Download ν =4/7 - Osaka University

Spin Polarization of
Fractional Quantum Hall States
ICTF16@Dubrovnik
Oct. 14, 2014
OR 77
Center for Advanced High Magnetic Field Science, Graduate School of Science, Osaka University, Japan
Shosuke Sasaki
The FQHE is the famous phenomena. However there are some questions.
One of them is the spin-polarization. We examine it in this talk.
z
Quantum Hall Device
Source
Magnetic Field
y
Drain
x
Current I
width d
Poential Voltage Vpotential
length of 2DES
Hall Voltage VH
1. Vpotential is nearly equal to zero at FQHE.
 No
 electron scattering.
 VH I where VH  I
2. Hall resistance
QHE creates no heat.
3. FQHE are purely eigen-value problem of electrons.
4. The value of VH is extremely larger than Vpotential.
Therefore the gradient of VH cannot be ignored.
Hamiltonian and its eigen-states
in single electron system
Potential
W(z)
-eVH
Fig.6
Gate
Potential
-eV1
W(z)
z
Potential for z-direction (Potential width is very narrow)
Potential
Probe
U(y)
Magnetic field
-eV2
y=0
Hall
Probe
y
Fig.7 Potential U(y)
y=d
Hamiltonian of single electron
2
H 0  p  eA  2m  U y   W z A   yB, 0, 0, rot A  (0,0, B)
H 0  E
This narrow potential realizes the appearance of only the ground state for z-direction as follows:
 x, y, z  
1
exp ikx  y  z  where the variables separated.

We obtain the eigen equation for the y-direction
 k  eBy 2  2  2
 1
1










U
y
exp
ikx

y

E


exp ikx  y 


2
2m
2m y


 
k  eBy 2  (eB) 2  y  k /(eB)2
 (eB) 2  y  c 
2
The eigen state is Landau wave function as follows: 
where the center position c is proportional to
ikx   y c 2
e e
 z 
for level L=0
the momentum
k 2
c

 integer
eB eB
Wave function in Many electron system
Landau wave function of electron for L=0
Gate
ikx   y c 2
 e e
Potential
Probe
Magnetic field
y
Hall
Probe
b
 z 
k
2
c

 integer
eB
eB
The position c is proportional to the momentum.
 Attention please.
x
Hall
Probe
Potential
Probe
Hall voltage is extremely larger than potential
voltage for IQHE and FQHE.
Accordingly there is no symmetry between x
and y directions.
electron orbital
When the momentum decreases,
increases,
This
is the
manymoves
electron
state
the center
position
to the
right.
left.
Total Hamiltonian of many-electron system
References: S. Sasaki, ISRN Condensed Matter Physics Volume 2014 (2014), Article ID 468130, 16 pages.
S. Sasaki, Advances in Condensed Matter PhysicsVolume 2012 (2012), Article ID 281371, 13 pages.
N


H T   H 0 xi , y i , z i  H C
We separate HT into H T  H D  H I
where HD is the diagonal term and HI is non-diagonal part.
i1
HD 
 k ,, k  W k ,, k  k ,, k 
where k1, 
, kN  
,k  E k C k , ,k
1
N
1
N
1
N
k1 ,, k N

W k1 ,
N
N
  0 i  1
N

i1
1

N! 
where C(k1•••kN) expresses the diagonal part of the Coulomb
interaction Hc which is called “classical Coulomb energy”.


