Electronic, magnetic, and transport propertiesof diluted magnetic semiconductors:

(Ga,Mn)As as a case study

J. Kudrnovsky

Institute of Physics, ASCR, Prague, Czech Republic

in collaboration with

G. Bouzerar, L. Bergvist, O. Eriksson, I. Turek,K. Carva, V. Drchal, and J. Masek

Motivation

• Curie temperature Tc and conductivity σ: relevant character-istics of diluted magnetic semiconductors (DMS)

• Close internal relation of Tc and σ: both depend on1. carrier concentration: σ - directly, Tc - through the Fermisurface topology2. carrier lifetime due to disorder: σ - relaxation time, Tc -spatial damping of exchange integrals

• Experimental motivation: Nottingham (Edmonds − 2002)• Model approach: Jpd-model ⇒ fitted parameters• Ab-initio approach: carrier concs/lifetime determined on the

same footing from LSDA-Hamiltonian ⇒ predictive power

Aim of the study

• Theoretical estimation of Tc vs σ ratio for both as-grown andannealed samples and comparison with the experiment

• Case study: (Ga,Mn)As - best studied, highest Tc ≈ 190 K(IoP Prague)

2

General remarks

Zincblende A3B5/A2B6 semiconductors diluted with few % ofmagnetic atoms allow for carrier and spin control ⇒ spintronics

Mechanisms of moment formation

• Mn2+ ⇒ Ga3+ adds a hole into valence band which becomesitinerant for small but finite x(Mn) converting semiconductorinto metal

• Mn-spins interact indirectly via holes creating long-range fer-romagnetic (RKKY-like coupling) necessary for the collectivemagnetic arrangement of Mn-spins

• Hybridization of Mnt2-impurity states with Asp-host states (va-lence band) at k=0 (Γ point) are responsible for strong Mn-host coupling ⇒ Jpd-model (Dietl, MacDonald, Jungwirth)

3

• Jpd-model features: empical parameters (Jpd-coupling and thecarrier concentration) allows to understand many propertiesqualitatively ⇒ it captures properly dramatic dependence oncarrier concentration

• Jpd-model limitations: Tc and conductivity are calculated withthe different set of parameters and even models; difficult toindividualize character of other impurities; oversimplified treat-ment of spin-fluctuations (MFA) ⇒ limited predictive power

• First-principle studies are quantitative and relevant quantities(Tc, conductivity) are determined in the framework of the sameparameter-free model ⇒ test present physical models in detail

• Other (theoretical) candidates: InAs:Mn, GaP:Mn - lower Tc;GaN:Mn-broad gap; ZnTe:Cr, ZnO:Co (A2B6); XO2: vac,K,Cu(X=Ti,Zr,Hf) or CaO/MgO:vac,C,N (sp-magnetism); CaAs orGaAs:Ti - Ef inside flat band; LiZnAs:Mn - n-type + co-doping;ZrO2:Mn - co-doping, etc.

4

Formalism

• Density functional theory (DFT) in the framework of local spin-density approximation (LSDA): TB-LMTO method, empty spheresat interstitial sites for a good space filling

• Disorder due to substitutional and interstitial Mn-atoms andAs(Ga)-antisites: CPA ⇒ correctly reproduces carrier concsand carrier lifetimes (relevant for both Tc and transport)

Curie temperature Tc

• Two-step approach proposed by Lichtenstein generalized torandom magnetic systems:

1. Total LSDA energies of low-lying excitations are mappedonto random classical Heisenberg Hamiltonian

2. Statistical study of this Hamiltonian: spin-fluctuations re-ducing magnetization with temperature and randomness inMn-impurity positions ⇒ beyond MFA and ALM

5

• Mapping: classical random Heisenberg Hamiltonian

Heff = −∑

RR′

JRR′ ηReR · eR′ ηR′ , JRR = 0

R - site index, ηR=1/0 if mag./non-mag. atom is at ReR - unit vectors (directions of local magnetic moments)JRR′ - exchange integrals between magnetic-atoms

JRR′ =1

4πIm

∫

CtrL

[

∆R(z) g↑RR′(z) ∆R′(z) g↓

R′R(z)]

dz ,

L = (ℓ,m), C - closed contour in the complex energy plane,

∆R(z) = P ↑R

(z) − P ↓R(z) - difference of potential functions,

gσRR′(z) (σ = ↑, ↓) - conditionally averaged Green-functions

evaluated in the framework of the CPA

6

Comments on the mapping:

