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5. Magnetic properties and disorder; recent developments; questions for the future

 Magnetic properties and disorder • Anisotropic RKKY interaction • Effect of

weak

disorder on the magnetization • Disorder and magnetization: Strong disorder approach • Lightly doped, strongly disordered DMS: Percolation picture • Stability of collinear magnetic state, magnetic fluctuations • Spin-charge coupling: possible stripe order • Magnetic order and transport: Resistive anomaly  Recent experiments – new puzzles  Magnetic semiconductors – what next?

Magnetic properties and disorder

Complicated system of coupled carriers and spins

Experiments:

Potashnik

et al.

(2001) Park

et al.

, PRB

, 085210 (2003)

(Ga,Mn)As Mn

-implanted

GaAs:C

 more disorder → straight/convex magnetization curves  annealing seems to

reduce

disorder (curves more mean-field-like)  magnetization for

0

is significantly less than saturation

Anisotropic RKKY interaction

Reminder: local moment polarizes carrier band, other local moments see oscillating magnetization ( “integrate out the carriers”) RKKY interaction for local carrier-impurity exchange

pd and parabolic band (Lecture 3) is isotropic in real and spin space isotropic in real space isotropic in spin space  Zaránd & Jankó, PRL

, 047201 (2002): local

pd →

and spherical (4-band) approximation with spin-orbit interaction RKKY isotropic in real space, highly anisotropic in spin space → frustration  Brey & Gómez-Santos: PRB

, 115206 (2003): non-local

pd → with Gaussian form, realistic 6-band weakly anisotropic in real and spin space

Hamiltonian

Attempt at more realistic description: C.T. & MacDonald, PRB

, 155206 (2005) (a) Start from Slater-Koster tight binding theory for

GaAs

with spin-orbit coupling [Chadi (1977)] 16 bands (b) Incorporation of

Mn

d-orbitals hybridization with

GaAs

sp-orbitals: photoemission (Okabayashi) ,

ab-initio

(Sanvito) interactions: Hubbard

and Hund‘s 1st rule

J H

to preserve spherical symmetry of d-shell in real and spin space: Parmenter, PRB

, 1273 (1973) – 5-orbital Anderson model does

not

J H

for single

Mn

Method: Chao

et al.

, PRB

, 3453 (1978) , related to Schrieffer-Wolff transformation in (

+

), out (

–

) unitary transformation does not change the physics  expand in   choose

(hermitian) to make linear order vanish  approximation: truncate after 2nd order  set 

= 1

 approximation: project onto ground state with

= 5

= 5/2

E F

projected energies:

Inserting one finds

dependent ↔ nonlocal in real space

spin scattering

with

d

5 !

d

4 For small

´

only 

=



´ = p

x,y,z

 antiferromagnetic  correct order of magnitude

d

5 !

d

(d) RKKY interaction  two

Mn

impurities at

0

and

→ canonical transformation  integrate out the carriers

J G G J

2  oscillating  ferromagnetic at small

 highly anisotropic in real space (from bands)  anisotropic in spin space for larger

(spin-orbit)

Can the spin anisotropy lead to non-collinear magnetization reduction of average magnetization at

= 0

and (Zaránd & Jankó) ?

For magnetization in

direction typical effective field in transverse (

) direction is given by With our results for

J ij

→ this is small even at no strong non-collinearity

= 0.01

or reduction due to

this

mechanism Complementary

ab-initio

approach: Do LDA (

+

) – usually without spin-orbit coupling – for supercell with 2

Mn

→ impurities in various positions, extract

(

)

from total energy highly anisotropic in real space, isotropic in spin space (no spin-orbit!) Typically overestimates

(

)

and thus

T c

(while we underestimate it)

Effect of weak disorder on the magnetization

Results for magnetization and

T c

in Lecture 3 did not include disorder Inclusion of disorder: Coherent potential approximation (CPA) Takahashi & Kubo, PRB

, 153202 (2002); also Bouzerar, Brey etc.

impurity sites  only

local

Coulomb disorder potential (or zero in most other works)  (local

, simple semicircular DOS assumed)

CPA:

Approximate true scattering by multiple scattering at a single impurity embedded in an effective medium – good for low impurity concentration Cannot describe localization, difficult to treat extended scatterers (Coulomb)

Typical for CPA and DMFT calculations with point-like impurities:

T c

decreases for high hole concentrations (weak compensation)

= 0.05

Origin: Impurity band broadens for higher spin polarization (since it is mostly due to large

) → spin polarization unfavorable for band filling &

1/2

1 hole per

Mn (Ga,Mn)As

: Takahashi & Kubo Problem: Always requires unphysically large

to obtain reasonable

T c

, since no long-range Coulomb potentials → incorrect impurity-band physics

Disorder and magnetization: Strong disorder approach

Zener model with potential disorder: Method: mean-field theory with disorder convex → concave for less disorder

Lightly doped, strongly disordered DMS: Percolation picture

For low concentrations

of magnetic impurities in

III-V

DMS Kaminski & Das Sarma, PRL

, 247202 (2002); PRB

, 235210 (2003), Alvarez

et al.

