FQHE Graphene Alexandru Bratu Minor Thesis

7/31/2019 FQHE Graphene Alexandru Bratu Minor Thesis

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th

Fractional Quantum Hall Effect on graphene

Minor - thesis

by

4 year Theoretical Physics undergraduate

Alexandru Bratu



( )Graphene monolayer of graphite was chosen due to its pseudospin degenaracy of Fermi points

that do not couple directly to the B field, i.e.magnetic field. The effect will be studied at the 0

L

n − =

( )andau level , the second Landau level, the behaviour of the variational ground state wavefun-

ction for GaAs due to its similarity in the zero Zeeman energy limit that comes close to that of graphene

a

LL

nd the analysis in terms of gauge theory.

FQHE at the n = 0 Landau level of graphene Graphene has quite an unusual eletronic dispersion around where the electromagnetic field 0.

This can be observed in the Landau level compositions and the Hall conductance staircase. The wav

E =

e func-

tions of the electrons at the 0 level are similar to the ones of GaAs. Thus at a high magnetic field the

0 Landau level of graphene identifies itself to the FQHE behaviour of electrons in G

n

n

== aAs with zero

Zeeman energy.

( )

Due to this corelation we have that physical behaviour of 0 LL of graphene matches that

of GaAs. At this level we have that each electron captures an even number 2 of quantized vortices to

n

p

=

( )become a composite fermion. At the Berry phases we have that the vortices cancel out

effectively a small part of the B field, such that the composite fermions are being governed by a do

explained below

− wn-

grade of the magnetic field of the form

*

0= 2 ,B B pρφ−

0

particle density in the 0 LL

whereflux quantum of constant value

n

hc

e

ρ

φ

→ = →

Berry phase ( )If a quasiparticle encircles a closed contour in the momentum space" ", a phase

shift known as Berry's phase is gained by the quasiparticle's wavefunction. The phase can be observed

due to

p momentum ∫

( )rotation of pseudospin quasicharge confined to quasiferromagnetic region , when a quasiparticle

repetitively moves between different carbon sublattices.

⇓

1



*

Then we observe that the 0 LL of electrons splits away into levels that behave in the same

manner as the composite fermions, which have a filling factor , that measures the efficiency of a

n

v

=

bsorp-

tion, in terms of the electron filling, where

*

*.

2 1

v v

pv =

±

( ) *The IQHE of the fermions for givesintegrated quantum hall effect v n = : .

2 1

n v

pn =

±

( )The origin of the gap, i.e. the amount of energy need to boost the fermion a higher

CF Landau level takes different takes for n of even and odd values.composite fermion −

n even − ( )The ground state is a pseudospin quantum state with zero spin with CF Landau level

2for each component of pseudospin occupied.

n singlet −

n odd − No quantum hall effect would be present if the CFs did not interact, however a residual interac-

tion between them would open a gap.

The collisions occuring would be weak in nature and they would be expressed like this , ,p

p q q

∈ ⇒»

( ) ( ) ( ) ( )with gcd , gcd , 1 2 1 , 2 , , , , ,

a ba r b r a n odd b m even a b r n m

r r ⇒ < = = = + = ∈ »

At the zero Zeeman energy level limit for GaAs FQHE the energies of excitation apply for

graphene FQHE at 0 level. In terms of the far-separated charged quasiparticle/quasihole pair

1at ob

3

n

v

=

= ( ) ( )2

2

2tained from inter CF interactions the gap is 0.07 and for the unpolarized

5

respectively gap of 0.04electron charge

, where fractional coefficient

magnetic length

B

B

B

e v

l e

l e

c l

eB

− =

→ → → =

.