• adiabatic approximation: local moments - slow, electrons -fast ⇒ justified for rigid moments, transversal spin-fluctuationsdominate while longitudinal (Stoner) excitations are negligible

• magnetic force theorem, Lloyd formula for two impurities, andvertex-cancellation theorem used to derive explicit expressionfor exchange integrals

• accurate evaluation of JRR′ even for large distances• Ab-initio vs models: direct-, indirect-, double-exchange-, and

superexchange-interactions are included in calculated JRR′ ⇒not easy to separate /possible in model approaches

• applicability to systems without 3D-translational symmetry⇒ real-space formulation

• tested successfully for various magnetic systems: conventionaland f-feromagnets, TM magnetic alloys, Heusler alloys, mag-netic overlayers on nonmagnetic substrates, recently for DMS

7

• Experiment: lattice constant increases with Mn-concs

Why: at. radii of RMn/RGa=0.017/0.125 nm? ⇒ native defects

0.556

0.558

0.560

0.562

0.00 0.05 0.10

Lat

tice

para

met

er (

nm)

Impurity concentration

MnGaMniAsGa

• Interpolation formula: (Ga1−x−yMnxAsy)AsMnix

a(x, y, z) = ao + 0.02 xMn + 0.69 yAs + 1.05xMni

• If lattice constant of GaMnAs increases with xMn

⇒ large number of native defects is present!

8

0.05

0.06

0.07

0.08

0.09

0.10

0.11

0.00 0.02 0.04 0.06 0.08 0.10 0.12

impu

rity

form

atio

n en

ergy

(R

y)

x(Mn)

As-antisite

y(As)=0.00

y(As)=0.01

y(As)=0.02

y(As)=0.03

y(As)=0.04

y(As)=0.05

FE =

(

∂

∂x,

∂

∂y,

∂

∂z

)

Etot(alloy|x , y , z)

• Formation energy (FE) of As-antisites/Mn-interstitials decreaseswith increasing content of Mn-atoms⇒ as-grown GaAs:Mn tends to be self-compensated with stronglyreduced number of holes needed to mediate FM-coupling

9

Spin-resolved local density of states of GaAs:Mn and GaN:Mn

-40

-30

-20

-10

0

10

20

30

40

-0.6 -0.4 -0.2 0 0.2

Loc

al D

OS

on m

agne

tic a

tom

s (s

tate

s/sp

in/R

y)

Energy - EF (Ry)

(Ga0.95 Mn0.05)As

(Ga0.95 Mn0.05)N

maj

min

0

5

10

15

20

-0.6 -0.4 -0.2 0 0.2 0.4

GaAs crystal

• Effect of Mn-disorder on the majority valence states• Fermi energy in the gap of minority states: halfmetal

10

GaMnAs - total DOS: effect of disorder

-40

-30

-20

-10

0

10

20

30

40

-0.6 -0.4 -0.2 0 0.2

DO

S (s

tate

s/sp

in/R

y)

Energy (Ry)

FM-(Ga0.95 Mn0.05) As alloys

majority

minority

tot

Mn

• Valence majority states strongly perturbed, minority statesperturbed only weakly

11

0

5

10

15

20

25

-0.6 -0.4 -0.2 0

A* ( k

,E)=

A(k

,E)/

(10+

A(k

,E)

(st

ates

/spi

n/R

y)

Energy (Ry)

FM-(Ga0.95 Mn0.05)As: majority

0

5

10

15

20

25

-0.6 -0.4 -0.2 0Energy (Ry)

Band-structure and

Bloch spectral function

(effect of disorder)

FM-(Ga0.95 Mn0.05)As

reference GaAs-crystal

k=L

Λ

k=Γ

∆

k=X

-0.6 -0.4 -0.2 0

Energy (Ry)

FM-(Ga0.95 Mn0.05)As: minority

k=L

Λ

k=Γ

∆

k=X

-0.6

-0.4

-0.2

0

0.2

L Γ X Γ

12

Exchange integrals: effect of disorder

-30

-20

-10

0

10

20

30

5 15 25 35 45 55 65 75

(d/a

)3 JM

n,M

n (m

Ry)

d/a along (110)-direction

fcc-Cu1-x Mnx x=0.005

x=0.02

x=0.05

-200-150-100-50

050

100150200

10 20 30 40 50 60 70 80 90

(d/a

)3 JM

n,M

n (

mR

y) Pair of Mn in fcc-Cu

13

Exchange integrals: effect of halfmetallicity

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.5 1.0 1.5 2.0

JMn,

X, X

=M

n, N

i (m

Ry)

