Ferromagnetic SemiconductorsFerromagnetic Semiconductors

Gergely Zaránd

Budapest Univ. Technology

Collaborators:Greg Fiete (Santa Barbara) Boldizsár Jankó (Notre Dame)Pawel Redlinski (Notre Dame)Jacek Furdyna (Notre Dame)Pascu Moca Catalin (Nagyvarad/Oradea)

• Introduction / Motivation

• (Ga,Mn)As and its simple picture

• (Ga,Mn)As in realityband structure + SO couplingimpurity band formationfrustration effectslocalization effects

OutlineOutline

Motivation:Motivation:

Combine semiconductor technology with MAGNETISM

Control magnetism through electricity(e.g., write bits through electric current)

transfer information through spin current ? Spin-base quantum computation ????....

Physics:

“Spintronics”:

localization + magnetism… anomalous Hall effect…

strong spin-orbit effects…

Difficulty: III-V: low solubility of Mn ions …

Solution:

•Annealing methods[Hayashi et al., APL 78, 1691 (2001),

…]

Wang et al., AIP Conf. Proc. 772, 333 (2005)

• Low-temperature growth of (Ga,Mn)As [Ohno, Science 281, 951 (1998)]

Goal: produce a semiconductor that can be integrated with standard technology and is a soft magnet, but has high TC

III-V MaterialsIII-V Materials

AsMnIn

SbMnGa

SbMnIn

xx

xx

xx

1

1

1

Ga1 xMnx AsNMnIII ),(

PMnIII ),(Carrier-mediated ferromagnetism

???

Examples of applicationsExamples of applications

[Ruester et al. PRL 91, 216602 (2003)]

• Spin polarized light emitting diode[R. Fiederling et al., Nature 402, 787 (1999)]

• Field effect control of ferromagnetism[H. Ohno et al., Nature 408, 944 (2000)]

• Light induced ferromagnetism[Koshihara et al., PRL 78, 1019 (2000)]

(Ga,Mn)As: The simplest picture (Ga,Mn)As: The simplest picture

Mn ions • MnGa replace Ga ions

Crystal structure + Mn ionsCrystal structure + Mn ions

Ga

As

Many holes in it … !!!

Mn ions • MnGa replace Ga ions

Crystal structure + Mn ionsCrystal structure + Mn ions

Ga

As

Many holes in it … !!!

Mn ions • MnGa replace Ga ions • MnI sit in holes…

Good MnGa ionsGood MnGa ions

MnGa ions:

12 44: psGa

25 43: sdMn heMn 32

eGa 33

• gives SPIN: S = 5/2, g=2 d 5 configuration• dopes hole

• negatively charged (strong scatterer!!!)• couples antiferromagnetically to holes

Sea of happyholes

Bad MnI ionsBad MnI ions

• Kill MnGa spins !• take away 2 holes !• expands lattice

• positively charged Bind to MnGa ions !

[Jungwirth et al. PRB 72, 165204 (2005)]

Sea of (partially)

happy holes

One can anneal them away !

One can anneal away !

AnnealingAnnealing

[Potashnik et al. APL 79, 1495 (2001)]

Simplest modelSimplest model

Mn

MnR

RMnpd SRr

J

mp

H )(22 *

23nmmeV54pdJ

Scaling not satisfied experimentally(exchange corrections,

spin fluctuations, disorder …)

Mean field theory (neglecting disorder):

0S

SJNh pdMneff

holesMnpdspin SNJH

2

SNJT

SSS holesMnpd

2

3)1( 3/1~~ pxNT holesMn

MFC

[Dietl et al. Science 287, 1019 (2000); Konig et al. PRL 84, 5628 (2000)]

(Ga,Mn)As: The reality (Ga,Mn)As: The reality

Complications Complications

• Several p-bandscomplicated band structure

• Large spin-orbit coupling magnetic anisotropies, spin relaxation etc.

• Very large disorderlocalization effects, impurity band, acceptor states

• Random spin positions• Large electron-electron interaction

Band structure and SO couplingBand structure and SO coupling

Electron structure IElectron structure I

123 44 psGaGa 325 44 psAsAs

s ( l = 0 )

p ( l = 1 )

s ( l = 0 )

j = 3/2

hoppingSO-

coupling

p ( l = 1 ) j = 1/2

j = 3/2

j = 1/2

j = 1/2

j = 1/2 valenceband

conductionbandGa

As 8 e-

[J.S. Blakemore, J. Appl. Phys. 53, R123 (1982)]

• Strong spin-orbit interaction• Holes have spin j=3/2 character• GaMnAs is degenerate Fermi system

Electron structure of GaAs: SO effectsElectron structure of GaAs: SO effects

eV42.1gapE

eV34.0SO

Cubic symmetry determines

,3/2abbaab pppp

•

•

• Luttinger parameters

[J.M. Luttinger, W. Kohn, PR 97, 869 (1955)]

,3/)1()(21 jjjjjjJ ababbaab

i

Holes have J = 3/2 spin that couples strongly to their orbital motion:

