2. Hall 2. Hall resistance resistance I V I V where H H Source Drain z y x length of 2DES width d Magnetic Field Current I Quantum Hall Quantum Hall Device Device Spin Polarization of Spin Polarization of Fractional Quantum Hall Fractional Quantum Hall States States ntial is nearly equal to zero at FQHE. No is nearly equal to zero at FQHE. No electron scattering. electron scattering. Center for Advanced High Magnetic Field Science, Graduate School of Center for Advanced High Magnetic Field Science, Graduate School of Science, Osaka University, Japan Science, Osaka University, Japan Shosuke Sasaki Shosuke Sasaki QHE creates no heat. QHE creates no heat. 3. FQHE are purely eigen-value problem 3. FQHE are purely eigen-value problem of electrons. of electrons. 4. 4. The value of The value of V VH is extremely larger than is extremely larger than Vpotential . . V Hall Voltage VH Poential Voltage Vpotential The FQHE is the famous phenomena. However there are The FQHE is the famous phenomena. However there are some questions. One of them is the spin-polarization. some questions. One of them is the spin-polarization. We examine it in this talk. We examine it in this talk. ICTF16@Dubrovni ICTF16@Dubrovni k Oct. 14, 2014 Oct. 14, 2014 OR 77 OR 77
Transcript
2. Hall resistance 2. Hall resistance IVIV
where HH
SourceDrain
z
y
x
length of 2DES
width d
Magnetic Field
Current I
Quantum Hall DeviceQuantum Hall Device
Spin Polarization of Spin Polarization of Fractional Quantum Hall StatesFractional Quantum Hall States
1. 1. Vpotential is nearly equal to zero at FQHE. Nois nearly equal to zero at FQHE. No electron scattering.electron scattering.
Center for Advanced High Magnetic Field Science, Graduate School of Science, Osaka University, JapanCenter for Advanced High Magnetic Field Science, Graduate School of Science, Osaka University, Japan
Shosuke SasakiShosuke Sasaki
QHE creates no heat. QHE creates no heat.
3. FQHE are purely eigen-value problem of 3. FQHE are purely eigen-value problem of electrons. electrons. 4. 4. The value of The value of VVHH is extremely larger than is extremely larger than Vpotential. . Therefore the gradient ofTherefore the gradient of VVHH cannot be ignored.cannot be ignored.
Hall Voltage VHPoential Voltage Vpotential
The FQHE is the famous phenomena. However there are some questions. The FQHE is the famous phenomena. However there are some questions. One of them is the spin-polarization. We examine it in this talk. One of them is the spin-polarization. We examine it in this talk.
Hamiltonian and its eigen-statesHamiltonian and its eigen-states in single electron system in single electron system
Magnetic field
Gate
Potential Probe
Hall Probe
integer2
eBeB
kc
The eigen state is Landau wave function as follows:
H0 p eA 22m U y W z
Hamiltonian of single electron
Fig.6Potential for z-direction (Potential width is very narrow)
z
P o ten tia l
W (z )W (z )
y
P o ten tia l
U (y )
Fig.7 Potential U(y)
-eV 1
-eV 2
y = 0 y = d
),0,0(rot ,0 ,0 , ByB AA
EH 0
yikxEyikxyUymm
eByk
exp1
exp1
22 2
222
This narrow potential realizes the appearance of only the ground state for z-direction as follows:
222 )/()( eBkyeBeByk
the momentum
22)( cyeB
We obtain the eigen equation for the y-direction
where the center position c is proportional to
zee cyikx 2 for level L=0
zyikxzyx exp1
,,
where the variables separated.
-eV-eVHH
Wave function in Many electron system Wave function in Many electron system
Magnetic field
Gate
Potential Probe
Hall Probe
Hall Probe
electron orbital
Potential Probe
y
x
b
zee cyikx 2
ckeB
2eB
integer
Landau wave function of electron for L=0
Attention please.Attention please.
Hall voltage is extremely larger than potential Hall voltage is extremely larger than potential voltage for IQHE and FQHE. voltage for IQHE and FQHE.