(d/a)

Mn-Mn

Mn-Ni

NiMnSb alloy

0 0.5 1 1.5 2(d/a)

Mn-Mn

Mn-Ni

Ni2MnSb alloy

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0 2 4 6 8 10 12 14 16 18

(d/

a)3 J

Mn,

Mn (

mR

y)

(d/a) along [110]-direction

NiMnSb Ni2MnSb

Ni1.75MnSb

CURIE TEMPERATURES:

NiMnSb__________________________________

Ni2MnSb

MFA: 1106 K 575 K

RPA: 880 K 360 K

MCS: 910 K 380 K

RPA(*) 852 K 356 K

Exp: 732 K 363 K===============================

MCS and RPA(*)are obtained byneglecting Mn-Ni interactions

14

Exchange intergrals: example for (Ga,Mn)As alloys

-0.5

0

0.5

1

1.5

2

2.5

0.5 1 1.5 2 2.5 3 3.5

JMn,

Mn (

mR

y)

(d/a)

-10

-7.5

-5

-2.5

0

2.5

1 2 3 4 5 6 7 8 9 10 11

ln |

(d/a

)3 JM

n,M

n | (

mR

y)(d/a) along [110]-direction

(Ga0.95-y Mn0.05 Asy)As

y=0.0

y=0.01

• Combined effect of halfmetalicity disorder• Superexchange(SE): EF moves towards unoccupied Mn-band⇒ JMn,Mn

1 is reduced (SE is localized and AFM-like)⇒ JMn,Mn

1 is irrelevant for Tc

15

Effect of reference state: FM vs DLM

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

0.5 1 1.5 2 2.5 3 3.5

JMn,

Mn (d

) (m

Ry)

(d/a)

(Ga95 Mn5)As alloy

FM

DLM

• Scf-approach ⇒ no dependence on reference state (iterativeimprovement of the reference state evaluated at T=Tc

• FM/DLM ⇒ full/no spin-spin correlation (magnetic phase pre-diction vs Tc estimate) ⇒ FM: induced moments (???)

16

Statistical part: Tc - estimates

• Transversal spin-fluctuations: included in RPA (present) orMonte Carlo methods ⇒ beyond MFA !

• Dilution: long-range ferromagnetism for low concentration ofmagnetic impurities with limited spatial extend of JRR′ ⇒magnetic percolation effects

• Dilution included via Monte Carlo sampling ⇒ averaged latticemodel overestimates Tc

• Recent estimates of Tc for well annealed samples which takeinto account magnetic percolation effects (2004):Bergqvist et al : LMTO-CPA + MC for both spin-fluctuationand alloy disorderSato et al : KKR-CPA + MC for both spin-fluctuation and alloydisorderBouzerar et al : LMTO-CPA, RPA for spin-fluctuations and MCfor alloy disorder

17

Local scf RPA (LRPA): basics

Set of equation of motion for the Heisenberg modelin the real space:

(E − heffi ) gij(E) = 2〈ez

i 〉δij − 〈ezi 〉∑

k

Jik gkj(E)

• 〈ezi 〉 − local magnetic ’moment’ at site i

• 〈...〉 − statistical average

• heffi =

∑

i Jij〈ezj〉 is the local effective field

Callen approach (the RPA approximation) is appliedto above set of equations and the classical limit is employed atthe end. The set of equations is solved in the real space for alarge set of magnetic sites ( ≈ 20 000-sites) generated at random.The average over about 100 configurations is employed.

18

kB Tc =1

3Nimp

∑

i

1

Fi, Fi =

∫ Aii(E)

EdE ,

and

Aii(E) = −1

2πIm

gii(E)

λi

, λi = limT→Tc〈ez

i 〉

• large number of shells can be included (not in MCS)

• two-orders of magnitude faster then MCS

• the accuracy of both scf-LPRA and MCS are comparable al-though the MCS is more general (spin-canting)

• clustering effects can be included

• Nscf version: Hilbert & Nolting, 2004 (underestimete Tc

19

Effect of magnetic percolation: toy model

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

x

T cMC (

x) /

T cMC (

x=1)

MC simulation of diluted spin system on a fcc lattice with J ij ≈ 1/ r3

1

2

3

5

8

12

16

20

VCA

20

Tc of annealed (Ga,Mn)As: no compensating defects,magnetic percolation included (Bouzerar et al, 2004)

0 0.02 0.04 0.06 0.08 0.1 0.12XMn

0

50

100

150

200

250

Tc (

K)

TheoryExp. annealed, Edmonds et al.Exp. as grown, Edmonds et al.Exp. Matsukura et al.Exp. Chiba et al.