Є

Є

Є

Assured that the electrons remain in the 0 LL, the Dirac nature of the zero-field dispersion,

gives rise to some consequences withing the FQHE. The CF cyclotron energy tends linearly with

res

n =

−*pect to , as expected from the parabolic dispersion of the CFs.B

2



For GaAs, the FQHE states finish as in a CF , i.e. all possible energy2 1

1levels of electrons, at , where the effective magnetic field vanishes. For Zero Zeeman energy

2

n v n Fermi sea

pn

v p

= → ∞±

= variatio-

nal calculations give favor to the singlet Fermi sea, so that the graphene may have a pseudospin singlet

1CF Fermi sea at . For the CF Fermi sea of GaAs it has been modeled as a simple Fer

2

v

p

−

= mi sea with

parabolic dispersion, which allows for deducing an effective mass for CFs. The same physical concept apply

also for the graphene, although electrons in graphene produce a Dirac Sea at 0.B = ⇓

"Fermi Sea"

FQHE at the second Landau level of graphene

Theoretically by nature FQHE depends much on the Haldane pseudopotentials, which are usually

denoted by the , where is the relative angular momentum in Hilbert space, is the

cen

M

mM V mM v m M

mM mM =

ter of mass angular momentum and is the central potential. In GaAs , the FQHE is mainly restric-

ted to lowest LL, very low fractional observations in 1 and close to none in the higher levels. Diff

V

n = eren-

ces are expected at the higher Landau levels for graphene and GaAs.

3



1 1 2 21 1 2 2 , , , 1 1 2 2

We must now evaluate the matrix elements within the graphene Landau level.Denote the state

, ; , ;...; , for the product state ... and , ; , ;...; , forN N

th

N N n m n m n m N N

n

n m n m n m n m n m n m η η η⊗ ⊗( ) ( )1 1, ,

. ThenN N n m n m Ψ ⊗ Ψ

1 2 3 4 1 2 3 4

1 2 3 4

1 2 3 4

4 , ; , , ; , , ; , , ; ,

1, ; , 1, ; ,

+ , ; 1, , ; 1,

n m n m V n m n m n m n m V n m n m

n m n m V n m n m

n m n m V n m n m

=

+ − −

− −

1 2 3 41, ; 1, 1, ; 1,n m n m V n m n m + − − − −

1 2 3 4,Conservation of angular momentum implies the matrixelements all proportional to .m m m m δ + + The graphene then tends to behave like the Landau levels of GaAs at the lowest levels,

the pseudopotentials for the effective interaction then becomes

th n

( ) ( ) ( ) ( )1 , 11

2 ,4

n graphene n n n n

m m m m V V V V

− −

= + +

where, ( )

( )( )

22 2 2

, 1 2

12

2,

2 22

n n k

m n n m

d k k k V L L e L k

k

π

π

− −−

= ∫

( )and is the effective pseudopotetial for the LL in GaAs and L denoted the eigenstate of

the angular momentum. By standard methods the pseudo-behaviour in Coulomb interaction behaves

n th th

m j V n j

( ) ( )the

following way in terms of the potential and wavefunctionV Ψ :

4



0 1Pseudopotentials for n and n Landau levels in graphene and GaAs = =

⇓

To understand how the interactions work in the reaction we have to numerically diagonalize finite

systems where electrons are moving along the surface of a sphere which at the center containN

( )s a mag-

tic "monopole" Maxwell's laws they do not exist but proven to exist in gauge invariance of 4-vectors .

2The monopole has a strength of size that gives out a flux through the surface o

by

Qhc Q

e Φ = f the sphere.

( )The pseudopotentials are used to give exact value for the very large systems and thus

provide good approximation.

n graphene

m V

For the pseudo-polarized sector we seen that for 1 graphene LL behaves very similarly to the

0 LL. Hence FQHE for the bigger LLs of graphene are more robust than in GaAs.

n

n

=

=

( )We have to compare the ground states for systems of graphene pseudospin degrees of freedom

, with actual spin being frozen, to a GaAs system of zero Zeeman splitting.

dof

5



Orbital angular momentum and spin/pseudospin of the ground states of finite systems on a sphere

1 2 2 3at , , and in the 0 and 1 Landau levels of graphene and GaAs. D is the dimension

