1. Magnetism in diluted semiconductors 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. Georges Bouzerar Institut Néel -MCBT CNRS

2. Collaboration Josef Kudrnovsky (Institut of Physics-Prague) Richard Bouzerar (University of Amiens) Olivier Cepas (CNRS Paris/ Neel Institute CNRS) Lars Bergqvist (Jullich)

3. Magnetic semiconductors: Early 60’s (non dilute magnetic materials): EuO (T c =70K), EuS (17K) and the spinels as CdCr 2 S 4 (84 K) or HgCr 2 Se 4 (106 K) 70’s and 80’s: D iluted M agnetic S emiconductors II-VI materials CdTe, ZnTe and ZnSe) II  Mn 2+ (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 T C < 3K. Late 80’s: III-V materials MBE  (In,Mn)As films on GaAs substrates FM with T C =35 K Late 90’s: MBE  (Ga,Mn)As films on GaAs substrates FM with T C =110 K I- Introduction

4. 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?? Need for materials with relatively high Curie temperature (beyond room temperature). DMS based materials are promising candidates for Spintronic devices. An exemple of spintronic application: Magnetic Random Access Memory (non volatile) based on GMR (nobel prices of Fert and Gruenberg)

5. 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? Understand experimentally and theoretically which physical quantities control T C

6.   In III-V materials the substitution of Ga 3+ by Mn 2+ introduces simultaneously a localized spin S=5/2 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 1- GaMnAs is one of the most studied diluted III-V material, it exhibits relatively high T C : x Mn =5%, T C =110 K and x Mn =7.5%, T C =175 K. 2- GaMnN is more controversial: some pretends to observe paramagnetism and other ferromagnetism with very high Curie temperature. 1- III-V compounds.

7. 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. 2- Disorder and Compensation defects. Trapped holes

8. 3- Some experimental results GaMnAs with Highest T C =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).

9. Resistivity at zero field as a function of T Curie temperature (Matsukura et al. PRB 1998). Inset effect of H. Resistivity at zero field as a function of temperature for various annealing time (Potashnik et al. 2001). metal isolant Resistivity after annealing: Hayashi et al., APL 78, 1691 (2001)

10. Optical conductivity as a function of frequency  for various concentration.The samples with x=0.052 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) before after Peak in the conductivity is inconsistent with Valence Band picture

11. Mathieu et al., PRB 68, 184421 (2003) Annealing dependence of magnetization curve  magnetization curves change straight/convex (upward curvature) → concave (downward curvature)  degradation for very long annealing (precipitates?) Potashnik et al., APL 79, 1495 (2001)

12. 4- Predictions based on Zener mean Field theory. Dietl et al. Science 2000 (cited 3000 times!!) Is the Zener mean field theory really appropriate to describe ferromagnetism in diluted magnetic semiconductors?? Includes a realistic bande structure for the host material

13. The theory should allow the study the magnetic properties : Curie temperature, magnetic excitations spectrum, local magnetizations,...) of a system of N imp interacting magnetic impurities with spin S (quantum or classical) randomly distributed on a given lattice. 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). II- Theory for disordered Heisenberg model. p i =1 if the site is occupied otherwise 0

14. 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) 1- The Mean Field-VCA theory

15. Includes collective excitations. Fulfils both theorems of Mermin-Wagner and Goldstone (in the absence of anisotropy T C =0 for1D and 2D) exact Very good agreement with exact Monte Carlo calculations in the absence of disorder (x=1). RPA : improvement of thermal and transverse fluctuations 2- The Random Phase Approximation -VCA theory

16. 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 3- Proper treatment of disorder and thermal fluctuations

17. Thermodynamic and dynamical properties within the real space SC-LRPA. Choose a disorder configuration: positions of impurities. For a given temperature T, after convergence one can calculate: The distribution of the local magnetizations m i (T) (Reflectrometry measurements). The spectral function A(q,w,T) and magnetic excitations (Inelastic Neutron Scattering exp.. The spin-spin correlation functions and magnetic susceptibility. Etc. ……… convergence Determination of the thermodynamic and dynamical properties.