 k1 xN , yN , zN
x1, y1, z1
 kN xN , yN , zN
 k1 x1, y1, z1
kN
filled,empty,filled




y
At ν=2/3 , the most uniform configuration
is created by repeating (filled,empty,filled).
This configuration gives the minimum
x
value for W.
Unit cell
At ν=3/5 , most uniform configuration is the
repeat of (filled,empty,filled,empty, filled).
Also this configuration yields minimum value for W.
Unit cell
The dashed lines
indicate the empty states
Spin Polarization
I.
V. Kukushkin, K. von Klitzing and K. Eberl have measured
the spin polarizations for twelve filling factors.
Magnetic field
z
II. Their results give the very important knowledges
for the polarization in FQHE shown below.
y
Source
x
Drain
width d
length of 2DES
I. V. Kukushkin, K. von Klitzing, and K. Eberl, Phys. Rev. Lett. 82, (1999) 3665.
Special transitions
via the Coulomb interactions
before
transition
Total momentum
conserves.
electron
A
1
electron electron
B
B
Momentum
B decreases. 1
Transition
to the left
2
electron orbital
This interaction is equivalent to
the spin exchange with next form:
after
Momentum A
increases by 2
electron
A
2
D
1 2 3
D
C
1 2 3
electron orbitals
 hermite conjugate
Electron distribution after
transition is exactly the same
as that before transition.
Therefore this partial Hamiltonian
should be solved exactly because of
the energy-degeneracy.
transition
C
.
  1  2 
Equivalent interaction
  1  3 
 hermite conjugate
Most effective Hamiltonian and its equivalent form
Number of electrons per unit cell
   
For n=2/3
Number of orbitals per unit cell
c3 c4
c1 c2
The strongest interaction is  because of the nearest pair.
Also the second strongest interaction is  because of the second nearest pair.
Let us find the equivalent Hamiltonian by using the following mapping:
 0,
H
  c*n 0
  c
 2   2 1   c1*c2  c2*c1 

1

c  c *2 j c 2 j 1  c *2 j c 2 j 1  c *2 j 1c2 j  
*
2 j 1 2 j
j 1,2,3



i1,2,3
B g 1 *
B 2c i c i  1
0 2
Zeeman energy
Renumbering of operators give the diagonalization of H
We introduce new operators
a1, j
a2 , j
c1 c2
c3 c4
c5 c6
a1,1 a2,1
a1, 2 a2, 2
a1,3 a2,3
Cell 2
Cell 3


 


 
j is the cell number
We calculate the Fourier
transformation of H, then
Cell 1


H    a1* pa2 p  a*2  pa1 p e ip a*2 pa1 p  e ip a1* pa2 p
p

p
B g 1
B 2a1*  pa1 p  a*2  pa2  p 2
0 2
This Hamiltonian can be exactly diagnalized by using the
eigen-values of the matrix M
 B g * B   e ip 

M  
ip
*




e

g
B
B


The exact eigen-energies yield polarization.
Energy spectra for ν=2/3, 3/5 and 4/7
ν=3/5
ν=2/3
 B g * B   e ip 

M  
ip
*
B g B 
   e
energy
ν=4/7
 B g * B

0

e ip 
 ip
 B g * B


e


*




g
B

0


B
M  
B g *B
  M 

*
0


g
B

B
 eip
* 



B g B 
* 
 eip

0


g
B
B


energy
Wave number
Energy spectrum of ν=2/3
1 1
e  
d 2
energy
Wave number
Energy spectrum of ν=3/5
Wave number
Energy spectrum of ν=4/7

 d

tanh s  p  2kBT 
 dp 
s 1

where
polarizati on   e
dimension of matrix  d
Polarization of the present theory
Present theory
Polarization
Polarization
ν =4/7
ν =3/5
Applied
magnetic field
Applied magnetic field
Applied magnetic field
He wrote as follows: For spinful composite fermions, we write n* = n*+ n* , where = n* and n*  are the filling factors of up and down spin
composite
fermions.
possible
spin polarizations
of theAvarious
FQHE states
then predicted
by analogy
to the IQHE
of spinful
electrons.
Recently
J.K.The
Jain
has written
the article.
note contrasting
two are
microscopic
theories
of the fractional
quantum
Hall effect
For example, the 4/7 state maps into n* = 4, where we expect, from a model that neglects interaction between composite fermions, a spin singlet
Indian
of Physics
2014,
88,npp
summarized
theory.
state at
very Journal
low Zeeman
energies
(with
* =915-929
2 + 2), a He
partially
spin polarizedthe
statecomposite
at intermediatefermion
Zeeman energies
(n* = 3 + 1), and a fully spin
polarized state at large Zeeman energies (n* = 4 + 0).
Polarization of Composite Fermion theory
Effective magnetic
field
ν =4/7
?
Spin direction is
opposite against usual
electron system.
Ratio of critical
field strength =
Finite
temperature
1
9
ν =3/5
?
Theoretical curve of Spin Polarization
Red points indicate the experimental data by Kukushkin et al.
Blue curves show our results. Thus the theoretical results are in good agreement with the
experimental data.
Shosuke Sasaki, Surface Science 566 (2004) 1040-1046, ibid 532 (2003) 567-575.
n23
Small shoulder
n 3 5
n 4 7
Small shoulder
   0.25, kBT   0.2

