Transcript Slide 1

Unbiased Numerical Studies of
Realistic Hamiltonians for
Diluted Magnetic
Semiconductors.
Adriana Moreo
Dept. of Physics and ORNL
University of Tennessee, Knoxville,
TN, USA.
Collaborators: Y. Yildirim, G. Alvarez and E.Dagotto.
Supported by NSF grants DMR-0443144
and 0454504.
Motivation: Spintronics
• Charge Devices
– Transistors
– Lasers
– CPU, processors
• Magnetic Devices
– Non-volatile memory
– Storage
– Magneto-Optical
devices
Electron has spin and charge:
• Spin stores information
• Charge carries it
New Possibilities:
• Spin transistor
• High spin, high density nonvolatile memory
• Quantum information computers using spin states
Mn Doping of GaAs
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Mn replaces Ga
Holes are doped into the system but
due to trapping the doping fraction p
tends to be smaller than 1.
Random Magnetic impurities with
S=5/2 are introduced.
x~10% is about the maximum
experimentally achieved doping.
x>2% is necessary for collective FM.
A metal-insulator transition occurs at
x~3.5%.
S=5/2
d orb.
Ga1 x Mnx As
Ohno et al., 1996.
(x=.035, Tc=60K)
Ga :Ar  3d 10 4s 2 4 p1
Mn :Ar  3d 5 4s 2
Experimental Properties
Potashnik et al,
MF regime; x=0.08, p=.7
0.02<x<0.085
Tc increases with p
(Ku et al.)
Okabayashi et al., PRB (2001)
Ohno et al,
X=.053
Different approaches appear to be
needed in different regimes
Correct interaction, phenomenological band
Impurity Band Approach
J
Bhatt, Zunger,
Das Sarma, …
Max Tc
RKKY collective
Mean Field
Correct band structure, approx.
interaction
Valence Band
approach:Dietl,
Mac Donald
0
0.1
Carrier Density (p)
1
Mac Donald et al.
Nature Materials ‘05
Numerical Calculations
• First unbiased MC
H  t  ci, c j ,  h.c.  J  S I sI
calculation
i , j ,
I
considered one single
orbital in a cubic
lattice. (Alvarez et al.,
PRL 2002).
• Unifies the valence
and impurity band
pictures.
Two Band Models
• MC and DMFT (Popescu
et al., PRB 2006).
H    tl cl,i , cl , j ,  h.c.   J l S I sl , I
l , i , j ,
l ,I
– Tc maximized by:

 Maximum overlap between
bands
 p =>1
 J/t ~4 when impurity band
overlaps with valence band.
j=3/2
j=1/2

New Approach: Numerical simulation of a realistic Model
Fcc lattice
Diamond Lattice
• Bonding p orbitals located at Ga sites will provide the valence band.
• 6 degrees of freedom per site: 3 orbitals px,py and pz and 2 spins.
•3 nearest neighbor hopping parameters from tight binding formalism.
t

xx
t
||
xx
t xy
Hoppings’ values
2
 1  4 2   1.82eV
t 
2
2ma
2

 1  8 2   1.20eV
t xx 
2
2ma
3 2
t xy 
  2.08eV
2 3
ma
||
xx
 p,     j , m j 
Similar results obtained by Y. Chang PRB’87
Values obtained from comparison with
Luttinger-Kohn Model for III-V SC.
6 bands
J=3/2, j=1/2
4 band approximation
• Keep states with
j=3/2.
• mj=+/-3/2, +/-1/2.
H


1




t
c
c

h
.
c
.

J
s
.
S
 a, 'b i,a i   , 'b

I
I
2 i ,  , , , ',a ,b
I ,
LK
J=0
Our results
Results
Tc well reproduced in metallic regime. Longer runs being performed to improve
shape of curve (work in progress).
Tc increases with p
What value of J?
Tc in agreement with
experiments.
Tc is very low.
Metallic regime corresponds to valence band picture.
Density of States and Optical
Conductivity.
Metallic behavior.
Drude peak
Splitting of majority and minority
bands.
How high can Tc be?
Dietl et al., Science (2000)
Mean Field approach.
J
 III V
a3
Assuming
 III V  0.215eVnm3
J GaAs  1.2eV
(Okabayashi et al., PRB (1998))
Tc is expected to increased for materials with smaller a, i.e., larger J such as GaN.
Conclusions
• Numerical simulations of models in which valence band
holes interact with localized magnetic spins provide a
unified answer to a variety of theoretical approaches
which work for particular regimes.
• Mn doped III-V compounds appear to be in the weak
coupling regime.
• Is room temperature Tc possible? (Ga,Mn)N seems
promising.
• Work in progress:
– Obtain impurity band and observe MIT as a function of x for fixed
J.
– 6 orbitals model being studied.
Band Structure of GaAs
Heavy holes
Valence
Band
Light holes
Split-off
Williams et al., PRB (1986)
Luttinger-Kohn Valence Band for
GaAs
Light holes
Heavy holes
Theoretical Pictures
• a) Valence Band Holes: MacDonald, Dietl,
et al. (Zener model). Mean field
approaches of realistic models.
• b) Impurity Band Holes: Bhatt, Zunger,
Das Sarma et al. Numerical approaches
with simplified models.
a)
b)
Impurity Band Picture
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Chemical potential lies in impurity band.
Disorder plays an important role.
Band structure depends strongly on x.
Accurate at very small x.
Supported by ARPES, Optical Conductivity.
Good Tc values.
Modeled with phenomenological Hamiltonians:
– Holes hop between random
Mn sites (impurity band)
– Interaction between localized and mobile
spins is LR.
Valence Band Picture: Zener Model
• Chemical potential lies in valence band
• The band structure is rather independent of the
amount of Mn doping.
• Holes hop in the fcc lattice.
• FM caused by hole mediated RKKY interactions.
• Good Tc values for metallic samples.
• Mean Field approaches; disorder does not play
a role. Impurity spins are uniformly distributed.
• Supported by SQUID measurements.
Valence Band
||
ci c j  ci ck   4t xx c j ck
Ti  4t xx
Tx  t xy sxy  t xy sxz
 t xy sxy Ty  t xy s yz
 t xy sxz  t xy s yz Tz
 Ak x2  B(k y2  k z2 )

