Unbiased Numerical Studies of Realistic Hamiltonians for Diluted Magnetic Semiconductors.

1
Unbiased Numerical Studies of Realistic
Hamiltonians for Diluted Magnetic Semiconductors.

• Adriana Moreo
• Dept. of Physics and ORNL
• University of Tennessee, Knoxville,
• TN, USA.
• Collaborators Y. Yildirim, G. Alvarez and
  E.Dagotto.

Supported by NSF grants DMR-0443144 and 0454504.
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Motivation Spintronics

• Charge Devices
• Transistors
• Lasers
• CPU, processors

• Magnetic Devices
• Non-volatile memory
• Storage
• Magneto-Optical devices

• Spin stores information
• Charge carries it

Electron has spin and charge

• New Possibilities
• Spin transistor
• High spin, high density nonvolatile memory
• Quantum information computers using spin states

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Mn Doping of GaAs
Ohno et al., 1996. (x.035, Tc60K)

• Mn replaces Ga
• Holes are doped into the system but due to
  trapping the doping fraction p tends to be
  smaller than 1.
• Random Magnetic impurities with S5/2 are
  introduced.
• x10 is about the maximum experimentally
  achieved doping.
• xgt2 is necessary for collective FM.
• A metal-insulator transition occurs at x3.5.

S5/2
d orb.
4
Experimental Properties
0.02ltxlt0.085 Tc increases with p (Ku et al.)
Ohno et al, X.053
Okabayashi et al., PRB (2001)
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Different approaches appear to be needed in
different regimes
Correct interaction, phenomenological band
Impurity Band Approach
J
Max Tc
Bhatt, Zunger, Das Sarma,
RKKY collective
Correct band structure, approx. interaction
Valence Band approachDietl, Mac Donald
1
0
0.1
Carrier Density (p)
Mac Donald et al. Nature Materials 05
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Numerical Calculations

• First unbiased MC calculation considered one
  single orbital in a cubic lattice. (Alvarez et
  al., PRL 2002).
• Unifies the valence and impurity band pictures.

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Two Band Models

• MC and DMFT (Popescu et al., PRB 2006).
• Tc maximized by
• Maximum overlap between bands
• p gt1
• J/t 4 when impurity band overlaps with valence
  band.

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New Approach Numerical simulation of a
realistic Model

• Bonding p orbitals located at Ga sites will
  provide the valence band.
• 6 degrees of freedom per site 3 orbitals px,py
  and pz and 2 spins.
• 3 nearest neighbor hopping parameters from tight
  binding formalism.

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Hoppings values
Values obtained from comparison
with Luttinger-Kohn Model for III-V SC.
6 bands J3/2, j1/2
Similar results obtained by Y. Chang PRB87
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4 band approximation

• Keep states with j3/2.
• mj/-3/2, /-1/2.

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Results
Tc well reproduced in metallic regime. Longer
runs being performed to improve shape of curve
(work in progress).
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Tc increases with p
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What value of J?
Tc in agreement with experiments.
Tc is very low.
Metallic regime corresponds to valence band
picture.
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Density of States and Optical Conductivity.
Metallic behavior. Drude peak
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Splitting of majority and minority bands.
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How high can Tc be?
Dietl et al., Science (2000) Mean Field approach.
Assuming
(Okabayashi et al., PRB (1998))
Tc is expected to increased for materials with
smaller a, i.e., larger J such as GaN.
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Conclusions

• Numerical simulations of models in which valence
  band holes interact with localized magnetic spins
  provide a unified answer to a variety of
  theoretical approaches which work for particular
  regimes.
• Mn doped III-V compounds appear to be in the
  weak coupling regime.
• Is room temperature Tc possible? (Ga,Mn)N seems
  promising.
• Work in progress
• Obtain impurity band and observe MIT as a
  function of x for fixed J.
• 6 orbitals model being studied.

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Band Structure of GaAs
Heavy holes
Valence Band
Light holes
Split-off
Williams et al., PRB (1986)
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Luttinger-Kohn Valence Band for GaAs
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Theoretical Pictures

• a) Valence Band Holes MacDonald, Dietl, et al.
  (Zener model). Mean field approaches of realistic
  models.
• b) Impurity Band Holes Bhatt, Zunger, Das Sarma
  et al. Numerical approaches with simplified
  models.

a)
b)
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Impurity Band Picture

• Chemical potential lies in impurity band.
• Disorder plays an important role.
• Band structure depends strongly on x.
• Accurate at very small x.
• Supported by ARPES, Optical Conductivity.
• Good Tc values.
• Modeled with phenomenological Hamiltonians
• Holes hop between random
• Mn sites (impurity band)
• Interaction between localized and mobile
• spins is LR.