0H

H0 1

p2

2m

1

m[ 2 Jaa paa

a 3 Jab pab

ab ]

Kohn-Luttinger HamiltonianKohn-Luttinger Hamiltonian

Approximate: .)(2

)4(2210 Hpjp

mH

Eigenstates are chiral:

)5.02

2/3ˆ2

mmmk

pj hh

( ,

)07.02

2/1ˆ2

mmmk

pj ll

( ,

k

n heavy hole ≈ 10 nlight hole

[A. Baldereschi and N.O. Lipari,

Phys. Rev. B 8, 2697 (1973)]SU(2) invariant

Spherical approximationSpherical approximation

Dilute limit Dilute limit

.)(||

)(2

0int V S CC rr

ersJH

Single Mn ionSingle Mn ion

int0 HHH Hamiltonian:

Spectrum for :

meVEb 110 4

00 J

Valenceholes

Localized hole with spin J=3/2

For :00 J

Mn spin and couple to form a spin triplet

S

J

1 SJF

A10~

Polaron hopping picture :

[Berciu, M., and R. N. Bhatt, PRL. 87, 107203 (2002);G. Fiete, GZ, K. Damle, PRL 91, 097202 (2003);Kaminski, A., and S. Das Sarma, Phys. Rev. Lett. 88, 2472002 (2002);Durst, A. C., R. N. Bhatt, and P. A. Wolff, PRB 65, 235205 (2002)]

Study Mn2 ion Study Mn2 ion

2211

,,;2,1

452

,2,1

)()(

.].[)(2

FSGFSG

ccREFRK

chccRtH

ZZ

Z

ZZ

Z

Z

FiFiFi

Z

FFF

FeffMn

Energy shift

Spin-dependent hopping

Local spin-anisotropy for holes

Obtain effective Hamiltonian (spherical approx): Compute low-lying spectrum of 2 Mn ions

[P. Redlinski, GZ, B Janko, cond-mat/0505038 ; G. Fiete, GZ, K. Damle, PRL 91, 097202 (2003)]

Computed parameters:Computed parameters:

Hopping F=3/2 fermions coupled to local classical spins:

,',

,,',,,

2/3

2/3,min )'()( RR

RRRRR

MnR

hhRRthJhSGH

sites

• Spin-hopping direction coupled matrix elements:

)()()()( )2/3()2/3(ijijijji nDrCnDrrt

diagonalmatrix

spin 3/2 rotation matrix

Minimum model (dilute limit)Minimum model (dilute limit)

Band structure of a relaxed Mn systemBand structure of a relaxed Mn system

( xactive=0.01, f=0.1 )

Impurity band in small concentration limit

ARPES:H. Asklund, et al., PRB 66, 115319 (2002). J. Okabayashi, et al. PR B 64, 125304 (2001);Physica E 10, 192 (2001).STM:B. Grandidier, et al., APL 77, 4001 (2000);T. Tsuruoka, et al. APL 81, 2800 (2002);OPTICAL CONDUCTIVITY:

E. J. Singley, et al PRL, 89, 097203 (2002); Phys. Rev. B 68, 165204(2003).ELLIPSOMETRY: K. S. Burch, et al. PRB 70, 205208 (2004).

( xactive < 0.01 )

Non-collinear magnetic statesNon-collinear magnetic states

( xactive=0.01, f=0.3 )

[G. Fiete, G.Z., and K. Damle, 2003, PRL 91, 097202 (2003)]

Distribution of angles

[see, e.g. : B. Grandidier, et al. APL 77, 4001 (2000).]

Experiments: small fields induce substantial increase of magnetization in small concentration unannealed samples

Metallic limit Metallic limit

RKKY interaction: non-collinear states ?RKKY interaction: non-collinear states ?

Neglect disorder, and compute effective spin-spin interaction

[GZ, and B. Janko, PRL 89, 047201 (2002)]

21

||2

||1|| )()()( SSRKSSRKRH eff

Non-collinear States ?

RKKY interaction RKKY interaction

[Brey, L., and G. Gomez-Santos, PRB 68, 115206 (2003);G. Fiete, GZ, B. Janko, et al., PR B 71, 115202 (2005);Timm, C., and A. H. MacDonald, PRB 71, 155206 (2005)]

Almost collinear states for x > 0.03

Ab initio calculations Ab initio calculations

[G. Bouzerar, G., T. Ziman, and J. Kudrnovsky, Europhys. Lett. 69, 812 (2005)]

Bergqvist, et al. PRL 93, 137202 (2004); Hilbert, S., and W. Nolting, PR B 71, 113204 (2005);Xu, J. L., M. van Schilfgaarde, and G. D. Samolyuk, PRL 94, 097201 (2005);G. Bouzerar, G., T. Ziman, and J. Kudrnovsky, Europhys. Lett. 69, 812 (2005)

Transport properties Transport properties

Resistivity anomalies in Resistivity anomalies in

GaMnAs data from P. Schiffer’s group

Sea also Potashnik et al., APL 79, 1495 (2001)Matsukura et al., PRB 57, R2037 (1998)Edmonds et al, APL 81, 4991 (2002)

AsMnGa xx1

21~ lkF

Possible explanations for the peak?Possible explanations for the peak?

