Unbiased Numerical Studies of Realistic Hamiltonians for

Diluted Magnetic Semiconductors.

Adriana MoreoAdriana MoreoDept. of Physics and ORNLDept. of Physics and ORNL

University of Tennessee, Knoxville, University of Tennessee, Knoxville, TN, USA.TN, USA.

Collaborators: Y. Yildirim, G. Alvarez and E.Dagotto.Collaborators: Y. Yildirim, G. Alvarez and E.Dagotto.

Supported by NSF grants DMR-0443144and 0454504.

Motivation: Spintronics

• Charge Devices– Transistors– Lasers– CPU, processors

• Magnetic Devices– Non-volatile memory– Storage– Magneto-Optical

devices

Electron has spin and charge:

New Possibilities:• Spin transistor• High spin, high density nonvolatile memory• Quantum information computers using spin states

• Spin stores information• Charge carries it

Mn Doping of GaAs• 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%.

25 43: sdArMn

d orb.

1210 443: psdArGa

Ohno et al., 1996.(x=.035, Tc=60K)

AsMnGa xx1

S=5/2

Experimental Properties

Potashnik et al,MF regime; x=0.08, p=.7

0.02<x<0.085Tc 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

Mean Field

Impurity Band Approach

RKKY collective

Carrier Density (p)

J

Mac Donald et al. Nature Materials ‘05

10 0.1

Max Tc

Valence Band approach:Dietl, Mac Donald

Bhatt, Zunger, Das Sarma, …

Correct band structure, approx.interaction

Correct interaction, phenomenological band

Numerical Calculations

• First unbiased MC calculation considered one single orbital in a cubic lattice. (Alvarez et al., PRL 2002).

• Unifies the valence and impurity band pictures.

,,,, ..

ji IIIji sSJchcctH

Two Band Models

• MC and DMFT (Popescu et al., PRB 2006).– Tc maximized by:

Maximum overlap between bands

p =>1 J/t ~4 when impurity band

overlaps with valence band.

IlIjil Il

ljlill sSJchcctH ,,,, ,

,,,, ..

j=3/2

j=1/2

• 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.

Diamond Lattice Fcc lattice

xxt

||xxt xyt

New Approach: Numerical simulation of a realistic Model

Hoppings’ values

eVma

t

eVma

t

eVma

t

xy

xx

xx

08.23

20.182

82.142

32

2

212

2

212

2||

Similar results obtained by Y. Chang PRB’87

jmjp ,, 6 bandsJ=3/2, j=1/2

Values obtained from comparison withLuttinger-Kohn Model for III-V SC.

4 band approximation

• Keep states with j=3/2.

• mj=+/-3/2, +/-1/2.

I

bai IIbiaiba SsJchcctH ...

2

1

,,',,,, ,',,',

LK

Our results

J=0

Results

Tc well reproduced in metallic regime. Longer runs being performed to improveshape of curve (work in progress).

Tc increases with p

What value of J?

Metallic regime corresponds to valence band picture.

Tc in agreement with experiments.

Tc is very low.

Density of States and Optical Conductivity.

Metallic behavior. Drude peak

Splitting of majority and minority bands.

How high can Tc be?

33

215.0 eVnma

J VIIIVIII

Dietl et al., Science (2000)Mean Field approach.

eVJGaAs 2.1Assuming

(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

Valence Band

Heavy holes

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

• 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

zyzxyxzxy

yzxyyxyxy

xzxyxyxyx

Tstst

stTst

ststT

)(

)(

)(

222

222

222

yxzzyzx

zyzxyyx

zxyxzyx

kkBAkkCkkCk

kCkkkBAkkCk

kCkkCkkkBAk

2sin

2sin4 ji

ij

akaks

Luttinger-Kohn

2cos

44 ||

ii

kjxxkijixxi

akc

cctcccctT

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

• 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.

I

bai IIbiaiba SsJchcctH ...

2

1

,,',,,, ,',,',

)2(2/1

)4(2/3,;,,

statesj

statesjmjppp jzyx

HH and LH bands

Split-off band (we discard these states)

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?

Alvarez et al., PRL 89, 277202 (02). See also Mayr et al., PRB 2002

Mn-doped GaAs; x=0.1;Tc = 150K. Spintronics? Model: carriersinteracting with randomly distributed Mn-spins locally

Clustered state, insulating

FM state,metallic

Monte Carlo simulationsvery similar to those formanganites.

JMn spin

carrier

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.

MF regime; x=0.08, p=.7

Ohno et al,X=.053

Van Esch et al.

x=.07x=.087

Outline

• 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

1210 443: psdArGa 3210 443: psdArAs

3210 443: psdArAs

Diamond Structure

Jancu et al. PRB57, 6493 (’98)

Band Structure:

First Brillouin Zone

Ga

As

1210 443: psdArGa

Luttinger-Kohn Model

• Based on symmetry• Only p orbitals are

considered• Spin-orbit interaction

jmjp ,, Change of base due to S-O interaction

j=3/2

j=1/2

Light holes

Heavy holesCaptures the behavior of the hh, lh,and so bands around Gamma point

• 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.

AsMnGa xx1

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 (non-

metallic 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.