Accordingly there is no symmetry between x Accordingly there is no symmetry between x and y directionsand y directions..
the center position moves to the right. the center position moves to the right. This is the many electron stateThis is the many electron stateWhen the momentum increases,When the momentum increases,
The position c is proportional to The position c is proportional to the momentum.the momentum.
the center position moves the center position moves to the left.to the left.When the momentum When the momentum decreases,decreases,
Total Hamiltonian of many-electron systemTotal Hamiltonian of many-electron system
HT H0 xi, yi, zi i1
N
HCWe separate HT into
where HD is the diagonal term and HI is non-diagonal part.
W k1,,kN E0 ki i1
N
C k1,,kN
HT HD H I
At ν=2/3 , the most uniform configuration is
created by repeating (filled,empty,filled).
This configuration gives the minimum value
for W.
y
x
Unit cell
At ν=3/5 , most uniform configuration is the
repeat of (filled,empty,filled,empty, filled).
Unit cell
Nkk
NNN kkkkWkkH,,
111D
1
,,,,,,
k1, ,kN 1
N !
k1x1, y1, z1 k1
xN , yN , zN
kNx1, y1, z1 kN
xN , yN , zN where
where C(k1•••kN) expresses the diagonal part of the Coulomb interaction Hc which is called “classical Coulomb energy”. filled,empty,filled
The dashed lines indicate the empty states
Also this configuration yields minimum value for W.
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.
Spin Polarization Spin Polarization
I. V. Kukushkin, K. von Klitzing, and K. Eberl, Phys. Rev. Lett. 82, (1999) 3665.
I. V. Kukushkin, K. von Klitzing and K. Eberl have measured the spin polarizations for twelve filling factors.
II. Their results give the very important knowledges for the polarization in FQHE shown below. Source
Drain
z
y
x
length of 2DES
width d
Magnetic field
Special transitions Special transitions via the Coulomb interactionsvia the Coulomb interactions
before after
conjugate hermite
31
Equivalent interactionEquivalent interaction
electron orbital
11 2 2
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.
electron A
electron B
electron orbitals
transition
1
DC
2 31 2 3
conjugate hermite
21
This interaction is equivalent to the spin exchange with next form:
electron A
transition
electron B
Total momentum conserves.
CD
Momentum A increases by .2
Momentum B decreases.Transition to the left
Most effective Hamiltonian and its equivalent formMost effective Hamiltonian and its equivalent form
H c2 j 1* c2 j c2 j
* c2 j 1 c2 j* c2 j1 c2 j1
* c2 j j1,2,3
Bg0
B12
2c i*c i 1
i1,2,3
For=2/3
1*22
*11221 cccc
0 , cn* 0
Let us find the equivalent Hamiltonian by using the following mapping:
1c 2c 3c 4c
Zeeman energy
The strongest interaction is because of the nearest pair.Also the second strongest interaction is because of the second nearest pair.
Number of orbitals per unit cell
Number of electrons per unit cell
Renumbering of operators give the diagonalization of Renumbering of operators give the diagonalization of HH
We introduce new operatorsWe introduce new operators
jj aa ,2,1
H a1* p a2 p a2
* p a1 p e ipa2* p a1 p e ipa1
* p a2 p p
Bg
0
B1
22 a1
* p a1 p a2* p a2 p 2
p
We calculate the Fourier We calculate the Fourier transformation of transformation of H,H, then then
3,23,12,22,11,21,1 aaaaaa
654321 cccccc
j is the cell number
This Hamiltonian can be exactly diagnalized by using the This Hamiltonian can be exactly diagnalized by using the eigen-values of the matrix eigen-values of the matrix MM
The exact eigen-energies yield polarization.The exact eigen-energies yield polarization.