18. The Curie Temperature. Within SC-LRPA one can derive a semi-analytical expression for the Curie temperature in disordered ferromagnetic systems. This expression is the generalisation of RPA to disordered ferromagnetic systems! F i depends on both the eigenvalues and eigenstates of the non symmetric effective Hamiltonian H eff

19. 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) III- Ab initio based studies for diluted magnetic semiconductors

20. ZnO ZnS ZnSe ZnTe Which possible diluted ferromagnetic semiconductors ? 1- Some ab-initio results II-VI materials

21. GaN GaP GaAs GaSb III-V materials

22. Density of states in (Ga,Mn)As with 3.125% Mn Wierzbowska et al., PRB 70, 235209 (2004)  LSDA : d-orbital weight at E F, VB top and CB bottom: absent in photoemission experiment and SIC calculations. LSDA+ U : phenomenological procedure to improve the tretament of elevtron-electron correlations d orbitals  Mn d-orbital weight shifted away from E F, better agreement (especially for the feature at -4 eV)  similar results from other methods going beyond LSDA : GGA, SIC-LDA LSDA LSDA +U

23. Not RKKY couplings! Zn 1-x Cr x Te Ga 0.95 Mn 0.05 As Couplings in diluted magnetic semiconductors (data from J.Kudrnovsky) RKKY couplings

24. 1- Determination of the Ground state properties and exchange integrals J ij 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. The two step approach calculations (TSA): How to proceed?

25.  The Mean field VCA underestimate the fluctuations, and overestimate strongly the Curie temperature (for 5% Mn T C =300 K).  The «Ising» calculations: Effects of disorder included but not the transverse fluctuations lead to non realistic T C.  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]. [1] L. Bergqvist et al. Phys. Rev. Lett., 93 137202 (2004) [2] K. Sato et al. Phys. Rev. B, 70 201202 (2004) 2- Role of the disorder and thermal fluctuations. Disorder but no transverse fluctuations Mean field + VCA SC-LRPA

26. LSDALSDA+U Courtesy to L. Bergqvist 3- Monte Carlo vs SC-LRPA. GaMnAs

27. 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 ). Remark: (1) The TB-LMTO calculations to estimate the couplings and (2) the treatment of disorder and fluctuations within SC-LRPA are reliable. 4- Comparison between experiment and theory for GaMnAs. 4- Comparison between experiment and theory for GaMnAs. As grown samples

28. ( Data from Edmonds et al. Nottingham ). Variation of T C for a fixed total density of Mn 2+ after different annealing treatment for Ga 1-x Mn x As. The total Mn concentration is: n h (10 20 cm -3 ) T C (K) détails 4.1 +/- 0.50 67 As grown 9.0 +/- 0.90 113 6hrs@180 o C 10.0 +/- 1.0 126 15hrs@180 o C 13.5 +/- 2.0 143 140hrs@180 o C 4.6 +/- 0.50 76 As grown Conclusion : T C varies strongly from 67 K to 143 K depending on the annealing conditions. a- Experimental results. 5- Effects of compensating defects.

29. We assume that Mn(Ga) density is fixed x and we vary the density y of As anti-sites (Ga 1-x-y Mn x As y )As. As on Ga sublattice is double donor, the density of holes is, Weak variation of T C with  for  > 0.60 For  < 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 T C = x 4/3  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. b- Effects of As anti-sites. We introduce the reduced variable Instability of Ferro GS ??

30. 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 c- Effects of Mn interstitials (Mn(I)).

