a space scale like the mean free path in
metals with significant disorder, one expects smooth q  0 limit
leading for an isotropic time reversal invariant (no magnetic field)
system to
 ij , q    ij    O  q 2 
For a very clean metal the situation might in principle
become complex since on the free electron level there is no
obvious scale. The strongly screened interactions might be
sometimes neglected (or “renormalized away” in the Fermi
liquid theory), but the electric response generally becomes
nonlocal.
qi q j 
qi q j

 ij , q     ij  2   T    2  L    O  q 2 
q 
q

Knows it all…
However it is possible to show that in any
free lattice model conductivity is local
since the difference can be again written
Hˆ   aˆkA† hkAB aˆkBs  c.c.
k
as a full derivative and thus vanishes
 nl     L   T
e2



i D q, qi
B.R., H.C. Kao, in preparation (2016)
 2


1
G  q,   
  G  q,     2G  q,   
qi


Long range interactions are required for a conductor to ne
nonlocal
Short range interactions metals or semi-metals do not change
the situation, the screening length providing the necessary
length scale.
However in Weyl semi – metals (at neutrality point), despite the
pseudo-Ohmic conductivity, Coulomb interaction remains long
range. This creates a unique situation of the interaction
correction creating the nonlocal electric response characterized
in the rotation, time reversal invariant case by two scalar
constants
qi q j
 ij , q    ij T    2  nl    O  q 2 
q
 nl   L   T  0
For actual computation the trace of the conductivity tensor
generally is simpler
qi qi
 xx  , q    yy  , q    ii T    2  nl    2 T     nl  
q
with an exact relation to polarization being

1
  , q   2 qi ij  , q  qi
2
q
q
qi q j

1 
 2 qi  ij n    2  a    qi   T     nl  
q
q


The naïve locality assumption

1
2


,
q




O
q






ii
2
q
2
is thus violated
u and
h,
From the perturbative values

q
2
  , q    0 1  C2 g  ; ii    2 0 1  C1 g 
one obtains
 T     0 1   g  C2  2C1     0 1   g 
 nl    2 0 g  C2  C1    g 0
ij  , k   T  ij  nl
1
T 
T
ki k j
k2
 nl
nl 
 T  T   nl 

0
u
, h and
Extention to other Dirac semi-metals and universality
For a Weyl semi-metal the contributions to nonlocality due to
interactions is sum of all the nodes: graphene – like, double –
graphene like… The integrals converge well in continuum after
the appropriate surface term is identified.
For the double layer graphene node
one obtains:
  kn
k
1 e2
0 
;C1   g ;C2  0.27 g
8
1
 nl    0 g
3
No known “topological explanation for these numbers
How this additional term affects physics?
Potentials for the nonlocal electrodynamics
Generally a vector field can be decomposed using two potentials
u, h
fl
sol
Ji  r   Ji  Ji  iu  r    ij  j h  r 
Ei  ij J j  T J i  nl  i u
One deduces the voltage between two points via sources
2 u
r
r
r, z
| z0 s
r
. #
zJz
i Ji 
u
r 1 
log|r r|s
r
, #
2 r flake
V  r1 , r2   E  ds  Vloc  nl u  r2   u  r1   .
4. Experimental observation
Anomalous dependence on geometry of electrical resistance
in a rectangular flake
The correction to the electrical resistance becomes dependent on
geometry of the source, drain and points between which the
voltage is measured
V21
Source
Drain
r2
r1
Without nonlocal electrodynamics due to
interactionsinteraction
1 L
R
0 W
The correction to the electrical resistance becomes site dependent
y
C
X=0
X=0
D
y=W/2
A
B
X=L
x
y=- W/2
R BA
L

