Magnetism in diluted semiconductors

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Magnetism in diluted semiconductors
Georges Bouzerar Institut Néel -MCBT CNRS

• I- Introduction.
• II- Theory for disordered Heisenberg models.
• III- Ab initio based studies for diluted magnetic
  semiconductors.
• IV- Magnetic excitations and phase diagram of the
  diluted III-V compound GaMnAs.
• V- Model approach the non perturbative V-Jpd
  model.
• VI- Conclusions.

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Collaboration
Josef Kudrnovsky (Institut of Physics-Prague) Rich
ard Bouzerar (University of Amiens) Olivier Cepas
(CNRS Paris/ Neel Institute CNRS) Lars Bergqvist
(Jullich)
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I- Introduction
Magnetic semiconductors

• Early 60s (non dilute magnetic materials) EuO
  (Tc70K), EuS (17K) and the spinels as CdCr2S4
  (84 K) or HgCr2Se4 (106 K)
• 70s and 80s Diluted Magnetic Semiconductors
• II-VI materials CdTe, ZnTe and ZnSe) II ? Mn2
  (isoelectric)
• easy to incorporate Mn
• direct Mn-Mn AFM exchange interaction
• ?PM, AFM, or SG (spin glass) behaviour
• codoping (holes) has led to FM for TC lt 3K.
• Late 80s III-V materials MBE ? (In,Mn)As films
  on GaAs substrates
• FM with TC35 K
• Late 90s MBE ? (Ga,Mn)As films on GaAs
  substrates FM with
  TC110 K

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DMS based materials are promising candidates for
Spintronic devices.

• Idea combination of semiconductor technology
  (charge) with magnetism (spin) should give rise
  to new devices
• Long spin-coherence times ( 100 ns) have been
  observed in semiconductors
• Use the charge to Control the magnetism (Write
  bits through electric current)
• Quantum computing??

An exemple of spintronic application Magnetic
Random Access Memory (non volatile) based on GMR
(nobel prices of Fert and Gruenberg)
Need for materials with
relatively high Curie temperature (beyond
room temperature).
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Understand experimentally and theoretically which
physical quantities control TC

• Role and effects of the disorder?
• Importance of thermal fluctuations?
• Nature of the magnetic couplings?
• Influence of the band structure of the material?
• Dependence with the carrier density and magnetic
  impurity concentration?
• Why carrier codoped II-VI compounds have much
  smaller TC than III-V materials?
• Effects of intrinsic defects (compensation)?
• Disagreement between different theoretical
  approaches?

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1- III-V compounds.
1- GaMnAs is one of the most studied diluted
III-V material , it exhibits relatively high TC
xMn5 , TC110 K and xMn7.5 , TC175 K. 2-
GaMnN is more controversial some pretends to
observe paramagnetism and other ferromagnetism
with very high Curie temperature.

• In III-V materials the substitution of Ga3 by
  Mn2 introduces simultaneously a localized spin
  S5/2 and an itinerant carrier (hole) in the
  valence/impurity band.
• The exchange between impurities is of
   generalized RKKY-type  (polarization of the
  gas of carrier effects of resonances). Standard
  RKKY couplings are inappropriate to describe
  ferromagnetism in these materials

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2- Disorder and Compensation defects.

• 1- First source of disorder the random position
  of the Mn impurities and possibly formation of
  inhomogeneities.
• 2- Second source of disorder defects of
  compensation.
• These defects appear during the MBE growth
• of the samples. These defects have a drastic
• effects on both TRANSPORT and MAGNETISM.
• There are two types of such defects
• As anti-sites which a priori only affect
• the density of carrier.
• Mn(I) affect both the hole density and
• the density of magnetically active Mn.

Trapped holes
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3- Some experimental results
GaMnAs with Highest TC 172 K
T dependence of the Magnetization and inverse
susceptibility (Wang et al. 2005)
Curie temperature and hole concentration (p) as a
function of the impurity concentration. Full
symbol (metallic samples) (Matsukura et al. PRB
1998).
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Resistivity at zero field as a function of
temperature for various annealing time (Potashnik
et al. 2001).
Resistivity at zero field as a function of T
Curie temperature (Matsukura et al. PRB 1998).
Inset effect of H.
isolant
metal
Resistivity after annealing Hayashi et al., APL
78, 1691 (2001)
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Peak in the conductivity is inconsistent with
Valence Band picture
before
after
Optical conductivity as a function of frequency w
for various concentration.The samples with
x0.052 are as grown and annealed (Burch et al.
PRL 2006).
Magnetization and charge density profile in
GaMnAs before and after annealing (Kirby et al.
PRB 2004)
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Annealing dependence of magnetization curve
Mathieu et al., PRB 68, 184421 (2003)
Potashnik et al., APL 79, 1495 (2001)

• magnetization curves change straight/convex
  (upward curvature) ? concave (downward curvature)
• degradation for very long annealing
  (precipitates?)