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Valence Band Picture Zener Model

• Chemical potential lies in valence band
• The band structure is rather independent of the
  amount of Mn doping.
• Holes hop in the fcc lattice.
• FM caused by hole mediated RKKY interactions.
• Good Tc values for metallic samples.
• Mean Field approaches disorder does not play a
  role. Impurity spins are uniformly distributed.
• Supported by SQUID measurements.

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Valence Band
Luttinger-Kohn
Expanding around k0 we obtain the hoppings in
terms of Luttinger parameters. There is a similar
3x3 block for spin down.
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Change of Base

• The on-site orbitals are labeled by the four
  values of m_j (/-3/2 and /-1/2)
• The nearest neighbor hoppings between the
  orbitals are linear combinations
• of the hoppings obtained earlier.
• The Hund interaction term has to be expressed in
  the new base. J is obtained
• from experiments or left as a free parameter.

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Results

• Non-interacting case reproduces L-K

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IB vs VB in metallic regime (large x and large p)
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T in diluted magnetic semiconductors as well?
Mn-doped GaAs x0.1Tc 150K. Spintronics?
Model carriers interacting with randomly
distributed Mn-spins locally
Monte Carlo simulations very similar to those
for manganites.
Clustered state, insulating
carrier
J
FM state, metallic
Mn spin
Alvarez et al., PRL 89, 277202 (02). See also
Mayr et al., PRB 2002
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Experimental Properties

• Metal-Insulator transition at x3.
• Tc increases with p. (Ku et al.)

0.02ltxlt0.085
Dietl et al. (Zn,Mn)Te
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Experimental Properties

• Impurity band in insulating regime (x lt0.035)

Okabayashi et al., PRB (2001)
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Experimental Properties

• Magnetization curves resemble the ones for
  homogeneous collinearly ordered FM. For large x
  (Potashnik et al.)
• Highest Tc170K.

Van Esch et al.
x.07 x.087
Ohno et al, X.053
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Outline

• Motivation
• Experimental Properties
• Theoretical Results
• New approach
• Results
• Conclusions

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Motivation 2. DMS

• What kind of materials can provide polarized
  charge carriers?
• III-V semiconductors such as GaAs become
  ferromagnetic when a small fraction of Ga is
  replaced by Mn.
• Can the ferromagnetism be tuned electrically?
• How do the holes become polarized?
• What controls the Curie temperature?

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III-V Semiconductors GaAs
Band Structure
Diamond Structure
First Brillouin Zone
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Luttinger-Kohn Model

• Based on symmetry
• Only p orbitals are considered
• Spin-orbit interaction

Captures the behavior of the hh, lh, and so bands
around Gamma point
Change of base due to S-O interaction
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• xgt2 is necessary for collective FM
• x10 is about the maximum experimentally
  achieved doping.
• The number of holes per doping fraction p should
  be 1 but until recently smaller values of p were
  experimentally achieved due to trapping of holes.

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Values of k in Finite Lattices
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Theoretical Results (I)

• Valence band context
• Reasonable Tc values.
• Good magnetization curves in metallic regime.
• Some transport properties.
• Fails to capture high Tc in insulating regime.
• MF treatment of realistic Hamiltonians.

Dietl, MacDonald,
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Theoretical Results (II)

• Impurity Band context
• Explains non-zero Tc in the low carrier
  (non-metallic limit).
• Percolative transition.
• Fails to provide correct
• M vs T in metallic regime.
• Phenomenological Hamiltonians

Bhatt, Zunger, Das Sarma,
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New Approach Numerical simulation of a
realistic Model

• Real space Hamiltonian
• Valence band tight binding of hybridized Ga
  and As p orbitals on fcc lattice. (Slater).
• Interaction AF Hund coupling between
  (classical) localized spin and hole spin.
• Only j3/2 states kept.
• Numerical Study
• Exact diagonalization and TPEM technique
  (Furukawa).
• 4 states per site and 4 sites basis per cube.
• 4x4xLxLxL number of a states in a cubic lattice
  with L sites per side. It contains 4xLxLxL Ga
  sites.