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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