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4- Predictions based on Zener mean Field theory.
Includes a realistic bande structure for the host
material
Dietl et al. Science 2000 (cited 3000 times!!)
Is the Zener mean field theory really appropriate
to describe ferromagnetism in diluted magnetic
semiconductors??
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II- Theory for disordered Heisenberg model.
The theory should allow the study the magnetic
properties Curie temperature, magnetic
excitations spectrum, local magnetizations, ...)
of a system of Nimp interacting magnetic
impurities with spin S (quantum or classical)
randomly distributed on a given lattice.
pi1 if the site is occupied otherwise 0

• The approach should be able to
• Treat disorder effects beyond a simple Virtual
  Crystal Approximation (VCA).
• Treat properly thermal and quantum fluctuations
  beyond Mean Field (MFT).

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1- The Mean Field-VCA theory

• MFT has several shortcomings.
• Underestimates both the effects of thermal
  fluctuations and disorder.
• Collective excitations are not included.
• Violate both theorems of Mermin-Wagner and
  Goldstone (finite Curie temperature for 1D et 2D)

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2- The Random Phase Approximation -VCA theory
RPA improvement of thermal and transverse
fluctuations

• Includes collective excitations.
• Fulfils both theorems of Mermin-Wagner and
  Goldstone (in the absence of anisotropy TC0
  for1D and 2D)

Very good agreement with exact Monte Carlo
calculations in the absence of disorder (x1).
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3- Proper treatment of disorder and thermal
fluctuations

• RPA-CPA (Coherent Potential Approximation) (G.B
  P. Bruno PRB 2002) effective medium theory
  
• improvement of the disorder
  treatment and thermal fluctuations.
• Does not include short range
  correlations, clusters,.
• Monte Carlo calculations exact but too heavy et
  very costly in CPU, especially when the exchange
  couplings are long-ranged.

Generalization of RPA (accurate for clean
systems) exact treatment of the
disorder effects the Self-Consistent Local RPA
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Thermodynamic and dynamical properties within
the real space SC-LRPA.

Choose a disorder configuration positions of
impurities.



convergence
Determination of the thermodynamic and dynamical
properties.

• For a given temperature T, after convergence one
  can calculate
• The distribution of the local magnetizations mi
  (T) (Reflectrometry measurements).
• The spectral function A(q,w,T) and magnetic
  excitations (Inelastic Neutron Scattering exp..
• The spin-spin correlation functions lt Si .Sj gt
  and magnetic susceptibility.
• Etc.

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The Curie Temperature. Within SC-LRPA one can
derive a semi-analytical expression for the Curie
temperature in disordered ferromagnetic systems.
Fi depends on both the eigenvalues and
eigenstates of the non symmetric effective
Hamiltonian Heff
This expression is the generalisation of RPA to
disordered ferromagnetic systems!
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III- Ab initio based studies for diluted magnetic
semiconductors

• Allow the determination of the possible
  good candidates (not enough to find the materials
  with the highest TC
• Allow a direct quantitative comparison, we
  choose an approach with no adjustable
  parameters.
• Includes in the most realisitic way the
  band structure and non perturbatively the effects
  of the substitution of an impurity (d orbitals
  and hybridization exactly treated).
• Material specific difficult to draw
  general conclusion.
• Only valid for the Ground state properties
  (T 0 K)

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1- Some ab-initio results
Which possible diluted ferromagnetic
semiconductors ?
II-VI materials
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III-V materials
GaN
GaP
GaAs
GaSb
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Density of states in (Ga,Mn)As with 3.125 Mn
LSDA
d orbitals

• LSDA d-orbital weight at EF, VB top and CB
  bottom absent in photoemission experiment and
  SIC calculations.

LSDAU phenomenological procedure to improve
the tretament of elevtron-electron correlations
LSDA U

• Mn d-orbital weight shifted away from EF,
  better agreement (especially for the feature at
  -4 eV)
• similar results from other methods going
  beyond LSDA GGA, SIC-LDA

Wierzbowska et al., PRB 70, 235209 (2004)
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Couplings in diluted magnetic semiconductors
(data from J.Kudrnovsky)
RKKY couplings
Not RKKY couplings!
Zn1-xCrxTe
Ga0.95Mn0.05As
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How to proceed?
The two step approach calculations (TSA)
1- Determination of the Ground state properties
and exchange integrals Jij from first principle
methods (here LDA Tight Binding -Linear Muffin
Tin Orbitals) 2- Proper treatment (Monte Carlo
or SC-LRPA) of (i) thermal fluctuations and (ii)
disorder in the treatment of the disordered
effective Heisenberg Hamiltonian to calculate the
magnetic properties.
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2- Role of the disorder and thermal fluctuations.