3 3 5 5of

n n L S

n n ν = = =

0the Hilbert space in the 0 sector; is the dimension of the 0 sector. The last column

gives the overlaps between the 0 and 1 graphene ground states.z z

L S D L

n n

= = =

= =

⇓

6



The table above shows we have that for the ground state quantum numbers, the orbital angular

momentum , and spin or pesudospin , for the 0 and 1 LLs at several fractions. We observe

that

L S n n = =

for 0 identical results are obtained for GaAs and graphene. Very close overlaps mean that ,

at the 1 LL graphene has strong resemblence to 0 graphene LL.

n

n n

== =

1 2 2 3The state of is fully polarized, however the other are pseudospin singlet. At

3 5 3 5the spin the ground state is different from that of the lowest LL at 8; due to the puzzling nature

an

&

N =d existence of FQHE .

PSEUDOSKYRMIONS

In the quantum wells of GaAs, we have that the excitations of the 1 state for exactly zero

Zeeman splitting are not just basic particle hole excitations but spin textures called

qua

skyrmions

ν =

−

( ) 1siparticles corresponding to topological twists or kinks in a spin space , where of the spins are

2reversed. However the size of the skyrmions tend to decrease as the Zeeman energy increases. Tech-

nically by experiment they have 3 5 flipped spins. No skyrmions occur at 3,5,... thus the compo-

1site fermion skyrmions are thought to be relevant near with very small Zeeman energies.

3

ν

ν

− =

=

( )

In constrast, for the state in which one of the two degenerate levels of the 0 graphene manifold

is fully occupied which produces zero Hall conductance . The Hall conductance is given by

n =

2

,e

h σ ν =

where,

( )

'

2 1 2 3 5, , , , , ...

7 3 5 5 2

e elementary charge

h Planck s constant

fraction usually of the form etc ν

→ − → →

At zero Hall conductance the excitations ought to be pseudoskyrmions.By diagonalization we seethat quasiparticles can also be in the 1,2 Landau levels of graphene, where a surplus of one pan = rticle

or hole of the completely polarized state produces a pseudospin singlet state.However there is no such

occurence for 3, as in this case the fully polarized pseudospin on the2

N n S qu

≥ − = (hole-quasi-

particle) side and has a single pseudospin reversed 1 on the quasiparticle side.2

asihole

N S

= −

7



The diagram depicts the dependence of the gap for creating a pair of pseudoskyrmion

and antipseudoskyrmion, computed by exact diagonalization.

N −

⇓

( ) ( ) ( ) ( ) ( )( ) ( )

0 1

1 1

2

1

Extrapolation to the thermodynamic limit explained below yields 0.606 15 , 0.126 7

and 0.18 1 , where are average level spacing, with

system's linear extent

entropy dwhere,

,d sL

L

s

e −

∆ = ∆ =

∆ = ∆ ∆

→

→

∼

( )

ensity

dimension of the space, i.e thinking in terms of space

To acount for this we must explore the behaviour of the entropy .

d d

→ −

s

Thermodynamic Limit

Thermodynamics holds only for macroscopic systems. We have seen also that the ideal gas law

only becomes exact in the limit .Since we are dealing with very large systems it is convenient toN → ∞

take the limit .We must do this in a sensible way however. If we take the keeping the

volume fixed we end up with an infinite density.Therefore we must also let in such a way that

where

N N

V

N

V ρ

→ ∞ → ∞

→ ∞

→ is the density or equivalently where is the specific volume.This is called the

.

V v v

N thermodynamic limit

ρ →

8



( )Let , be the entropy per particle considered as a function of and in the thermodynamic

limit, i.e.

s v e v ε

( ) ( )1, lim , ,

N n s v S N N

N ε ε

Λ→∞=

( )0

0

where is a sequence of boxes with volume . We can define also the internal energy per particle inthe thermodynamic limit as a function of and . We let , be the value of for which

N vN u s v u s v

s s

ε

ε

Λ

= ,( ) ( )( )0 0, that is, , , . We can also define the absolute temperature and pressure of

the system by

v s u s v v s T P =

,u u

T P s v

∂ ∂ = = − ∂ ∂ .

v s

The free energy per particle in the thermodynamic limit is then

( ) ( )