 0W
2

2

 
W
W
2
L




ln    1 
1   g  arctan   


2
L

L
W











R DC
L

 0W

1
 L  2  
W
W


ln    1 
1   g  arctan   
 L  4L  W 
 


An anomalous Hall effect
y
C
X=0
X=0
D
y=W/2
A
B
X=L
x
y=- W/2
R CA

  W 2


   1  
1

L
L
W 
W  W




 g   arctan    2 arctan    
ln
  W 2

 0W

L
2
L
2

L









4






 L 


The asymmetric flake set up
VTS
Vd
VBS
Vg
graphene on SiO2
substrate
IDS
Mobility: 4000 ~10000
cm2/V-s
Constant Current
Source &
300Meter
K
Voltage
T
S
Current
(mA)
Volage
(mV)
Voltage
(mV)
Total
(mV)
(VDS=9.04)
10
VTS=4.87
VDT=4.09
8.96
10
VBS=4.53
VDB=4.47
9
D
B
T
S
D
B
Nonlocality voltage
near the Dirac Point
s13
32
60
s12
50
e /h
40
2
2
e /h
24
16
20
10
8
-4
-3
-2
1.2
-1 0 1 2
12
-2
n (10 cm )
S13
3
0
-5
4
-4
1.2
T=300K
-3
-2 -1 12 0 -2 1
n (10 cm )
2
3
4
S12
T=80K
VTB (mV)
0.9
VTB (mV)
30
0.6
0.3
0.9
0.6
0.3
0.0
-4
-3
-2
-1 12
0 1-2 2
n (10 cm )
3
4
0.0
-5
-4
-3
-2 -1 12 0 -2 1
n (10 cm )
2
3
4
Temperature Dependence
1.4
1.2
VTB (mV)
1.0
0.8
0.6
0.4
300
260
220
220
180
180
160
140
120
100
90
80
0.2
0.0
lowT
-60
-40
-20
0
20
40
60
Vg (V)
temperature destroys the nonlocality
70
50
60
50
2
2
e /h
30
20
40
30
20
10
10
-5
-4
-3
-2 -1 0
12
-2
n (10 cm )
1
2
0
-5
3
-4
2.0
S18
0.9
-3
-2 -1 0 1
12
-2
n (10 cm )
2
3
4
-2 -1 0 1
12
-2
n (10 cm )
2
3
4
S17
0.6
 V (mV)
 V (mV)
1.6
0.3
1.2
0.8
0.4
0.0
-5
-4
-3
-2 -1 0
12
-2
n (10 cm )
1
2
0.0
-5
3
weak nonlocality
S18
0.5
-4
-3
S17
1.0
strong nonlocality
0.5
0.4
 V (mV)
 V (mV)
Disorder also
destroys the
nonlocality
e /h
40
wide carrier range
0.3
0.0
-0.5
narrow carrier range
-1.0
0.2
-5
-4
-3
-2 -1 0
12
-2
n (10 cm )
1
2
3
-5
-4
-3
-2 -1 0 1
12
-2
n (10 cm )
2
3
4
Anomalous reflection and transmission
The additional term changes in a different way the two
polarizations of the electromagnetic wave passing (bouncing off)
a graphene layer
x
y
z
 
The p  Ex  and the s E y polarization components have the
following reflection coefficients
 T
 cos 
rp 
 T   nl  ;rs  
c
c cos    T
5. 3D Dirac semi-metals
Topological transition from TI to nontopological insulator was
shown to result in 3D Dirac semi – metal:
BiO2
Na3Bi
Liu ZK et al, Science (2014)
Dirac or Weyl points in 3D appeared
occasionally in band structure calculations
away from Fermi surface, but not always.
Properties of a (poor) metal with strong
spin – orbit, Bi, led to its (approximately)
Dirac Hamiltonian with pseudospin
replaced by spin with all three Pauli
matrices now presentpoint theory
For free quasi-particles in addition to the real part
NW e 2
 0   

12 hv
2
v
 0 '     '0   log


B.R., Lewkowicz, PRB 88, 045108 (13)
In many cases of interest
the imaginary part
dominates over small
dissipation.
Electrodynamics of such a material becomes rather peculiar
The interaction correction
The correction as in graphene renormalizes coefficients of
both the real and the imaginary parts


 2

2 
 L   0 1    log  C   O  

 3



5
C
3
More importantly the difference between the longitudinal
and the transverse conductivities is.
2

 nl   0
log
3

B.R., Lewkowicz, PRB 88, 045108 (13)
The charging phenomenon
Fluxon induces charges
proportional to  nl
Incident light splits into the transversal and the longitudinal
ones, both very weakly attenuated.
z
L
T
Moreover at certain
angle and frequency the
monochromatic wave
appears that is both
transverse and
longitudinal.
y
x
For certain angle the reflection disappears: total absorption
phenomenon
Summary
1. Dirac semi-metal like graphene at Dirac point is a neutral
plasma that has finite conductivity, but does not makes the
interactions between quasiparticles short range.
2. There is no complete scale separation of the energy scales
also in the interacting case and some quantities like the
conductivity tensor are nonanalytic. Low energy continuum
Weyl model requires regularization that is sufficiently precise
to avoid effects of chiral symmetry violation by the cutoff.
3. Transport and electromagnetic properties of WSM are
unusual including interaction effects leading to nonlocal
electrodynamics resulting in several optical and electric
peculiarities.
A radial and a rotational current in a graphene layer with
Corbino geometry exhibits different electrical resistance:
S
D
1


/2
R
g
out
R

ln
rad

R
0
in
B(t)
Rcirc 
1
0