• The Mean field VCA underestimate the
  fluctuations, and overestimate strongly the Curie
  temperature (for 5 Mn TC300 K).
• The Ising calculations Effects of disorder
  included but not the transverse fluctuations lead
  to non realistic TC.
• The SC-LRPA lead to much smaller values for the
  Curie temperature.
• Good agreement with Monte Carlo for 5 Mn and
  the same couplings (see next slide) 130 K 1 et
  103 K 2.

Disorder but no transverse fluctuations
Mean field VCA
SC-LRPA
1 L. Bergqvist et al. Phys. Rev. Lett., 93
137202 (2004) 2 K. Sato et al. Phys. Rev. B, 70
201202 (2004)
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3- Monte Carlo vs SC-LRPA.
GaMnAs
LSDA
LSDAU
Courtesy to L. Bergqvist
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4- Comparison between experiment and theory for
GaMnAs.

• We observe a very good agreement between
  experiment and theory for optimally annealed
  samples (calculations are done for non
  compensated systems).
•  As-grown  samples exhibit much smaller Curie
  temperature strong reduction due to the presence
  of compensating defects.
• The calculations provide the maximum value of the
  Curie temperature for each concentration of Mn.
• Below 1 no ferromagnetism (percolation
  threshold ).

As grown samples
Remark (1) The TB-LMTO calculations to estimate
the couplings and (2) the treatment of disorder
and fluctuations within SC-LRPA are reliable.
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5- Effects of compensating defects.

(Data from Edmonds et al. Nottingham). Variation
of TC for a fixed total density of Mn2after
different annealing treatment for Ga1-xMnxAs.
The total Mn concentration is
a- Experimental results.
nh (1020 cm -3) TC (K) détails
4.1 /- 0.50 67 As grown
9.0 /- 0.90 113 6hrs_at_180 oC
10.0 /- 1.0 126 15hrs_at_180 oC
13.5 /- 2.0 143 140hrs_at_180 oC
4.6 /- 0.50 76 As grown
Conclusion TC varies strongly from 67 K to 143
K depending on the annealing conditions.
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We assume that Mn(Ga) density is fixed x and
we vary the density y of As anti-sites
(Ga1-x-yMnx Asy )As . As on Ga sublattice is
double donor, the density of holes is,
b- Effects of As anti-sites.
We introduce the reduced variable

• Weak variation of TC with g for
• g gt 0.60
• For g lt 0.55 the ferromagnetism becomes
  unstable, the super-exchange dominate in the
  nearest neighbour exchange (frustration??? See
  following).
• Simple theory RKKY MF-VCA predicts that TC x
  4/3 g 1/3 inconsistent with our results and
  experiments
• We can not explain the experimental results by
  assuming that As anti-sites is the dominating
  mechanism of compensation.

??
Instability of Ferro GS
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c- Effects of Mn interstitials (Mn(I)).

• It is energetically more favourable for Mn(I)
  to be located near Mn(Ga). The coupling is
  strongly antiferromagnetic reduction of Mn(Ga)
  magnetically active.
• Mn(I) is a double donor reduction of the
  carrier density.
• After annealing Mn(Ga)-Mn(I) pairs break and
  release carriers and increase active Mn(Ga).
• The problem reduces to an effective model of
  interacting
  active Mn
• with a hole density

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d- Comparison between experiment and theory.
For each sample to calculate TC we use the
following expressions

• Samples with highest TC are in very good
  agreement with the calculations done for g 1.
• For  as-grown  samples we also reproduce Tc by
  taking into account that g lt 1.
• This study confirms that the dominating mechanism
  of compensation is due to Mn interstitials.
• The theoretical curve (g 1) allows to give a
  good estimate value for TC for each sample
• TC649 ( xeff - 0.0088)1/2

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6- Comparison between different III-V materials.

• GaMnAs is the III-V material with the highest
  Curie temperature.
• Instead of the theoretical predictions based on
  MF-VCA calculations (Dietl et al.) that
• TC 600 K for xMn 6 .GaMnN has a very small
  TC. In spite of a very strong nearest neighbour
  coupling (10 times that of GaMnAs)
• InMnAs exhibits intermediate Curie temperatures

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IV- Magnetic excitations spectrum and phase
diagram of GaMnAs
The magnetic spectral function is directly
accessible via Inelastic Neutron Scattering
experiment , it reads,
The index (c) denotes the configuration average!
The eigenvalues (magnon mode) and respectively
left and right eigenvectors are ?a , ? La ? ,
? Ra ? .
For more details see G.B. ,EPL (2007).
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The Mn density is x0.03

• Excitations are well defined in a very small
  region around the G point
• ?(q) D q 2 in the limit of long wave length
  (Goldstone mode)
• Second moment ?(q) Cq it does not correspond
  to the real width of the excitation.