Ck x k y


Ck x k z

 ak 
 ak   ak 
ci  cos i  sij  4sin i sin j 
 2   2 
 2 
Ck x k y
Ak  B(k  k )
2
y
2
x
Ck y k z


Ck y k z

Ak z2  B(k x2  k y2 ) 
Ck x k z
2
z
Luttinger-Kohn
Expanding around k=0 we obtain the hoppings in terms of Luttinger parameters.
There is a similar 3x3 block for spin down.
Change of Base
HH and LH bands
j  3 / 2 (4 states)
 px , p y , pz ;    j, m j   j  1 / 2 (2 states)
Split-off band
(we discard these
states)
• The on-site “orbitals” are labeled by the four values of m_j (+/-3/2 and +/-1/2)
•The nearest neighbor hoppings between the “orbitals” are linear combinations
of the hoppings obtained earlier.
• The Hund interaction term has to be expressed in the new base. J is obtained
from experiments or left as a free parameter.


1




H
t
c
c

h
.
c
.

J
s
.
S


a , 'b i ,a i    , 'b
I
I
2 i ,  , , , ',a ,b
I ,
Results
• Non-interacting case: reproduces L-K
LK
Our results
IB vs VB in metallic regime (large x
and large p)
T* in diluted magnetic semiconductors as well?
Mn-doped GaAs; x=0.1;Tc = 150K. Spintronics? Model: carriers
interacting with randomly distributed Mn-spins locally
Monte Carlo simulations
very similar to those for
manganites.
Clustered state,
insulating
FM state,
metallic
Alvarez et al., PRL 89, 277202 (02). See also Mayr et al., PRB 2002
carrier
J
Mn spin
Experimental Properties
• Metal-Insulator transition at x~3%.
• Tc increases with p. (Ku et al.)
0.02<x<0.085
Dietl et al. (Zn,Mn)Te
Experimental Properties
• Impurity band in
insulating regime (x
<0.035)
Okabayashi et al., PRB (2001)
Experimental Properties
• Magnetization curves resemble the ones
for homogeneous collinearly ordered FM.
For large x (Potashnik et al.)
• Highest Tc~170K.
Van Esch et al.
x=.07
x=.087
MF regime; x=0.08, p=.7
Ohno et al,
X=.053
Outline
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Motivation
Experimental Properties
Theoretical Results
New approach
Results
Conclusions
Motivation: 2. DMS
• What kind of materials can provide
polarized charge carriers?
• III-V semiconductors such as GaAs
become ferromagnetic when a small
fraction of Ga is replaced by Mn.
– Can the ferromagnetism be tuned electrically?
– How do the holes become polarized?
– What controls the Curie temperature?
III-V Semiconductors: GaAs
Band Structure:
Ga : Ar  3d 10 4s 2 4 p1
As
As :Ar  3d 10 4s 2 4 p 3
Ga
Diamond Structure
Ga :Ar  3d 10 4s 2 4 p1
Jancu et al. PRB57, 6493 (’98)
As :Ar  3d 10 4s 2 4 p 3
First Brillouin Zone
Luttinger-Kohn Model
j=3/2
• Based on symmetry
• Only p orbitals are
considered
• Spin-orbit interaction
j=1/2
Light holes
Captures the behavior of the hh, lh,
and so bands around Gamma point
 p,     j , m j 
Heavy holes
Change of base due to S-O interaction
Ga1 x Mnx As
• x>2% is necessary for collective FM
• x~10% is about the maximum
experimentally achieved doping.
• The number of holes per doping fraction p
should be 1 but until recently smaller
values of p were experimentally achieved
due to trapping of holes.
Values of k in Finite Lattices
Theoretical Results (I)
• Valence band context:
– Reasonable Tc values.
– Good magnetization curves in metallic
regime.
– Some transport properties.
– Fails to capture high Tc in insulating regime.
– MF treatment of realistic Hamiltonians.
Dietl, MacDonald, …
Theoretical Results (II)
• Impurity Band context:
– Explains non-zero Tc in the low carrier (nonmetallic limit).
– Percolative transition.
– Fails to provide correct
M vs T in metallic regime.
– Phenomenological Hamiltonians
Bhatt, Zunger, Das Sarma, …
New Approach: Numerical
simulation of a realistic Model
• Real space Hamiltonian
– Valence band : tight binding of hybridized Ga and As
p orbitals on fcc lattice. (Slater).
– Interaction: AF Hund coupling between (classical)
localized spin and hole spin.
– Only j=3/2 states kept.
• Numerical Study
– Exact diagonalization and TPEM technique
(Furukawa).
– 4 states per site and 4 sites basis per cube.
– 4x4xLxLxL: number of a states in a cubic lattice with
L sites per side. It contains 4xLxLxL Ga sites.