Transcript Historical Development of Localization Theories in

Dept. of Physics, Fudan University
Shanghai, 15 May 2012
From Impurity Bands to
High Temperature Superconductivity:
The Important Role of Spin Effects
Tokyo University of Science
Hiroshi Kamimura
Outline of the talk
• In this talk I would like to show how the electronelectron interactions play an important role as spineffects in doped semiconductors and copper oxide
superconductors.
• In the first part I will show how the interplay of disorder
and electron-electron interactions creates the
remarkable spin-dependent behavior in the Andersonlocalized regime of the impurity band in doped
semiconductors.
• In the second part I will show how the interplay of Mott
physics and Jahn-Teller physics creates mysterious
electronic properties in copper oxide high-temperature
superconductors by choosing the simplest system of
cuprates; i.e. copper oxides.
Part 1. Impurity Bands
in doped semiconductors
1. An isolated impurity state
When a group-V element such as P, Sb, and As is
doped into Si or Ge, the hydrogen 1s-like shallow
impurity state appears just below a conduction band.
A schematic picture of the ground state of an
isolated donor impurity are shown in the next
two slides.
A donor atom (P) in a host crystal (Si). The spatial extension
of donor wavefunction (aB*) is shown for a phosphorus in
silicon.
+e
-e
(Effective Bohr radius =33 Å for P in Si)
Donor 1s level in an energy gap of a semiconductor
Ground state energy
ε1s = m*e4/2κh2
( = 30 meV for Si)
m* = effective mass,
Κ = dielectric constant
(12 for Si)
2. Impurity Bands
This slide shows how the hydrogen 1s-like shallow donor
state becomes an impurity band, when the impurity
concentration increases.
Various energy levels appear corresponding to different
impurity distributions. Finally an impurity band is formed,
as shown below.
. . . ...
Conduction
band edge
... .
.. .
.
.
.
.
.
. . ..
E
Impurity band (D band)
Donor 1s state
(ε1s)
D (E )
3. Impurity Conduction
• The impurity-band system shows very interesting
transport phenomena, called “Impurity
Conduction”.
• In 1958, Helmut Fritzsche reported that, in ntype doped semiconductors, there are three
regions of donor concentration; low, intermediate,
and metallic, according to the features of
electronic conduction, as shown in the next slide.
H. Fritzche
J.P.C.S. 6 (1959)
69
Resistivity of
Antimony-doped
Germanium
(compensated
Sample)
ε3
In low concentration,
each electron is trapped
around each donor
impurity, and hopping
conduction occurs only
in compensated samples,
with activation
energyε3.
As the impurity
concentration increases,
another activation
energy appears. It is
called the intermediate
concentration region
with the concentrationdependent activation
energy ε2 .
ε2
nc
Metal-insulator transition
occurs at n c.
High concentration region
above nc is a metallic
region.
Summarizing the above-mentioned
experimental results,
the conductivity is expressed in the form
of three terms;
σ＝σ1exp(-ε1 /kT ) + σ2exp(-ε2 /kT ) +
σ3exp(-ε3 /kT ) .
Here ε1 is the energy required to eject an
electron into the conduction band and ε3
the activation energy for hopping
conduction at low concentration regime.
Here we pay attention to the second term in a
previous slide: Impurity Conduction in the
intermediate concentration region
• At high temperatures, the conduction is of
activation type,
σ＝σ2exp(-ε2 /kT ).
The activation energy ε2 decreases
with increasing donor concentration.
It is seen to vanish at the critical impurity
concentration nc of metal-insulator transition,
as shown in the next slide.
Observed concentration dependence of ε2
This decreases sharply with donor concentration nD.
This experimental result
shows that an electron
correlation plays an
important role in the
impurity band.
nc
4. Let us investigate the origin of
concentration-dependent activation energy ε2 .
In the uncompensated case in which each donor has one
electron, the donor (D) band is completely filled by the
electrons. Then, let us consider a case in which an extra
electron is doped in this system.
For this purpose,
let us consider
whether an extra
electron can be
accommodated in
the 1s orbital in a
donor impurity, or
not.
4. The state of two electrons in a donor
• According to the first principles calculations of variational
method by Natori and Kamimura, the another electron with
opposite spin can occupy the 1s orbital with an expanded radius
(1s’ orbital ) in an isolated donor to form a stable a negative
donor ion, D- ion.
• ( see A. Natori and
H. Kamimura, JPSJ 43
(1977) 1270.)
This is called
D- state with
two electrons
in (1s) (1s’)
configuration.
phosphorus
The 2nd
electron
1s’ orbital
electron
5. Origin of the concentration-dependent
activation energy ε2
The donor D band is completely filled
in the uncompensated case.
An extra electron
in (1s’) orbital occupies
D- state.
o
This D- state forms a
D- band, when the donor
concentration increases.
We define the energy gap
between D and D- bands
as ε2.