( ) ( )

, ,

, ,

a s v u s v Ts

u u s v s s v

s

= − ∂ = − ∂

v

For the ideal gas

( ) 2, ln ln ln ln

2 2 2 22

d d h d d s v k v d

m ε ε

π

+ = + + − −

Therefore ( )2

ln ln , ln ln .2 2 2 22

d d h d d s k v u s v d

m π

+ = + + − −

Thus

( ) ( )( )2 ,

12 , 2 ,

u s v kd u kdT T

s kd u s v u s v

∂ = = ⇒ = ∂ v

and

( ) ( )( )2 ,1

.2 , 2 ,

u s v d u Pd P

v v vd u s v u s v

∂ = − = ⇒ = ∂ s

Therefore .

kT P

v =

9



It is sometimes convenient to use , the density, rather than . Letv ρ

( ) ( )1, lim ,

l l l l l

s S V V V

ε ρ ε ρΛ→∞

=

where is a sequence of boxes with volume where lim . is the entropy density and is the

energy density.Note that

l l l l V V s ε

→∞Λ = ∞

( ) 1, ,s s

εε ρ ρ

ρ ρ

= .

( ) ( ) ( )( )0 0 0Let , be the value of for which , , that is, , , . We also have the relationu s s s s u s s ρ ε ε ρ ρ ρ= =

( ) ( )1 1, , , ,

s u s u s or u s v vu

v v ρ ρ

ρ

= = .

1 2Let and denote partial differentiation with respect to the first and second variable respec-

tively. Then

D D

( )

( )

( )

1

1

1, ,

1 1,

,

= ,

s T u s v vu

s s v v

s vD u

v v v

D u s

u s s

ρ

ρ

∂ ∂ = = ∂ ∂ = ⋅

=

∂∂

v v

( )

( ) ( ) ( )

( )

1 22 2

1 2

1, ,

1 1 1 1, , ,

1, , ,

,

s P u s v vu

v v v v

s s s s u vD u D u

v v v v v v v v

s u s D u s D u s

v v

u s s

ρ ρ ρ

ρ

∂ ∂ = − = ∂ ∂ = − + ⋅ + ⋅

= − + ⋅ + ⋅

∂= − +

∂

s s

( ) ( ), ,u s u s

s

ρ ρ ρ

ρ

∂+

∂

.

Thus ,

u u u T P u s

s s ρ

ρ

∂ ∂ ∂ = = − + + ∂ ∂ ∂

s ρ ρ

and the free energy density in the thermodynamic limit is given by ( ) ( ) ( ) ( ), , , ,

u a s u s Ts u s s s

s ρ ρ ρ ρ

∂ = − = − ∂

ρ

10



For the Ideal Gas we finally have that :

( ) 2 2, ln ln ln ln

2 2 2 2 22

d d d h d d s k d

m ε ρ ρ ρ ε

π

+ + = − + − − .

Therefore the entropy is :

( )2 2ln ln , ln ln

2 2 2 2 22

d d d h d d s k u s d

m ρ ρ ρ

π

+ + = − + − − .

2

Now the gap present at the 0 Landau level si consistent with , which is half energy8

needed to produce a simple particle hole pair excitation. In the pseudospin texture the 0 LL can

b

B

e n

l

n

π=

− =Є

( )e observe directly by scanning tunneling microscopy explained below , since the pseudospin of the

electron determines on which sublattice it stays. The encapsulated paragraphs below will in turn alsoexpl ( )ain the flux probability transmission through the barriertunnel .

