Unbiased Numerical Studies of Realistic Hamiltonians for Diluted

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.

Motivation: Spintronics • Charge Devices – Transistors – Lasers – CPU, processors • Magnetic Devices – Non-volatile memory – Storage – Magneto-Optical devices Electron has spin and charge: • Spin stores information • Charge carries it New Possibilities: • Spin transistor • High spin, high density nonvolatile memory • Quantum information computers using spin states

Mn Doping of Ga. As • • • 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 S=5/2 are introduced. x~10% is about the maximum experimentally achieved doping. x>2% is necessary for collective FM. A metal-insulator transition occurs at x~3. 5%. S=5/2 d orb. Ohno et al. , 1996. (x=. 035, Tc=60 K)

Experimental Properties Potashnik et al, MF regime; x=0. 08, p=. 7 0. 02<x<0. 085 Tc increases with p (Ku et al. ) Okabayashi et al. , PRB (2001) Ohno et al, X=. 053

Different approaches appear to be needed in different regimes Correct interaction, phenomenological band Impurity Band Approach J Bhatt, Zunger, Das Sarma, … Max Tc RKKY collective Mean Field Correct band structure, approx. interaction Valence Band approach: Dietl, Mac Donald 0 0. 1 Carrier Density (p) 1 Mac Donald et al. Nature Materials ‘ 05

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.

Two Band Models • MC and DMFT (Popescu et al. , PRB 2006). – Tc maximized by: § Maximum overlap between bands § p =>1 § J/t ~4 when impurity band overlaps with valence band. j=3/2 j=1/2

New Approach: Numerical simulation of a realistic Model Fcc lattice Diamond Lattice • 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.

Hoppings’ values Values obtained from comparison with Luttinger-Kohn Model for III-V SC. 6 bands J=3/2, j=1/2 Similar results obtained by Y. Chang PRB’ 87

4 band approximation • Keep states with j=3/2. • mj=+/-3/2, +/-1/2. LK J=0 Our results

Results Tc well reproduced in metallic regime. Longer runs being performed to improve shape of curve (work in progress).

Tc increases with p

What value of J? Tc in agreement with experiments. Tc is very low. Metallic regime corresponds to valence band picture.

Density of States and Optical Conductivity. Metallic behavior. Drude peak

Splitting of majority and minority bands.

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 Ga. N.

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 observe MIT as a function of x for fixed J. – 6 orbitals model being studied.

Band Structure of Ga. As Heavy holes Valence Band Light holes Split-off Williams et al. , PRB (1986)

Luttinger-Kohn Valence Band for Ga. As Light holes Heavy holes

Theoretical Pictures • a) Valence Band Holes: Mac. Donald, 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)

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.

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.

Valence Band Luttinger-Kohn Expanding around k=0 we obtain the hoppings in terms of Luttinger parameters. There is a similar 3 x 3 block for spin down.

Change of Base HH and LH bands Split-off band (we discard these states) • 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.

Results • Non-interacting case: reproduces L-K LK Our results

IB vs VB in metallic regime (large x and large p)

T* in diluted magnetic semiconductors as well? Mn-doped Ga. As; x=0. 1; Tc = 150 K. Spintronics? Model: carriers interacting with randomly distributed Mn-spins locally Monte Carlo simulations very similar to those for manganites. Clustered state, insulating FM state, metallic Alvarez et al. , PRL 89, 277202 (02). See also Mayr et al. , PRB 2002 carrier J Mn spin

Experimental Properties • Metal-Insulator transition at x~3%. • Tc increases with p. (Ku et al. ) 0. 02<x<0. 085 Dietl et al. (Zn, Mn)Te

Experimental Properties • Impurity band in insulating regime (x <0. 035) Okabayashi et al. , PRB (2001)

Experimental Properties • Magnetization curves resemble the ones for homogeneous collinearly ordered FM. For large x (Potashnik et al. ) • Highest Tc~170 K. Van Esch et al. x=. 07 x=. 087 MF regime; x=0. 08, p=. 7 Ohno et al, X=. 053

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

Motivation: 2. DMS • What kind of materials can provide polarized charge carriers? • III-V semiconductors such as Ga. As 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?

III-V Semiconductors: Ga. As Band Structure: As Ga Diamond Structure Jancu et al. PRB 57, 6493 (’ 98) First Brillouin Zone

Luttinger-Kohn Model j=3/2 • Based on symmetry • Only p orbitals are considered • Spin-orbit interaction j=1/2 Captures the behavior of the hh, lh, and so bands around Gamma point Light holes Heavy holes Change of base due to S-O interaction

• x>2% is necessary for collective FM • x~10% 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.

Values of k in Finite Lattices

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, Mac. Donald, …

Theoretical Results (II) • Impurity Band context: – Explains non-zero Tc in the low carrier (nonmetallic limit). – Percolative transition. – Fails to provide correct M vs T in metallic regime. – Phenomenological Hamiltonians Bhatt, Zunger, Das Sarma, …

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 j=3/2 states kept. • Numerical Study – Exact diagonalization and TPEM technique (Furukawa). – 4 states per site and 4 sites basis per cube. – 4 x 4 x. Lx. L: number of a states in a cubic lattice with L sites per side. It contains 4 x. Lx. L Ga sites.