We can clearly see that the band gapε2 decreases with the increase of
donor concentration. We conclude that the appearance of a D- band
and of the energy gap ε2 is due to the electron correlation.
5. Anderson-Localized States
(Effects of disorder)
• In the intermediate concentration regime, Mott
pointed out that a wave-function of a localized
electron extends over a number of impurity sites.
• An envelope function of the wave function which
depicts an amplitude at each site decreases
exponentially, as shown below: We call such a
state “Anderson-localized state”, denoting |α>.
Exponential envelope is expressed as
v iα~ exp (- | ri – rα|/ξα)
ξα: localization length
ξα ~ | 1 – εα/ Ec|-ν
Ec : mobility edge
localization centre rα
6. Work with Professor Sir Nevill Mott
on the impurity band
• I worked with Sir Nevill Mott on the interplay of electron
interactions and disorder in the impurity band at the
Cavendish Laboratory, University of Cambridge from
October, 1974 to August, 1975.
• After I came back to University of Tokyo, I started to
formulate “Theoretical Model on the Interplay of
Disorder and Electron-Correlation” with my graduate
students.
Drink Party at the
garden of Prof.
Mott’s house (11
May 1975,
Sunday)
5. Effective Hamiltonian for the Interplay of
Disorder and Electron-Correlation
H. Kamimura, Progress of Theoretical Physics: Suppl. No. 72 (1982) 206231
We adopt the Anderson-localized states as a set of basis functions.
Then the Hamiltonian can be rewritten as follows:
Intrastate interaction
Interstate interactions
(1)
Anderson-localized
state
|i> is an impurity state at the i-th site.
.
envelope
v iα~ exp (- | ri – rα|/ξα)
ξα ~ | 1 – εα/ Ec-ν
(2)
Behavior of the envelop function as a function of εα
• v iα~ exp (- | ri – rα|/ξα)
• ξα ~ | 1 – εα/ Ec-ν|
εα
Mobility edge Ec
Deep energy
6. Intrastate Interaction Theory
(1) Effective Hamiltonian
• In the Anderson-localized regime, the interstate
interactions are small compared with intrastate
interactions. Thus in this regime we consider first the
intrastate interaction only. In the presence of a magnetic
field, the effective Hamiltonian is given by
(3)
• For each state α, there are four possible values of total energy E,
corresponding to four different electron occupancies:
α
(2) Effects of electron correlation U
• According to this intrastate interaction
Hamiltonian, an energy εα is required
for the first electron to occupy a state α
while an energy εα + Uα is necessary
for the second electron to occupy state α,
which is already occupied by another
electron of opposite spin.
εα + Uα
εα
This is a doubly occupied state (DO).
If the energy for the second electron εα + Uα is below
the Fermi level EF, the state α is occupied by two
electrons, while if εα + Uα is above the Fermi level, the
state α cannot accommodate two electrons
even if the one-electron energy εα is below EF .
The former is called a doubly occupied state (DO), while
the latter is a singly occupied state (SO).
εα + Uα
EF
εα + Uα
εα
εα
SO
DO
(3) Since the energy εα of Anderson- localized state |α>
takes a random value in the system of the impurity band,
the competition of disorder and electron correlation
plays an important role .
• In order to express the mode of disorder, we
assume a constant density of states for the
distribution of the energy εα. It ranges from ε= 0
to the mobility edge Ec with a width W.
• Thus the density of states for the first electron
in state |α> is expressed as
D1(E) = 1/W
for 0 ≤ E ≤ W.
W: a measure of disorder
Ū: Strength of electron correlation
Please remind:
Behavior of the envelop function as a function of εα
•v iα~ exp (- | ri – rα|/ξα)
•ξα ~ | 1 – εα/ Ec-ν|
εα
Mobility edge Ec
Deep energy
v iα
Uα
small
large
In the case of ratio W/Ū = 6, I will show how to calculate
Density of first-electron states per electron, D1(E), and the
density of second- electron states, D2(E) below.
Hatched region:
SO states.
Cross hatched
Region:
DO states.
SO (εβ)
DO (εα)
εα +Uα
εα
7. Correlation effects on the spin
susceptibility and the electronic specific heat
• The direct consequence of the singly
occupied (SO) states in the Intrastate
Interaction Theory leads to the appearance
of Curie law.
. Otherwise, the spin susceptibility becomes a
Pauli-type, since the density of states at EF
is finite in the present system (D1(EF)  1/W).
. The finite density of SO and DO states
results in the T –linear specific heat in low
temperatures.
7.1 Numerical results of spin susceptibility as a
function of temperature for various values of W/Ū
Weak correlation
Strong
correlation
In the strong correlation regime the Curie lawχ=μB2ns/kBT
appears. Here ns is the number of singly occupied states for
small W/U.