   0.25, kBT   0.2
n 7 5
n 1 2
kBT   5.5
  0.25, kBT   0.1
Small shoulder
  0.35, kBT   0.1
n 8 5
 0.1, kBT   0.1
These small shoulders exist certainly in the data.
 explain the small shoulders. We try it.
We should
Spin Peierls effect
We take account of the famous mechanism “spin Peierls effect” into consideration.
The
between
Landau
orbitalsinbecomes
widerand
in the
first and
third and
unit-cell
like this.
Theinterval
interval
becomes
narrower
the second
fourth
unit-cell
so on.
wider narrower wider narrower
n =2/3


' '


' '
We express this deformation by the parameter t.
The classical Coulomb energy is expressed by t as;
 W N   0 C t 2 where C is the parameter depending upon devices.
   0 1  t ,   0 1  t 
    0 1  t ,    0 1  t 
where  0 ,  0 are the coupling constants
for non-deformation, and are also
dependent upon devices.
Let us find the value t which gives the minimum energy.
Eigen-energy versus deformation t

 
 

 a1*  p a 2  p   a 2*  p a1  p    a 2*  p a3  p   a3*  p a 2  p  

H  

*
*
ip *
ip *
p 
   a3  p a 4  p   a 4  p a3  p     e a 4  p a1  p   e a1  p a 4  p  


   g B1 22a  p a  p   a  p a  p   a  p a  p   a  p a  p   4
*
B
*
1
1
*
2
*
3
2
3
*
4
p
energy
Classical Coulomb energy is proportional to t 2 .
Eigen-energy of the above Hamiltonian H is shown
by red curve.
The total energy becomes minimum at the
lowest point as follows:
Lowest point
4
Spin Polarization n = 2/3

'
'


'
'
 B g *B

0
 e ip 


Matrix of
*
B g B

0 
 
the Hamiltonian M 
 0

B g *B
  

*
*
*
*
 a1  p a 2  p   a 2  p a1  p    a 2  p a3  p   a3  p a 2  p  

  e ip
H  
0

 B g * B 


*
*
ip *
ip *


 

 
   a3  p a 4  p   a 4  p a3  p     e a 4  p a1  p   e a1  p a 4  p  
   B g * B1 2 2 a1*  p a1  p   a 2*  p a 2  p   a3*  p a3  p   a 4*  p a 4  p   4
p

p
Calculated total energy for n=2/3
Polarization of n=2/3

1 1
 4

e  
dp  tanh s  p  2k BT 

4 2   s 1

Spin Polarization : n =3/5
Electron configuration of n=3/5



' '
'