, PRL

, 277202 (2002), etc.

 at

= 0

: ferromagnetism if aligned clusters percolate  at

> 0

: two clusters align if the weak coupling

´ → between them is & express by

k B T T

-dependent BMP radius For lower

the aligned region growths and eventually percolates at

T c

weak link

´ (Kaminski & Das Sarma) Exponentially small for small hole concentration, but not zero (since no quantum fluctuations) – compare VCA/MF:

T c

n h

1/3

n i

Global magnetization is carried by sparse cluster at

T c

: small Percolation theory: Kaminski & Das Sarma (2002) Monte Carlo simulations: Mayr

et al.

, PRB

, 241202(R) (2002)  Highly convex magnetization curves (upwards curvature)  MC: magnetization at

0

strongly reduced compared to saturation For

clustered

→ defects: large ordered clusters exist for

T c

, percolate at

T c

more rapid (Brillouin-function-like) increase of bulk magnetization

Stability of collinear magnetic state, magnetic fluctuations

Question: stable Is the collinear state found by approximate (mean-field) methods against magnetic fluctuations? In particular with disorder?

Expand energy around mean-field solution → density of states of magnetic excitations Schliemann & MacDonald, PRL

, 137302 (2002) etc.: spin disorder, no Coulomb disorder  DOS at negative energies: state not stable  …collective spin excitations involving many spins (high participation ratio) Schliemann, PRB

, 045202 (2003): 6-band

model  solution not even stationary Goldstone mode

C.T., J. Phys.: Cond. Mat.

, R1865 (2003): Spin

and

Coulomb disorder, parabolic band  DOS at negative energies: state not stable  clustered defects: DOS shifts away from zero → stiffer magnetic order Collinear state unstable due to anisotropic magnetic interactions – but argument from RKKY interaction suggests that the deviation is small

Spin-charge coupling: possible stripe order

C.T., cond-mat/0509653 Carrier-mediated ferromagnetism → strong dependence of magnetism on hole concentration in

(In,Mn)As

(Ga,Mn)As

Landau theory for magnetization

carrier concentration 

: and excess Ohno

et al.

(2000) magnetization charge with Introduce electrostatic potential  : for p-type DMS

Spin- and charge-density waves (stripes) lowest energy periodic, anharmonic magnetization and carrier density for typical solutions

Magnetic order and transport: Resistive anomaly in dirty itinerant ferromagnets

Zumsteg & Parks, PRL

, 520 (1970)

Ni

Potashnik

et al.

, APL

, 1495 (2001)

(Ga,Mn)As

(

)

(

)

T c dR

/

dT T c

Theories for paramagnetic regime,

T c

(1)

de Gennes & Friedel, J. Phys. Chem. Solids

, 71 (1958)  scattering from magnetic fluctuations  close to

T c

: critical slowing down → static, elastic Approach equivalent to:  perturbation theory, similar to inverse quasiparticle lifetime, but transport rate involves factor  anomaly from small momentum transfers

Ornstein/Zernicke (sharp maximum)

(2)

Fisher & Langer, PRL

, 665 (1968) disorder damping for large length scales ↔ small

: electronic Green function decays exponentially on scale

(mean free path)  no de Gennes-Friedel singularity from small

 weak singularity from large

2

k F

, have to go beyond Ornstein/Zernicke/Landau theory:

small

anomalous specific-heat exponent Equivalent to Boltzmann equation approach (Lecture 4) , disorder and magnetic scattering treated on equal footing

Problem:

fails for magnetic correlation length 

(

)

(mean free path), magnetization variations are explored by diffusive carriers

Beyond the Boltzmann approach C.T., Raikh & von Oppen, PRL

, 036602 (2005) (a) Description of transport on large length scales (phase coherence length) 3D resistor network  magnetization ~ constant in cells  conductivity of network:  spatial average h

…

i  large system: equivalent to average over • quenched disorder • magnetization (thermal)

(b) Two spin subbands: ↑ , ↓ UCF UCF Correlation function spin ↑ , ↓ carriers have different Fermi energies but see same disorder universal conductance fluctuations (UCF)

is a scaling function of

= eff. Zeeman energy

£ (Stone 1985, Altshuler 1985, Lee and Stone 1985)

diffusion time

Correlations decrease with increasing Zeeman energy (c) With spin-orbit coupling: realistic case is a scaling function

(

)

= eff. Zeeman energy

spin-orbit time

, increases by factor of 2 in strong effective Zeeman field (d) Typical magnetization assuming Gaussian fluctuations: long-wavelength modes

with scaling function For Gaussian fluctuations: maximum at

T c

Beyond Gaussian fluctuations:

Stronger singularity

than in Fisher/Langer and de Gennes/Friedel theories

Condition:

Transport disorder-dominated at

T c

(low

T c

, strong disorder) –

(In,Mn)Sb

Recent experiments – new puzzles

Ferromagnetism in superdilute magnetic semiconductors

Dhar

et al.

