Ferromagnetic Semiconductors Gergely Zaránd Budapest Univ. Technology Collaborators: Greg Fiete (Santa Barbara) Boldizsár Jankó (Notre Dame) Pawel Redlinski (Notre Dame) Jacek Furdyna (Notre Dame) Pascu Moca Catalin (Nagyvarad/Oradea)

Introduction / Motivation (Ga,Mn)As and its simple picture (Ga,Mn)As in reality band structure + SO coupling impurity band formation frustration effects localization effects Outline

Motivation: Combine semiconductor technology with MAGNETISM Control magnetism through electricity (e.g., write bits through electric current) transfer information through spin current ? Spin-base quantum computation ????.... Physics: “Spintronics”: localization + magnetism… anomalous Hall effect… strong spin-orbit effects…

Difficulty: III-V: low solubility of Mn ions … Solution: Annealing methods [Hayashi et al., APL 78, 1691 (2001),…] Wang et al., AIP Conf. Proc. 772, 333 (2005) Low-temperature growth of (Ga,Mn)As [Ohno, Science 281, 951 (1998)] Goal: produce a semiconductor that can be integrated with standard technology and is a soft magnet, but has high T C

III-V Materials Carrier-mediated ferromagnetism ??????

Examples of applications [Ruester et al. PRL 91, (2003)] Spin polarized light emitting diode [R. Fiederling et al., Nature 402, 787 (1999)] Field effect control of ferromagnetism [H. Ohno et al., Nature 408, 944 (2000)] Light induced ferromagnetism [Koshihara et al., PRL 78, 1019 (2000)]

(Ga,Mn)As: The simplest picture

Mn ions Mn Ga replace Ga ions Crystal structure + Mn ions Many holes in it … !!!

Mn ions Mn Ga replace Ga ions Crystal structure + Mn ions Many holes in it … !!! Mn ions Mn Ga replace Ga ions Mn I sit in holes…

Good Mn Ga ions Mn Ga ions: gives SPIN: S = 5/2, g=2 d 5 configuration dopes hole negatively charged (strong scatterer!!!) couples antiferromagnetically to holes Sea of happy holes

Bad Mn I ions Kill Mn Ga spins ! take away 2 holes ! expands lattice positively charged Bind to Mn Ga ions ! [Jungwirth et al. PRB 72, (2005)] Sea of (partially) happy holes One can anneal them away !

One can anneal away ! Annealing [Potashnik et al. APL 79, 1495 (2001)]

Simplest model Scaling not satisfied experimentally (exchange corrections, spin fluctuations, disorder …) Mean field theory (neglecting disorder): [Dietl et al. Science 287, 1019 (2000); Konig et al. PRL 84, 5628 (2000)]

(Ga,Mn)As: The reality

Complications Several p-bands complicated band structure Large spin-orbit coupling magnetic anisotropies, spin relaxation etc. Very large disorder localization effects, impurity band, acceptor states Random spin positions Large electron-electron interaction

Band structure and SO coupling

Electron structure I s ( l = 0 ) p ( l = 1 ) s ( l = 0 ) j = 3/2 hopping SO- coupling p ( l = 1 ) j = 1/2 j = 3/2 j = 1/2 valence band conduction band 8 e -

[J.S. Blakemore, J. Appl. Phys. 53, R123 (1982)] Strong spin-orbit interaction Holes have spin j=3/2 character GaMnAs is degenerate Fermi system Electron structure of GaAs: SO effects

Cubic symmetry determines Luttinger parameters [J.M. Luttinger, W. Kohn, PR 97, 869 (1955)] Holes have J = 3/2 spin that couples strongly to their orbital motion: Kohn-Luttinger Hamiltonian

Approximate: Eigenstates are chiral: n heavy hole ≈ 10 n light hole [A. Baldereschi and N.O. Lipari, Phys. Rev. B 8, 2697 (1973)] SU(2) invariant Spherical approximation

Dilute limit

Single Mn ion Hamiltonian: Spectrum for: Valence holes Localized hole with spin J=3/2

For: Mn spin and couple to form a spin triplet

Polaron hopping picture : [Berciu, M., and R. N. Bhatt, PRL. 87, (2002); G. Fiete, GZ, K. Damle, PRL 91, (2003); Kaminski, A., and S. Das Sarma, Phys. Rev. Lett. 88, (2002); Durst, A. C., R. N. Bhatt, and P. A. Wolff, PRB 65, (2002)]

Study Mn2 ion Energy shift Spin-dependent hopping Local spin-anisotropy for holes Obtain effective Hamiltonian (spherical approx): Compute low-lying spectrum of 2 Mn ions [P. Redlinski, GZ, B Janko, cond-mat/ ; G. Fiete, GZ, K. Damle, PRL 91, (2003)]

Computed parameters:

Hopping F=3/2 fermions coupled to local classical spins: Spin-hopping direction coupled matrix elements: diagonal matrix spin 3/2 rotation matrix Minimum model (dilute limit)