GaMnAs

21

Tc of (Ga,Mn)As with As-]antisites:−abrupt transition into non-magnetic state? ⇒ spin-canting− Problem: Tc[RPA] ⇒ collinear spins (not in MC-simulations)

0.4 0.5 0.6 0.7 0.8 0.9 10

50

100

150

200

xMn=0.03xMn=0.05xMn=0.08

GaMnAs

γ

Tc(K)

22

-0.5

0

0.5

1.0

1.5

2.0

2.5

1 2 3 4 5 6 7 8 9 10 11

(d/a

)3 JM

n,M

n (m

Ry)

(d/a) along (110)-direction

(Ga0.95 Mn0.05)As

LDALDA+U

(a)

0

100

200

300

0 0.02 0.04 0.06 0.08 0.1

Cur

ie te

mpe

ratu

re (

K)

Mn-concentration

(Ga1-x Mnx) As

MFA

MC-r

MC-r : LDA+U

-2

-1

0

1

2

-5 -2.5 0 2.5

Mn-

LD

OS

(sta

tes/

spin

/eV

)

E (eV)

majority

minority

LDA+U

LDA• Correlations shift Mn-impurity band

to higher binding energies• electron correlations suppress

d-character at Fermi energy• Tc influenced only weakly by

electron correlations

23

Tc of (Ga,Mn)As: experiment (Edmonds et al.)γ = nholes/xMn - compensation ratiostrong dependence of Tc on annealing

0 0.2 0.4 0.6 0.8 1 1.20

50

100

150

200

0 0.2 0.4 0.6 0.8 1 1.20

50

100

150

200

γ

TC(K)

XMn=0.067

As-grown samples

24

Tc of (Ga,Mn)As: effect of As[Ga]-antisitesγ = nholes/xMn - compensation ratio− weak dependence of Tc on annealing: contradicts experiment

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1γ

0

100

200

300

Tc

(K)

RPA-disorderMF-VCA

GaMnAs: xMn=0.05

25

Tc of as-grown (Ga,Mn)As: (Bouzerar et al, PRB 72 (2005))

Effective problem with:

Mn(Ga)

Mn(I)

hole

x_eff=xtot−2x(I)

n_h=xtot−3x(I)

xtot=x(Ga)+x(I)

26

Tc of as-grown (Ga,Mn)As: comments

• Theoretical model is justified by recent ab initio calculations:Masek and Maca (2004), R. Wu (2005)

• Model requires the knowledge of exchange integrals correspond-ing to effective Mn-concentration xeff and effective carrier con-centration neff

• xeff and neff are obtained from experimental nominal Mn-concentration and the level of annealing (as characterized bycompensation ratio parameter γeff )

• Ab initio theory: xeff and neff are not independent !⇒ rigid-band model using frozen potentials for xeff

⇒ co-doping by nonmagnetic atoms used as purely computa-tional tool allows to estimate exchange integrals for a givenxeff and neff selfconsistently

• Examples: ZnGa-doping enhances neff , while SeAs- or AsGa-dopings reduce neff

27

Tc of as-grown (Ga,Mn)As: nominal xMn=0.067(Bouzerar et al, PRB 72 (2005))

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

50

100

150

200 Experimental dataTheory for γ=1Theory x Mn=0.035Theory x Mn=0.05

γ=0.60

γ=0.52

γ=0.50

xMn

Tc(K)

γ=0.80

γ=0.60

γ=0.56 ± 0.08

γ=0.52 ± 0.08

γ=0.80± 0.12

γ=0.85 ± 0.12

γ=0.96 ± 0.14

28

Spin-spin correlation function: Ga95Mn5As, MC study (Bergqvist)

0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

d (a)

G(r

ij)T=160 KT=190 K

• Real-space static spin-spin correlation function (HH model):G(rij) = 〈ei · ej〉, G(0) = 1, G(rij) → 0 for rij → ∞

decreases exponentially• G(rij) ⇒ scattering on thermodynamical fluctuations

29

Spin-spin correlation function: bcc-Fe, Ga95Mn5As - angle-distribution

0 50 100 1500

100

200

300

400

500

600

θ0 50 100 150

0

50

100

150

θN

NN−[0.5 0.5 0][1.5 1.5 0]