31. d- Comparison between experiment and theory. For each sample to calculate T C we use the following expressions: Samples with highest T C are in very good agreement with the calculations done for  = 1. For « as-grown » samples we also reproduce Tc by taking into account that  < 1. This study confirms that the dominating mechanism of compensation is due to Mn interstitials. The theoretical curve (  = 1) allows to give a good estimate value for T C for each sample : T C =649 ( x eff - 0.0088) 1/2 Experiment Theory

32. 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 T C = 600 K for x Mn = 6%.GaMnN has a very small T C. In spite of a very strong nearest neighbour coupling (10 times that of GaMnAs) InMnAs exhibits intermediate Curie temperatures 6- Comparison between different III-V materials.

33. For more details see G.B.,EPL (2007). 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: ω α, |Ψ L α ˃, |Ψ R α ˃.

34. Excitations are well defined in a very small region around the Γ point ω(q) = D q 2 in the limit of long wave length (Goldstone mode) Second moment γ(q) = Cq : it does not correspond to the real width of the excitation. The Mn density is x=0.03

36. Both T C 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 LaCaMnO 3. The first moment of A(q, w) does not correspond to the magnetic excitation.

37. 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 J AF, the pairs are disconnected x---> x eff Competition between extended ferro couplings and short range Superexchange. G. B. R. Bouzerar and O. Cepas, PRB (R. C.) (2007). Distribution of canting angle of the ground state for two different concentration of holes.

38. Phase diagram of GaMnAs. This phase may explain inconsistency with magnetization measurements at low hole concentration

39. Spins operators : s i : carrier S i : magnetic impurity t ij =t for (i,j) nearest neighbour V controls the impurity band position. Crucial term : source of scattering (Coulomb potential) due to the substitution of a cation by another one with a different charge: V i =V if i is occupied by a magnetic impurity. V- Model study: the V-J pd model

40. R. Bouzerar et al. EPL 2007 Choice of a configuration of disorder and set of parameters V and Jpd, x Calculation of the Mn-Mn magnetic couplings Diagonalization of the itinerant carrier Hamiltonian in each spin sectors Transport propertiesMagnetic properties Procedure for the study of transport and magnetism A.I. Liechtenstein et al., PRB (1995)

41. 1- The damped RKKY model (V=0).  No finite size effects (the same results for both 800 or 2000 impurities dilute on the fcc lattice).  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) damping is introduced R. Bouzerar et al. PRB 2006

42. 2- The effects of the cutt-off. The standard RKKY model is inappropriate to describe ferromagnetism in dilute magnetic semiconductors as GaMnAs or GaMnN.  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..

43. 3- The non perturbative treatment of the V-J pd model DOS as a function of V at fixed J pd S=5t Magnetic couplings (similar to ab initio) : (a) as a function of hole concentration (b) as a function of V at fixed J pd S=5t and fixed hole density

44.  An average over at least 200 configurations of disorder has been performed. The Jpd model captures the essential physics of the diluted magnetic semiconductors. Note the crucial role of the onsite potential!  Even at low carrier density we do not have: We find the opposite TC reduces with incerasing Jpd!.  Significant increase of the stability region of the ferromagnetic phase. 4- Curie temperature (in units of t) at fixed V=-2.4t.

45. 5- Comparison between TSA and full Monte Carlo treatment of the V-J pd model. (a) (b) (a) and (b) hole density per defect is respectively nh/x=0.125 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

46. (c) MF-VCA vs SC-LRPA Tc within MF-VCA is much larger and saturates for JS→ ∞ 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)

47. 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

48. Magnetic couplings as a function of the distance r for different system sizes (huge effects near the typical distance between impurities)

49. Calculated distributions or the Curie temperatures (huge fluctuations for the small systems! )

50. The case of non dilute systems: the double exchange model in presence of on-site disorder? see GB and O. Cepas,PRB 2007 The on site potential V i is chosen randomly in [-W/2,W/2], W is the disorder strength The TSA versus the full Monte Carlo calculations (a) no disorder W=0 and (b) the disordered case In contrast to the dilute case, a very good agreement is obtained for the non dilute case even in the presence of disorder!

51. Thank you