Band structure of a relaxed Mn system ( x active =0.01, f=0.1 ) Impurity band in small concentration limit ARPES: H. Asklund, et al., PRB 66, (2002). J. Okabayashi, et al. PR B 64, (2001); Physica E 10, 192 (2001). STM: B. Grandidier, et al., APL 77, 4001 (2000); T. Tsuruoka, et al. APL 81, 2800 (2002); OPTICAL CONDUCTIVITY : E. J. Singley, et al PRL, 89, (2002); Phys. Rev. B 68, (2003). ELLIPSOMETRY: K. S. Burch, et al. PRB 70, (2004). ( x active < 0.01 )

Non-collinear magnetic states ( x active =0.01, f=0.3 ) [G. Fiete, G.Z., and K. Damle, 2003, PRL 91, (2003)] Distribution of angles [see, e.g. : B. Grandidier, et al. APL 77, 4001 (2000).] Experiments: small fields induce substantial increase of magnetization in small concentration unannealed samples

Metallic limit

RKKY interaction: non-collinear states ? Neglect disorder, and compute effective spin-spin interaction [GZ, and B. Janko, PRL 89, (2002)] Non-collinear States ?

RKKY interaction [Brey, L., and G. Gomez-Santos, PRB 68, (2003); G. Fiete, GZ, B. Janko, et al., PR B 71, (2005); Timm, C., and A. H. MacDonald, PRB 71, (2005)] Almost collinear states for x > 0.03

Ab initio calculations [G. Bouzerar, G., T. Ziman, and J. Kudrnovsky, Europhys. Lett. 69, 812 (2005)] Bergqvist, et al. PRL 93, (2004); Hilbert, S., and W. Nolting, PR B 71, (2005); Xu, J. L., M. van Schilfgaarde, and G. D. Samolyuk, PRL 94, (2005); G. Bouzerar, G., T. Ziman, and J. Kudrnovsky, Europhys. Lett. 69, 812 (2005)

Transport properties

Resistivity anomalies in GaMnAs data from P. Schiffer’s group Sea also Potashnik et al., APL 79, 1495 (2001) Matsukura et al., PRB 57, R2037 (1998) Edmonds et al, APL 81, 4991 (2002)

Possible explanations for the peak? Critical fluctuations ? Magnetic polarons ? [Kasuya, Dietl and Spalek, P. Littlewood] Selfconsistent potantials ? [Nagaev’s theory] Only a kink at T C [Fischer-Langer] Maximum way above TC [P. Littlewood] Curves cross… “Spin disorder scattering” Diverges at T C … None of these works …

Proposal: Interplay of magnetization and localization Interplay with localization produces peak at Magnetic-ordering decreases effective disorder Resistance changes at microscopic scale [Similar ideas emerged for Manganites [Viret et al. PRB 55, 8067 (1997)] There Jahn-Teller effect provides localization Some conceptual difficulties ] [GZ, P. Moca, and B. Janko, PRL 94, (2005).]

Influence of spin on disorder: possible mechanisms Static spins, double exchange mechanism Spin splitting of bands [Lopez-Sancho and Brey, PRB 68, (2003)] Interference between magnetic and static scattering [Csontos et al, Nature Mat. 2005]

We need to know Metallic Phase: Finite conductivity Mott’s variable range formula Insulating Phase:

Single parameter scaling theory of localization (T=0) Typical dimensionless conductance of slab ~ L

Spin distribution changes disorder ! Insulator: Metal:

Beta function, Phase diagram To compute we need to solve a differential equation beta function extracted from model calculations [GZ, P. Moca, and B. Janko, PRL 94, (2005).]

Experimentally observed anomalies, localized fits GaMnAs data from P. Schiffer’s group Some fine-tuning is needed to fit the metallic data through variable range hopping [GZ, P. Moca, and B. Janko, PRL 94, (2005).]

Fitting through metallic expression [GZ, P. Moca, and B. Janko, PRL 94, (2005), and unpublished]

Best fit! More fits… Experiments on (Ga,Mn)As metal rings find similar behavior ! K. Wagner, et al. PRL 97, (2006)

Conclusions General review, GaMnAs: Jungwirth et al. cond-mat/ Carrier-mediated mechanism in GaMnAs: Dietl, T., 2003, condmat/ First principles calculations Sanvito, S., G. Theurich, and N. A. Hill, Journal of Superconductivity 15, 85 (2002); Sato, K., and H. Katayama-Yoshida, Semicond. Sci. Technol. 17, 367 (2002) II-VI materials Furdyna, J. K., and J. Kossut, Diluted Magnetic Semiconductors, volume 25 of Semiconductor and Semimetals (Academic Press, New York, 1988). Spintronics Zutic, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004). REVIEWS:

Transfer matrix / scaling analysis of Lyapunov exponents Lyapunov exponent Single parameter scaling: slabs Universal function Microscopic length scale

Single parameter scaling theory of localization II Consider a slab of size and conductance increases as we increase decreases as we increase

Test these ideas for a toy model Disordered Kondo lattice: Take+ classical spins Spins at mean field level Transfer Matrix Analysis (MacKinnon and Kramers, PRL, 1981) [Similar analysis in the context of manganites: Li et al., PRB 56, 4541 (1997)]

Beta function, Phase diagram