• DLM ⇒ rectangular distribution: average NN-angle θavNN=π/2

FM ⇒ Kronecker-delta like distribution at θ = 0 (θaviNN=0)• At T=1.1 Tc: bcc-Fe is close to DLM (θav

NN=75 degs); Ga95Mn5As(θav

NN=38 degs) but the θav at the averaged distance amongMn-atoms is closer to DLM

30

Conductivity: residual resistivity

Ab initio theory of residual resistivity is based on two steps:

• Selfconsistent electronic structure within the LSDA-CPA:the same as that used for mapping to the Heisenberg model

• Residual resistivity formulated in the Kubo-Greenwood linear-response theory with all quantities (matrix elements, Green-function elements) expressed in terms of Kohn-Sham orbitalsand one-electron Hamiltonian

• Disorder-induced vertex corrections are included• Theory neglects the effect of phonons as well as the effect of

thermodynamical fluctuations (spin-spin correlation function)⇒ impurity scatterings dominate the low-temperature limit

• All defects (Mnsubst, Mni, AsGa) contribute to the resistivity• Linear-response theory cannot describe properly transport in

(quasi)-localized impurity bands (hopping conductivity)

31

Bulk residual resistivity: TB-LMTO-CPA

• The conductivity tensor for spin λ (λ = ↑, ↓) (µ = x, y, z):

σλµν ∝ Tr 〈gσ(E+

F )〉Dν〈gσ(E−

F )〉Dµ + vertex part

where

Dµ = [Rµ, S] (µ = x, y, x) is the effective velocity

• present formulation leads to nonrandom velocity operator ⇒vertex part is obtained straightforwardly in the CPA method

• residual resistivity: ρµµ = 1/(σ↑µµ + σ↓

µµ)• Resistivity: concentrated metal alloys ⇒

ρ ≈ 0.1 ÷ 1 × 10−4 Ω cm• Resistivity: DMS (for (Ga,Mn)As with 6-7% of Mn) ⇒

ρ ≈ 3 ÷ 8 × 10−3 Ω cm

32

Example of residual resistivity; AgPd alloy

0

5

10

15

20

25

30

35

40

0 0.2 0.4 0.6 0.8 1

resi

dual

res

istiv

ity (

µΩ c

m)

Ag concentration

fcc Ag-Pd

TB-LMTO

exper.

KKR

33

CPP-layer residual conductivity: TB-LMTO-CPA

• DMS alloys are prepared as thin layers: how thick layer repre-sents bulk conductivity?

• Thickness dependence of the conductivity?

• The Kubo-Landauer conductance for spin σ (σ = ↑, ↓):

Cσ ∝ Tr 〈gσRL(E+

F )〉BσL〈g

σLR(E−

F )〉BσR + vertex part

BσL = iS01 [GL

σ(E+F ) − GL

σ(E−F )]S10 and

BσR = Bσ

L+ are the embedding potentials

• GσL/R(z) are surface Green functions describing non-random

leads L,R ⇒ L|sample|R• Disorder: 2D-lateral supercells (JK/ Kelly, 2001) or the CPA

formalism (Turek, 2006) ⇒ inclusion of disorder-induced vertexcorrections (relevant for the CPP transport!)

34

Residual resistivities in 10−5 Ωm: comparison of conductivitiesevaluated using bulk K-G approach and also by K-L method forCPP-trasport for system: Cr |Ga(1 − x)Mn(x)As[001]| Cr

Ga1−xMnxAsMn-conc ρCPP ρbulk ρexp

0.05-annealed 1.23 ± 0.04 1.20 1.490.06-annealed 1.06 ± 0.02 1.07 1.320.08-as grown 0.88 ± 0.01 0.89 2.87

• Only Mn-impurities are considered• CPP conductivity ⇒ slope of the dependence of conductance

on layer thickness (for thick samples) ⇒ effect of interfaceconductance (leads) is thus eliminated

• annealed samples ⇒ small concentrations of compensation de-fects (As-antisites, Mn-interstitials) agree well with experiment

• as grown samples ⇒ resistivity is underestimated

35

CPP-conductance: Cr‖n−Ga0.92Mn0.08As‖Cr(001)

• Spin polarization P=(C↑-C↓)/(C↑+C↓)• P=100% ⇒ halfmetallic case (only majority channel)• bulk behavior ⇒ for about 20-25 layers and more

36

Majority-spin CPP-conductance: effect of vertex corrections

• Incoherent part (vertex corrections) dominates for more than15-20 layers (strikingly different from bulk case!)