, PRL

, 037205 (2005); Sagepa

et al.

, cond-mat/0509198 (

GaN:Gd

with

= 8

10

-8 to

Gd

concentrations from

2

10

-4 )

7

10

15 to

2

10 cm

-3  wurtzite structure  formal valence

Gd

3+ : isovalent, configuration

4d

7 , local spin

= 7/2

 high concentration of native donors (

N

vacancies) expected Observations:  room-temperature ferromagnetism (

T c

360K

for

= 8

10

-8 )  highly insulating field-cooled sweep zero-field-cooled

 giant magnetic moment per

Gd

for low

 effective field acting on VB is reversed

absolute moment per Gd Magnetization must be carried by “something else” – native defects?

How does very little

Gd

induce magnetic ordering?

“

d

0 ferromagnetism” in oxides

Coey

et al.

, Nature Mat.

, 173 (2005)  ferromagnetism in

HfO

2 (no partially filled shells?)  magnetization extraplotates to nonzero value for strongly diluted DMS

Sn

x X x

O

Two species of substitutional

Mn

Kronast

et al.

(Ga,Mn)As

: XAS, XMCD results

(BESSY II Collaboration), submitted 

Mn

2+ with

~ 3d

5 configuration, large moment, orders magnetically 

Mn

3+ with

~ 3d

4 configuration, strong valence fluctuations, large moment, does not order Questions: What mechanism lifts the

d

5 Why does high-spin

Mn

3+ !

d

4 transition by several eV?

not participate in the ordering?

Magnetic semiconductors – what next?

Goals for DMS experiments:

 control of growth dependence, reproducability  unconventional DMS (oxides etc.), concentrated magnetic semiconductors  other magnetic probes: NMR/NQR/  SR and neutron scattering  dynamics: optical pump-probe and noise  crossover to antiferromagnetism, superconductivity, QHE…

…and theory:

 study of crossover between weak doping (BMP‘s) and band picture  unconventional DMS (oxides etc.): different mechanisms?

 better

ab-initio

methods to get hydrogenic impurity level of

Mn

GaAs

 detailed simulation of DMS growth to find defect distribution  selfconsistent theory of scattering and carrier-mediated magnetism

DMS/nonmagnetic semiconductor heterostructures

Transport, disorder & magnetism 1.

delta-doped layers, single layer

vs.

superlattice metallic or insulating?

magnetic properties?

FNF structure: RKKY coupling between layers, control by gate voltage – unlike metal structures 3.

DMS quantum dots • many local moments • few local moments 4.

DMS/nonmagnetic interfaces, spin injection MC simulation of interdiffusion

cf.

experiment: Kawakami

et al.

, APL

, 2379 (2000)

Electronic correlations & quantum critical points

 electronic correlations:

(Ga,Gd)N

 at least two quantum critical points: • ferromagnetic end point • metal-insulator transition  …with overlapping critical regions  Griffiths-McCoy singularities: rare regions relevant for poperties Galitski

et al.

, PRL

, 177203 (2004)

Vision:

DMS are ideal materials to study the interplay of disorder and electronic correlations: both are important and can be tuned Possible parallels to cuprates: indications that dopand-induced disorder is important in cuprates, McElroy

et al.

, Science

309

, 1048 (2005)

Diluted Magnetic Semiconductors

Prof. Bernhard Heß-Vorlesung 2005

I am grateful for discussions and collaborations with G. Alvarez, W.A. Atkinson, M. Berciu, L. Borda, G. Bouzerar, L. Brey, H. Buhmann, K.S. Burch, S. Dhar, T. Dietl, H. Dürr, S.C. Erwin, G.A. Fiete, E.M. Hankiewicz, F. Höfling, P.J. Jensen, T. Jungwirth, P. Kacman, J. König, J. Kudrnovský, A.H. MacDonald, L.W. Molenkamp, W. Nolting, H. Ohno, F. von Oppen, C. Paproth, M.E. Raikh, F. Schäfer, J. Schliemann, M.B. Silva Neto, J. Sinova, C. Strunk, G. Zaránd and others