7/31/2019 FQHE Graphene Alexandru Bratu Minor Thesis http://slidepdf.com/reader/full/fqhe-graphene-alexandru-bratu-minor-thesis 1/18 th Fractional Quantum Hall Effect on graphene Minor - thesis by 4 year Theoretical Physics undergraduate Alexandru Bratu
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7/31/2019 FQHE Graphene Alexandru Bratu Minor Thesis
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 :
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7/31/2019 FQHE Graphene Alexandru Bratu Minor Thesis
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
= −
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7/31/2019 FQHE Graphene Alexandru Bratu Minor Thesis
( )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 =
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7/31/2019 FQHE Graphene Alexandru Bratu Minor Thesis
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 < >
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7/31/2019 FQHE Graphene Alexandru Bratu Minor Thesis
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
− −
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7/31/2019 FQHE Graphene Alexandru Bratu Minor Thesis
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
= −
= =
.
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7/31/2019 FQHE Graphene Alexandru Bratu Minor Thesis
( ) ( )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− − =
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7/31/2019 FQHE Graphene Alexandru Bratu Minor Thesis