37

Tc and σ: MFA and As-antisites (Lyon, 2003)

0

50

100

150

200

250

300

0 200 400 600 800

Cur

ie te

mpe

ratu

re (

K)

Conductivity (Ω-1cm-1)

exp.

calc.

38

Tc and σ: various compensating defects (APL 91, 2007)

0

50

100

150

200

0 200 400 600 800

Cur

ie te

mpe

ratu

re (

K)

Conductivity (Ω-1 cm-1)

exp: Edmonds (as-grown/annealed)

xMntot = 0.067

theory: AsGa - onlytheory: Mni (yAs=0.0)theory: Mni + AsGa (yAs=0.005)

39

Tc of GaAs:Mn with As-antisites: canted spins

0

50

100

150

200

0 0.005 0.01 0.015 0.02 0.025

Cur

ie te

mpe

ratu

re (

K)

As-antisite concentration y

(Ga0.95-y,Mn0.05,Asy)As

MC

LRPA+canting

LRPA

FM c-FM

100% of M 90% of M

60% of M

• Comparison of various theories of Tc: MC. L-RPA, and L-RPAwith canted spins (Bouzerar & Cepas, 2007)

40

Total energy: effect of As-antisites

0

20

40

60

80

100

120

140

160

180

0 0.1 0.2 0.3 0.4 0.5

Eto

t - E

grou

nd (

µRy)

Order parameter r

(a)

(Ga95-y Mn5 Asy)As

FM DLM

-80

-60

-40

-20

0

20

40

60

J(q)

(m

Ry)

L Γ X W K Γ

(b) Mn-Mn

y=0.0

y=0.01

y=0.015

y=0.02

y=0.025

• Order parameter r=x−Mn/xMn, xMn=x+Mn+x−Mn

41

Spin canting: a simple theory

Heff = −∑

RR′

JRR′ ηReR · eR′ ηR′ +∑

n.n.(RR′)

JAF ηReR · eR′ ηR′

• Competition between long-range FM-coupling and superexchange(AFM) coupling ⇒ spin canting: ab initio mapping includesboth long-range and superexchange parts but does not allowto separate them explicitly

• Energy of two-spins in the field of all other spins:

E = JAF cos(θR − θR′) + hR cos(θR) + hR′ cos(θR′)

hR =∑

R 6=R′′

JRR′′ , hR′ =∑

R′ 6=R′′

JR′R′

and assuming for simplicity: hR = hR′ = h or θR = −θR′

cos(θ) = h/(2JAF) for JAF > h/2

θ = 0 otherwise

42

• For JAF → ∞ are spin anti-aligned and effectively decoupled

• Inherent defect of real-space RPA not present in Monte-Carlosimulations: assumption of collinear spin reference-state fromwhich TRPA

c is derived ⇒ instability with respect to negative(AFM) NN-couplings

• Cure for real systems:

JRR′ ⇒ JRR′ cos(θR) cos(θR′) ,

where angles θR are those from calculated θR-distribution (forspins having at least one nearest neighbor) as obtained fromthe L-RPA method

43

Spin canting: GaAs:Mn with As-antisites

0 20 40 60 800

0.02

0.04

0.06

0.08

0.1

0.4δ(θ)

P(θ)

y=0.0125

y=0.015

θ

Ga0.95-yAsyMnAs

• Distribution of canting angles between Mn-spins (having atleast one nearest-neighbor) evaluated in the L-RPA for T=0 K

44

Spin canting: GaAs:Mn with As-antisites

0 20 40 60 800

200

400

600

800

1000

1200

θ

Num

ber

of c

ount

s

y=0.125

y=0.150

• Distribution of canting angles between Mn-spins obtained fromcalculations based on the Monte-Carlo studies done at T=1K(Bergqvist 2007)

45

Conclusions

• The two-step procedure for exchange interactions and the Curietemperatures is suitable also for as-grown and partially an-nealed (Ga,Mn)As samples

• Curie temperature of as-grown samples is dominated by Mn-interstitials while As-antisites has only negligible effect

• The residual conductivity dominates the low-temperature be-havior of as-grown and annealed (Ga,Mn)As alloys

• All defects present in (Ga,Mn)As: Mnsubs, Mni, and AsGa con-tribute into to residual conductivity

• Curie temperatures and residual conductivities were determinedon equal footing from the same ab initio Hamiltonian withoutany adjustable parameters

• Good agreement between the theory and experiment (Edmondset al., 2002) represents a clear theoretical evidence for carrier-induced ferromagnetism in (Ga,Mn)As alloys

46