Bge
eBgM
ip
ip
*B
*B
Cell 1 Cell 2 Cell 3
Energy spectra for ν=2/3, 3/5 and 4/7 Energy spectra for ν=2/3, 3/5 and 4/7
Bge
Bg
Bg
eBg
M
ip
ip
*B
*B
*B
*B
0
0
0
0
Bge
eBgM
ip
ip
*B
*B
ν=2/3ν=2/3 ν=3/5ν=3/5
Bge
Bg
eBg
Mip
ip
*B
*B
*B
ν=4/7ν=4/7
d
sse Tkpp
d 1B2tanhd
2
11
de
matrix ofdimension
onpolarizati where
Energy spectrum of ν=2/3 Energy spectrum of ν=2/3 Energy spectrum of ν=3/5 Energy spectrum of ν=3/5 Energy spectrum of ν=4/7 Energy spectrum of ν=4/7
Wave number
energy
Wave number
energy
Wave number
energy
Polarization of Composite Fermion theory
ν =3/5
?
Polarization of the present theoryPresent theory
Effective magnetic field
Applied magnetic field
Spin direction isopposite against usual electron system.
Recently J.K. Jain has written the article. 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.
ν =4/7 ν =3/5
Finite temperature
9= 1Ratio of critical field strength
He wrote as follows: For spinful composite fermions, we write = + , where = and are the filling factors of up and down spin composite fermions. The possible spin polarizations of the various FQHE states are then predicted by analogy to the IQHE of spinful electrons. For example, the 4/7 state maps into = 4, where we expect, from a model that neglects interaction between composite fermions, a spin singlet state at very low Zeeman energies (with = 2 + 2), a partially spin polarized state at intermediate Zeeman energies ( = 3 + 1), and a fully spin polarized state at large Zeeman energies ( = 4 + 0).
ν =4/7
?
Polarization
Applied magnetic field Applied magnetic field
Polarization
Theoretical curve of Spin PolarizationTheoretical curve of Spin Polarization
0.25, kBT 0.2
2 3
0.25, kBT 0.2
3 5
4 7
0.35, kBT 0.1
kBT 5.5
1 2
7 5
8 5
0.25, kBT 0.1
0.1, kBT 0.1
Small shoulder
Small shoulder Small shoulder
We should explain the small shoulders. We try it.
Red points indicate the experimental data by Kukushkin et al. 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 Blue curves show our results. Thus the theoretical results are in good agreement with the experimental data.experimental data.
These small shoulders exist certainly in the data.
The interval becomes narrower in the second and fourth unit-cell and so on.
=2/3
Let us find the value t which gives the minimum energy.
tt
tt
1 ,1
1 ,1
00
00
20 tCNW
The classical Coulomb energy is expressed by t as;
where C is the parameter depending upon devices.
where are the coupling constants for non-deformation, and are also dependent upon devices.
00 ,
narrower narrowerwider wider
We express this deformation by the parameter t.
We take account of the famous mechanism “spin Peierls effect” into consideration.The interval between Landau orbitals becomes wider in the first and third unit-cell like this.
Eigen-energy versus deformation t
Lowest point
Classical Coulomb energy is proportional to t t 2 2 . .
Eigen-energy of the above Hamiltonian Eigen-energy of the above Hamiltonian HH is shown is shown by red curve.by red curve.
The total energy becomes minimum at the The total energy becomes minimum at the lowest point as follows: lowest point as follows:
energyenergy
p
pipip
papapapapapapapaBg
papaepapaepapapapa
papapapapapapapaH
4221 4*43
*32
*21
*1
*B
4*11
*43
*44
*3
2*33
*21
*22
*1
Spin Polarization Spin Polarization = 2/3 = 2/3
Calculated total energy for Calculated total energy for =2/3=2/3
Calculated total energy for Calculated total energy for =3/5=3/5
Polarization of Polarization of =3/5=3/5
pTkps
Bse d2tanh26
1 6
1
Matrix of the Hamiltonian
Theoretical curve of Spin PolarizationTheoretical curve of Spin Polarization
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.
S. Sasaki, ISRN Condensed Matter Physics Volume 2013 (2013), Article ID 489519, 19 pages
Summary of theoretical calculationSummary of theoretical calculationOur treatment is simple and fundamental Our treatment is simple and fundamental without any quasi-particle. without any quasi-particle.
We have found a unique electron-configuration with the minimum classical Coulomb energy. For 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. this unique configuration there are many spin arrangements which are degenerate.