Experimental results on the Curie-Weiss plots
of molar donor susceptibility of Si:P
(K. Andres et al., Phys. Rev. B 24 (1981) 244)
At low temperatures,
the observed susceptibility
is monotonically
increasing as if it were
to diverge with decreasing
temperature The downward
bending indicates
”interstate interactions”
which we have neglected
are important!!
• The solid straight line
indicates the free-spin
behavior of Curie law.
↑
8. Experimental results of specific heat of
Si:P as a function of temperature
(Nobuyoshi Kobayashi et al., Solid State Commun. 24 (1977) 67)
• nD = 1.71018cm-3 (nc = 4.51018cm-3)
In samples of intermediate
concentration, an anomaly
appears in very low
temperatures below 2K.
I will discuss an origin of
this anomaly later.
Pay attention to
the Anomaly.
is due to the
interaction between
spins of singly occupied
states.
This
9. Magnetoresistance
in the variable-range hopping
A. Kurobe and H. Kamimura, J.Phys. Soc. Jpn. 51 (1982) 1904
According to the Intrastate Interaction theory, there appear
the following four different kinds of hopping process,
corresponding to the three electronic states;
(1) unoccupied state (UO), (2) singly-occupied (SO) state
and (3)doubly-occupied (DO) state.
From SO to UO,
From SO to SO, From DO to UO, From DO to SO
9.1 Mechanism of positive magnetoresistance
When we apply a magnetic field, the magnetic moment in the state
|β> in the process (2) change its direction.
As a result, the directions of the moments in the |α> and |β> states
become parallel, and the hopping processes between SO states are
suppressed by the Pauli principle.
Similarly the hopping process (3) is also suppressed by fields. Thus
the resistance increases by magnetic fields. This gives rise to
positive magnetorresistance.
9.2-1 Calculated magnetoresistance
as a function of magnetic fields.
9.2-2 Resistivity at 300 mK in very dilute 2D system
of electrons in high-mobility Si-MOSFET as a function of
in-plane magnetic field for different electron densities
(K.M. Mertes, et al., PRB 60 (1999) R5093)
Insulating samples
at zero field
Conducting phase
9.3 Experimental result for
magnetoresistance in 1T-TaS2
(After Onuki et al. Physica 99b (1980) 177 )
Only negative
magnetoresitance
appears below 0.1 K.
How can we explain the negative
magnetoresistance below 0.1 K?
• For this purpose we have introduced
the spin-dependent interstate
interactions between Andersonlocalized states which we have
neglected do far.
• In the presence of interstate
interactions, the spins of the SO states
are coupled antiferromagnetically.
Suppression of the positive contribution
at low temperatures
Thus the hopping processes (2) and (3) are not
suppressed by magnetic fields. Thus the positive
contribution to magnetoresistance disappears in
the low temperature.
Calculated magnetoresistance of 1T-TaS2
in the presence of interstate interaction
as well as intrastate interaction
Conclusions for magnetoresistance
in variable-range hopping in the presence of
electron-electron interactions
• Features of magnetoresitance in the Anderson-localized
regime has been discussed based on the theory by Kurobe
and Kamimura.
• I have shown that the intrastate interactions gives rise to
the positive magnetoresistance.
• Further, I have mentioned that the interstate interactions
give rise to the negative magnetoresistance at low
temperatures.
Theory of magnetoresistance by Kurobe and Kamimura
• In recent decades the theory of magnetoresistance by
Kurobe and Kamimura has been used by a number of
experimental groups to analyze their experimental
results not only on semiconductors but also on
polymers, quasicrystals, etc. I will show some of
examples.
• (1) Electrical transport of spin-polarized carriers in disordered
ultrathin films, L.M. Hernandez et al, PRL 91 (2003) 126801
• (2) Insulating behavior of dilute two-dimensional holes in GaAs under
an in-plane magnetic field, H. Noh, J. Yoon, D.C. Tsui and M.
Shayeen, PRB 70 (2004) 241306
• (3)Magnetoresistance studies of polymer nanotube/wire pellets and
single polymer nanotubes/wires, Y. Long et al, Nanotechnology 17
(2006) 5903
Acknowledgements to the collaborators
in work of Part I
Hideo Aoki
Eiichi Yamaguchi
Tadashi Takemori
Akiko Natori
(University of Tokyo)
(Doshisha university)
(Tsukuba University)
(University of ElectroCommunications)
Tadashi Takemori (Tsukuba University)
Atsushi Kurobe
(Toshiba Semiconductor Co.)
Mikio Eto
(Keio University)
References of
Part I:
(Further reading)
Text book by
H. Kamimura
and Hideo Aoki
with Foreward by
Sir Nevill Mott
Comprehensive Semiconductor Science and Technology
Editors in Chief: P. Bhattacharya, R. Fornari, and H. Kamimura
Volume 1, Chapter 3. Impuritu Bands by M. Eto and H. Kamimura