Scanning Tunneling Microscopy In "tunneling" through a barrier whose height exceeds its total energy, a material particle is beha-

ving purely like a wave.Thus penetration of a clasically excluded region of limited width by a particle

be observed, in the sense that the particle can be observed to be a particle,of total energy less than the

potential energy in the excluded region, both before and after it penetrates t

can

he region. From the barrier potential we know that acceptable solutions to the time-independent Schrodinger

equation should exist for values of the total energy 0. The equation breaks up into 3 sall E ≥

( ) ( )eparate

equations for the 3 regions 0 , 0 , and

. In the regions to the left and to the right of the barrier the equations are those for a

x left of the barrier x a within the barrier x a(right of

barrier)

< < < <:

free particle

of total energy . Their general solutions areE : ( )

( )

, 0

,

I I

I I

ik x ik x

ik x ik x

x Ae Be x

x Ce De x a

ψ

ψ

−

−

= + <

= + >

where ( )0

2

I

m V E k

−=

0 0

In the region within the barrier, the form of the equation, and of its general solution, depends on

whether or .E V E V < >

11



( )0 0

0

Both of these cases have been treated in the previous sections.

In the first case, , the general solution is

E V & E V

E V

< ><

( ) , 0II II ik x ik x x Fe Ge x a ψ

−= + < < where,

( )02

II

m V E k

−=

Since we are considering the case of a particle incident on the barrier from the left, in the region to

the right of the barrier there can be only a transmitted wave as there is nothing in that region to produce a

reflection. Thus we can set

0D = More or less we can see that the probability density oscillates but has minimum values somewhat

greater than zero, as for 0. In the region 0 the wave function has components of both typex x a < < < s,

but it is principally a standing wave of exponentially decreasing amplitude, and this behaviour can be seen

in the evolution of the probability density in that region. The most interesting result of the calculation is the ratio T, of the probability flux transmitted

through the barrier into the region , to the probability flux incident upon the barrier.Thx a > is

transmission coefficient is foudn to be

( )

1 1

2 2

1

1

0 0 0 0

* sinh1 1 ,

*16 1 4 1

II II k a k a

II e e vC C k a

T v A A E E E E

V V V V

− −

−

− = = + = + − −

where,

0

20

21

II

mV E k a

V

= −

If the exponents are very large, this formula reduces to

2

0 0

16 1 II k a E E T e

V V

− −

12



The diagrams above show the gaps as functions dependendt on , number of particles for several

factors with graphene Landau Level. The composite fermion pseudoskyrmions are the particlesth

N

v n −

( )1

1with lowest energy excitations at . Their energy is about 0.017 , while for a pseudospin reversed

3particle hole pair of CFs. THe latter gap is greater than the one available in LLL lowest Landau

eV ν − =

− ( )( ) ( )1 0

level

@ 0 because in terms of pseudopotentials @ 1 LL @ 0 LL . The gap energies at

2 2 2involve pseudospin reversal for composite fermions. For the gaps have that5 3 5

n V n V n

& ν ν ν

= = − > = −

= = = ( ) ( )

( ) ( )

0

25

1

25

0.051 1

0.062 1

&

∆ =∆ =

,

which were determined from the trial wave funciotns of the composite fermion theory. Thus we may conclude that the LLL FQHE of graphene in the large Zeeman energy limit is equi-

valent to LLL FQHE in GaAs in zero Zeeman energy limit, ending in a pseudo-spin singlet "Fermi sea"

at half filling. The effective interaction as discusse above showed that compostie fermion formation was in

the 1 LL of graphene than in 1 LL of GaAs. The fractional quantum hall effect is predicten n = = d at

2due to the reverse flux attachment For GaAs we have that the skyrmion quasiparticles occur at

31, 3 in the 0, 1, 2 Landau levels.n

ν

ν

= −

= =

.

13



Behaviour of the Variational Ground State Wavefunction

& Analysis in terms of Gauge Theory

The wavefunction for the carbon compound must satisfy 3 constraints which, in conjuction,

assumed of Jastrow form, completely removes all variational freedom. The constraints are

−

:

) ( )The many-body wavefunction must be comprised solely of single-body wavefunctions lying in LLL,

if is analytic, i.e without loss of generality the function is assumed to be continuos and satisfy

Cau

f z

1

( )( ) { }

chy-Riemann equations around a closed region G differentiable region , i.e. :

/ 0

f

f z

∞ ∞

∞

∞ − → ⇒

∈ .

) ( ) ( ) ( )The wave function must be totally anti symmetric, i.e. f z f z f z − = − −2 .