Critical fluctuations ?

Magnetic polarons ?[Kasuya, Dietl and Spalek, P. Littlewood]

Selfconsistent potantials ? [Nagaev’s theory]

Only a kink at TC

[Fischer-Langer]

Maximum way above TC[P. Littlewood]Curves cross…

“Spin disorder scattering”

Diverges at TC …?)1( Fk

None of these works …

Proposal: Interplay of magnetization and localizationProposal: Interplay of magnetization and localization

Interplay with localization produces peak at

CT

Magnetic-ordering decreases effective disorder

Resistance changes at microscopic scale

[Similar ideas emerged for Manganites [Viret et al. PRB 55, 8067 (1997)]

• There Jahn-Teller effect provides localization• Some conceptual difficulties ]

[GZ, P. Moca, and B. Janko, PRL 94, 247202 (2005).]

Influence of spin on disorder: possible mechanismsInfluence of spin on disorder: possible mechanisms

•Static spins, double exchange mechanism )cos(1~ ijijt

• Spin splitting of bands

F

FF n

eke 4

4

4

~~1

])([21

~~ 2222 nnnnnnn

[Lopez-Sancho and Brey, PRB 68, 113201 (2003)]

• Interference between magnetic and static scattering pSVJJV z2~

1 22

[Csontos et al, Nature Mat. 2005]

We need to know )(TL

Metallic Phase:L

]),([)(),( 02

2

GTLgTLhe

HT d

pinin TDTL ~,~)(

Finite conductivity Finite conductivity T

Mott’s variable range formula )1/(1

0~ln dd TN

Insulating Phase:

)]([ 00 lGG

Single parameter scaling theory of localization (T=0) Single parameter scaling theory of localization (T=0)

)(Lg Typical dimensionless conductance of slab ~ L

L

LL 2'

LL

LgfLg'

),()'(

)(ln

)(lnLg

LdLgd

)(g

gln

cgln

0T

Spin distribution changes disorder !

),(}){( ThSP i

),(}){( 000 ThGSPGG i

)1/(1

0),(~ln dd TNTh

)],(),([)(),( 02 ThGTLGTLHT d

Insulator:

Metal:

Beta function, Phase diagramBeta function, Phase diagram

To compute we need to solve a differential

equation

)(ln

)(lnLG

LdLGd

beta function extracted from model calculations

G

[GZ, P. Moca, and B. Janko, PRL 94, 247202 (2005).]

Experimentally observed anomalies, localized fitsExperimentally observed anomalies, localized fits

GaMnAs data from P. Schiffer’s group

Some fine-tuning is needed to fit

the metallic data through variable range hopping[GZ, P. Moca, and B. Janko, PRL 94, 247202 (2005).]

Fitting through metallic expressionFitting through metallic expression

p

inin TDTL ~,~)(

)],(),([)(),( 02

2

THgTLgTLhe

HT d

)1(~ 200 mgg

[GZ, P. Moca, and B. Janko, PRL 94, 247202 (2005), and unpublished]

Best fit !

More fits…More fits…

Tin ~/1 Experiments on (Ga,Mn)As metal rings find similar behavior !

K. Wagner, et al. PRL 97, 056803 (2006)

Conclusions Conclusions

General review, GaMnAs:Jungwirth et al. cond-mat/0603380

Carrier-mediated mechanism in GaMnAs:Dietl, T., 2003, condmat/0306479.

First principles calculationsSanvito, S., G. Theurich, and N. A. Hill, Journal of Superconductivity 15, 85 (2002);Sato, K., and H. Katayama-Yoshida, Semicond. Sci. Technol. 17, 367 (2002)

II-VI materialsFurdyna, J. K., and J. Kossut, Diluted Magnetic Semiconductors, volume 25 of Semiconductor and Semimetals (Academic Press, New York, 1988).

SpintronicsZutic, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004).

REVIEWS:

Transfer matrix / scaling analysis of Lyapunov exponentsTransfer matrix / scaling analysis of Lyapunov exponents

Lyapunov exponent

,..)](/[ 0 WMGM

Single parameter scaling:

slabs

MM M

M

Universal function

Microscopic length scale

Single parameter scaling theory of localization IISingle parameter scaling theory of localization II

400lConsider a slab of size and conductance 0g

cgg 0g increases as we

increase L

2dL~)L(g

cgg 0g decreases as we

increase L

)/2exp(~)( LLg

Test these ideas for a toy modelTest these ideas for a toy model

Disordered Kondo lattice:

,,,,

),,(,,

iii

iiii

jiji SJcccctH

Take J + classical spinsSpins at mean field level

Transfer Matrix Analysis

(MacKinnon and Kramers, PRL, 1981)

[Similar analysis in the context of manganites: Li et al., PRB 56, 4541 (1997)]

Beta function, Phase diagramBeta function, Phase diagram

)1(~ 200 mgg