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The Mn density is x0.03
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• Both TC and the stiffness vanishes below the
  percolation threshold.
• In contrast to non disordered systems the ratio
  D/Tc is not constant.
• Values of D are as large as those reported in
  metallic manganites as LaCaMnO3.
• The first moment of A(q, w) does not correspond
  to the magnetic excitation.

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Competition between extended ferro couplings and
short range Superexchange.

• The ground state gets canted above a critical
  value of the SE coupling!
• The unpaired spins remains almost uncanted.
• In the limit of very large JAF , the pairs are
  disconnected x---gt xeff

Distribution of canting angle of the ground state
for two different concentration of holes.
G. B. R. Bouzerar and O. Cepas, PRB (R. C.)
(2007).
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Phase diagram of GaMnAs.
This phase may explain inconsistency with
magnetization measurements at low hole
concentration
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V- Model study the V-Jpd model
Spins operators si carrier Si magnetic
impurity
tij t for (i,j) nearest neighbour
Crucial term source of scattering (Coulomb
potential) due to the substitution of a cation by
another one with a different charge ViV if i is
occupied by a magnetic impurity.
V controls the impurity band position.
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Procedure for the study of transport and
magnetism
Choice of a configuration of disorder
and set of parameters V and Jpd, x
Diagonalization of the itinerant carrier
Hamiltonian in each spin sectors
Calculation of the Mn-Mn magnetic couplings
Transport properties
Magnetic properties
A.I. Liechtenstein et al., PRB (1995)
R. Bouzerar et al. EPL 2007
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1- The damped RKKY model (V0).
damping is introduced
? MF-VCA strongly overestimates the true Curie
temperature
? SC-LRPA -Variation of Tc is non
monotonic -Even with the relatively large damping
the tability region of ferromagnetim is very
narrow (inconsistent with experimental results)
? No finite size effects (the same results for
both 800 or 2000 impurities dilute on the fcc
lattice).
R. Bouzerar et al. PRB 2006
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2- The effects of the cutt-off.
? An average over at least 200 configurations of
disorder has been performed.
?Pure RKKY regime the region of stabilityfor
the carrier is reduced to 10 of the magnetic
impurity.
?Even for a relatively strong damping the region
of stability remains narrow..
The standard RKKY model is inappropriate to
describe ferromagnetism in dilute magnetic
semiconductors as GaMnAs or GaMnN.
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3- The non perturbative treatment of the V-Jpd
model
DOS as a function of V at fixed JpdS5t
Magnetic couplings (similar to ab initio) (a)
as a function of hole concentration (b) as a
function of V at fixed JpdS5t and fixed hole
density
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4- Curie temperature (in units of t) at fixed
V-2.4t.
? An average over at least 200 configurations of
disorder has been performed.
? Significant increase of the stability region
of the ferromagnetic phase.
? Even at low carrier density we do not
have We find the opposite TC reduces with
incerasing Jpd!.
The Jpd model captures the essential physics of
the diluted magnetic semiconductors. Note the
crucial role of the onsite potential!
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5- Comparison between TSA and full Monte Carlo
treatment of the V-Jpd model .
(a)
(b)
(a) and (b) hole density per defect is
respectively nh/x0.125 and 0.25

• Tc (SC-LRPA) exhibits a maximum in both cases
• At small values of within Sc-LRPA
• For ,Tc vanishes.
• Disagreement with the full Monte Carlo
  simulations
• (in the inset), for the amplitudes!

for details see R. Bouzerar and GB, submitted
(2009) condmat0902.4722
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(c) MF-VCA vs SC-LRPA

• Tc within MF-VCA is much larger and saturates for
  JS?8
• We observe only for small values of Jpd an
  agreement between both methods.
• For larger hole density the agreement for small
  values disappear (RKKY oscillations kills the
  ferromagnetic phase!)

(c)
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6-Origin of the disagreement between full Monte
Carlo and the two step approach?
Finite size effects on TC (within SC-LRPRA) an
average over at least 200 configurations of
disorder for the largest systems and few
thousands for the smallest systems is done.
See R. Bouzerar and GB, submitted (2009)
condmat0902.4722
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Magnetic couplings as a function of the distance
r for different system sizes (huge
effects near the typical distance between
impurities)
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Calculated distributions or the Curie
temperatures (huge fluctuations for the small
systems! )
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The case of non dilute systems the double
exchange model in presence of on-site disorder?
The TSA versus the full Monte Carlo calculations
(a) no disorder W0 and (b) the disordered case
The on site potential Vi is chosen randomly in
-W/2,W/2 , W is the disorder strength
In contrast to the dilute case, a very good
agreement is obtained for the non dilute case
even in the presence of disorder!
see GB and O. Cepas ,PRB 2007
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Thank you