We succeed to diagonalize exactly the partial Hamiltonian which We succeed to diagonalize exactly the partial Hamiltonian which includes the strongest and second strongest interactions. includes the strongest and second strongest interactions.
Then the results are in good agreement with the experimental Then the results are in good agreement with the experimental data.data.
The composite fermion theory has some difficulties for the spin The composite fermion theory has some difficulties for the spin polarization. polarization.
It is necessary to measure the polarization and its direction, It is necessary to measure the polarization and its direction, especially, at especially, at = 4/5 and 6/5. The shapes of polarization curves = 4/5 and 6/5. The shapes of polarization curves and the direction are very important to clarify the FQHE. and the direction are very important to clarify the FQHE.
AcknowledgementAcknowledgement
Professor Masayuki Hagiwara Professor Masayuki Hagiwara Center for Advanced High Magnetic Field Science, Graduate School of Science, Osaka University, Center for Advanced High Magnetic Field Science, Graduate School of Science, Osaka University, JapanJapan
Professor Koichi Katsumata, Professor Koichi Katsumata,
Professor Hidenobu Hori, Professor Hidenobu Hori,
Professor Yasuyuki Kitano,Professor Yasuyuki Kitano,
Professor Takeji KebukawaProfessor Takeji Kebukawa
andand
Professor Yoshitaka FijitaProfessor Yoshitaka FijitaDepartment of Physics, Osaka University, Toyonaka, Osaka 560-0043, JapanDepartment of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan
Thank you for your attentionThank you for your attention
For the detailed discussion of composite For the detailed discussion of composite fermions, please come after this talk.fermions, please come after this talk.
Effective magnetic fieldApplied magnetic field
+
Effective magnetic fieldApplied magnetic field
+
ν =8/5ν =4/3
ν =2 ν = - 2/3+ ν =2 ν = - 2/5+
Jain’s explanation for >1
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
Thus polarizations of these states are not clarified in the composite fermion theory.
Two flux quanta are attached to hole
Two flux quanta are attached to holeopposite
The IQH state of electrons is combined with the composite fermion state of holes (not electrons).
Polarization of the present theory
ν =4/3 ν =8/5
Our results are in good agreement with the experimental data.
Composite fermions for ν =3/5, 3/7, 4/7, 4/9 Recently J.K. Jain has written the article :, 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.
Effective magnetic fieldAppliedmagnetic field
Effective magnetic fieldAppliedmagnetic field
ν =4/9ν =3/7
Two flux quanta are attached to each electron
Blue dashed curves indicate the energiesof composite-fermion with up-spin.Red for down-spin
Note:
Red (down-spin) energy is higher than that of up-spin.
Magnetic field
B
energy
Magnetic field
B
energy
1 4 9 16
Effective magnetic field(opposite direction)Applied
magnetic field
Effective magnetic field(opposite direction)
Appliedmagnetic field
ν =3/5 ν =4/7The effective magnetic field is oppositeThe effective magnetic field is opposite to that with to that with ν =3/7, 4/9.
The polarization with The polarization with ν =3/5, 4/7 is opposite to that with is opposite to that with ν =3/7, 4/9.
down-spin up-spin
Detail Comparison for Polarization
Composite Fermion theory
Present theoryContenuous spectrum
Contenuous spectrum
Composite fermion result deviates from the experimental data.
22 3
2Ratio = 1
2Ratio of B = 1
2 22
opposite direction
Dashed curves indicate the empty levels.Solid curves indicate the levels occupied with composite fermions.
Effective magnetic field for composite fermion
Applied magnetic field
Composite fermion result deviates from the experimental data.
Spin direction isopposite against usual electron case.
Absolute zero temperature
Absolute zero temperature
Finite temperature
Finite temperature
ν =3/5 ν =4/7
ν =4/5Polarization for FHQ states with Polarization for FHQ states with ν ν =4/5 and 6/5=4/5 and 6/5
The polarization versus magnetic field should be measured. The polarization versus magnetic field should be measured.
+
Applied magnetic field Effective magnetic field
Effective magnetic field
+
Applied magnetic field
Electron bound with Electron bound with four flux quanta four flux quanta
Hole bound with four Hole bound with four flux quantaflux quanta
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