) ( )

1

Wave function is an eigenstate of total angular momentum,i.e. meaning that must be a

polynomial in the particle positions ,..., of degree , where is the total angular momentum.

N

j k j k

N

f z z

z z M M <

−∏3

( )Only function satisfying all 3 conditions is , with odd, where the function is obtained

from symmetric gauge theory.

m f z z m =

( )

For homogeneous magnetic field we have that 0 applied in the direction, and only the

correspoding spin in the + direction is considered 1 . Consider the vector potential is

expressedz

B z

z σ σ

> −

− = = A

in the symmetric gauge as

( )

1 1, ,

2 2x y

A A By Bx = = −

A .

Then becomesπ−

( ) 2 .2 2x y

eB eB i iy x z π−

= −∂ − ∂ + − − = − ∂ −z

Here we again used and . For the vector potential, the scalar potential becomesz x iy z x iy φ= + = −

( )2 21 1.

4 4

B x y Bzz φ = − + = −

( ) ( ) ( )1 1For the wave function , , , we make the ansatz index 1 means that a 1 state is usedx y z z e −Ψ = Ψ − :

( ) ( )1

41

, ,z z g z z e −

Ψ =eBzz

Then we obtain1

41

2 .z

e g π−

−Ψ = − ∂eBzz

14



( ) ( )1Because the lowest energy state fulfills 0, we obtain , , and thereforeg z z g z π

−Ψ = =

( ) ( )1

41

,z z g z e −

Ψ =eBzz

( ) ( )is analytic in , and writingg z z up to mormalization

( ) , where 0,1,2,...m

m g z z m = =

the number of allowed , . . the degree of degeneracy G, can be calculated as followsm i e : ( ) ( ) ( )

( ) ( ) ( )1

2

1

For a round system with radius , , 0 must be zero for z .

We consider , .

m

m

m

m

R z z g z R

z z z ρ

Ψ = = >

= Ψ

( )21

2 22, .

m

m z z z e f z ρ

− = = eB z

2 2 222

As a function of , is maximal at , and because this must be smaller than ,m

z f z z ReB

=

2 220 or 0

2

m eB R m R

eB ≤ ≤ ≤ ≤

must hold. Therefore, is given byG 2 .

2 2

eB eB G R S

π= =

( )

2Here, is the surface of the system. The times degenerate lowest energy state is the lowest

0 Landau level.

S r G

n

π −

=

In order to clarify the meaning of , we consider the component of the angular momentum operatorm z − : ( ) .z y x y x z z L xp yp i x y z z = − = − ∂ − ∂ = ∂ − ∂

Calculating the commutator of with , we obtainz

Lπ−

:

, 2 , 22 2z z z z z

eB eB L z z z z π π−

= − ∂ − ∂ − ∂ = ∂ + = −

and in the same manner

, .z

Lπ π− − =

15



Therefore, we obtain

( ), , , , 0z z z z

H L L L Lπ π π π π π π π π π− − − − − − − − − −

= = + = + − =

( ) ( )1

and we conclude that and can be diagonalized simultaneously. Indeed, it can be confirmed that

when acting with on , , the result is

z

m

z

L H

L z z Ψ

( ) ( )1 1

m m

z L m Ψ = Ψ . Therefore, is the component of the angular momentumm z − .

( ) ( )satisfies 3 constraints satisfies the 3 constraintsg z f z ⇒ ⇒

( )

( )

Existence of quasiparticels like the skyrmions and pseudoskyrmions

Validity of the analytic , anti-symmetric and Jastrow form function2

Prediction of FQHE at @ 03

f z

n i.ν −

= =

1.

2.

3.

Conclusion final remarks

( )How the zero Zeeman energy limit FQHE of GaAs approximates the FQHE of graphene at LLL.

e. LLL - lowest Landau levels

4.

⇓Structure - of -Graphene

16



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http://nano.tu-dresden.de/pubs/slides_others/2006_06_07_leonid_litvin

3

4

)

)

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8

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er

May 8, 2007 Mohammad Reza Ramezanali

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