Calculated total energy for n=3/5
 BgB  0
 
 0

0

 0
 e ip
Matrix of the Hamiltonian

0
0
0
 BgB  0


 BgB  0

0

 BgB  0
0
0

0
0
0
0
0
0

 BgB  0

e ip 

0

0

0



 BgB  0 
Polarization of n=3/5

1
 6







e 
tanh

p
2
k
T

dp

s
B

6  2   s 1

Theoretical curve of Spin Polarization
S. Sasaki, ISRN Condensed Matter Physics Volume 2013 (2013), Article ID 489519, 19 pages
These theoretical results are in good agreement with the experimental data.
Thus the spin Peierls instabilities appear in the experimental data of Kukushkin et al.
Summary of theoretical calculation
Our treatment is simple and fundamental without any quasi-particle.
We have found a unique electron-configuration with the minimum classical Coulomb energy. For
this unique configuration there are many spin arrangements which are degenerate.
We succeed to diagonalize exactly the partial Hamiltonian which
includes the strongest and second strongest interactions.
Then the results are in good agreement with the experimental data.
The composite fermion theory has some difficulties for the spin
polarization.
It is necessary to measure the polarization and its direction,
especially, at n= 4/5 and 6/5. The shapes of polarization curves
and the direction are very important to clarify the FQHE.
Acknowledgement
Professor Masayuki Hagiwara
Center for Advanced High Magnetic Field Science, Graduate School of Science, Osaka University, Japan
Professor Koichi Katsumata,
Professor Hidenobu Hori,
Professor Yasuyuki Kitano,
Professor Takeji Kebukawa
and
Professor Yoshitaka Fijita
Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan
Thank you for your attention
For the detailed discussion of composite
fermions, please come after this talk.
Jain’s explanation for n>1
ν =4/3
ν =8/5
J.K. Jain,
“A note contrasting two microscopic theories of the fractional quantum Hall effect”
Indian Journal of Physics 2014, Vol. 88, pp 915-929
The IQH state of electrons is combined with the composite fermion state of holes (not electrons).
ν =2
+
ν = - 2/3
Two flux quanta are
attached to hole
Effective magnetic field
Applied magnetic field
+
ν =2
+
ν = - 2/5
Two flux quanta are
attached to hole
opposite
Effective magnetic field
Applied magnetic field
+
Thus polarizations of these states are not clarified in the composite fermion theory.
Polarization of the present theory
ν =4/3
ν =8/5
Our results are in good agreement with the experimental data.
Composite
fermions
forarticle
ν =3/5,
3/7, 4/7, 4/9
Recently J.K. Jain
has written the
:,
A note contrasting two microscopic theories of the fractional quantum Hall effect Indian Journal of Physics 2014, 88, pp 915-929
He summarized the composite fermion theory.
Two flux quanta are attached to each electron
Effective magnetic field
Applied
magnetic field
Applied
magnetic field
Blue dashed curves indicate the energies
of composite-fermion with up-spin.
Red for down-spin
Effective magnetic field
energy
Magnetic field
ν =3/7
ν =3/5
Applied
magnetic field
ν =4/9
The effective magnetic field is opposite
to that with ν =3/7, 4/9.
ν =4/7
Effective magnetic field
(opposite direction)
Applied
magnetic field
Effective magnetic field
(opposite direction)
1
4
9
16B
Note:
Red (down-spin) energy is higher than
that of up-spin.
energy
Magnetic field
B
The polarization with ν =3/5, 4/7 is opposite to that with ν =3/7, 4/9.
Detail Comparison for Polarization
Absolute zero
temperature
ν =3/5
ν =4/7
Spin direction is
opposite against usual Composite fermion result deviates
electron case.
from the experimental data.
Finite temperature
Composite
Fermion theory
Composite fermion result deviates
from the experimental data.
Dashed curves indicate the empty levels.
Solid curves indicate the levels occupied with composite fermions.
Applied
magnetic field
Effective magnetic
field for composite
fermion
down-spin
2
Ratio of B = 1
Absolute zero
temperature
opposite
direction
Finite temperature
Present theory
Contenuous
spectrum
2
2
up-spin
Ratio = 1
2
2
2
Contenuous
spectrum
3
2
Polarization for FHQ states with ν =4/5 and 6/5
ν =4/5
ν =1
ν = - (1/5)
Applied magnetic field
Effective magnetic field
Hole bound with four
flux quanta
+
Applied magnetic field
Effective magnetic field
+
ν =1
ν =6/5
Electron bound with
four flux quanta
ν =1/5
The polarization versus magnetic field should be measured.