Ferromagnetism and Metal-Insulator Transition in Hubbard Model with Alloy Disorder

Transcription

1 Ferromagnetism and Metal-Insulator Transition in Hubbard Model with Alloy Disorder Krzysztof Byczuk Institute of Physics, Augsburg University Institute of Theoretical Physics, Warsaw University October 13th, 25

2 Main results New collective effects induced by correlation and disorder Enhancement of in binary alloy ferromagnets New Mott Hubbard metal insulator transition at n 1 K. Byczuk, M. Ulmke, D. Vollhardt Phys. Rev. Lett. 9, (23) K. Byczuk, W. Hofstetter, D. Vollhardt Phys. Rev. B 69, 4512 (24) K. Byczuk, M. Ulmke Eur. Phys. J. B 45, 449 (25)

3 Collaboration Walter Hofstetter - Aachen, Germany Martin Ulmke - FGAN - FKIE, Wachtberg, Germany Dieter Vollhardt - Augsburg University, Germany Plan of the talk 1. Introduction localized vs. itinerant FM Mott-Hubbard MIT binary alloy disorder alloy FM in Nature 2. Earlier result within DMFT on pure FM 3. Our results on binary alloy FM enhancement of magnetization and Curie Weiss law Mott Hubbard MIT at n 1 4. Conclusions

4 Ferromagnetism of local moments Exchange (Heisenberg) coupling H = X ij J ij Si S j J < FM appears at Curie temperature T > T < Saturated magnetization M(T = ) = µ B S

5 Itinerant Ferromagnetism Dynamical way to make FM To reduce interaction energy electrons prefer FM state U > ε due to Pauli principle E FM int = t < µ ε However, E FM kin < EPM kin! t < µ Saturated magnetization M(T = ) < µ B S spin up spin down FM stable if U t - intermediate coupling problem!!! Many itinerant FM are alloys

6 Mott-Hubbard MIT at n = 1 U t ij, p = U t ij, r = typical intermediate coupling problem U c t ij

7 Alloy Band Splitting Binary alloy disorder (alloys A 1 x B x, e.g Fe 1 x Co x ) P(ɛ i ) = xδ(ɛ i + 2 ) + (1 x)δ(ɛ i 2 ) E H = P i ɛ i + t P ij a i a j 1 x x alloy band splitting for W in any dimension DOS intermediate coupling problem!!! physical quantity: O = R dɛp(ɛ) < Ô(ɛ) >

8 Alloy Ferromagnets in Nature I Piryts: T 1 x (T+1) x S 2, T=Fe, Co, Ni, Cu, Zn t 6 2g en g with n =, 1, 2, 3, 4 Fe 1 x Co x S 2, max x.76 Jarrett et al., PRL 1968, Leighton 24

9 Alloy Ferromagnets in Nature II 11 (critical temperature: K) Theory Annealed for 14 days at 875 o C x (amount of Ge) MT 2 X 2 ternary intermetallic alloys UCu 2 Si 2 x Ge x, max x 1.6 Silva Neto et al., PRL 23 Si and Ge isovalent, only structural disorder

10 Alloy Ferromagnets in Nature III Fe weak FM, Co strong FM, bcc alloy Fe 1 x Co x, max x.5 Pratzer et al., PRL 23

11 Alloy Ferromagnets in Nature IV Alloy ruthenates SrRu 1 x Mn x O 3, FM Met AF Ins Cao et al., cond-mat/49157

12 Hubbard model to capture right physics ɛ + U t U t ɛ DOS: ρ(e) = P n R dx ψn (x) δ(e ɛ n ) n i n i H = X iσ ɛ n iσ + X ijσ long history, many contradictions t ij a iσ a jσ + U X i E U= E t= 2(n/2) exactly solvable in d = 1 exactly solvable in d = (DMFT) how to approximate in 1 < d <? U E F 2(1 n/2) ρ( E) ρ( E)

13 Physical picture, n = 1 E+U atomic levels t = t > UHB E+U E U LHB E at U = U c resonance disapears gaped insulator spin flip on central site dynamical processes with spin-flips inject states into correlation gap giving a quasiparticle resonance

14 Route to FM in one-band Hubbard (DMFT) H = P ta iσ a jσ + U P n i n i a=1 a=.98 a=.97 N (ε) a= a=.5 a=.9 a=.98 a=1 1.4 N (ε) ε DOS asymmetry - a.3.2 a=.97 a=.98.1 a=.99 a=1 T.6.5 F n U=8 U=6 U=4 P U = ε T.3.2 P Interaction - U.1 F Wahle et al n a =.98

15 FCC d = FM in one-band Hubbard 2ɛ 2 ] π(1+ 2ɛ) N (ɛ) = exp[ 1+ T.6.4 P U = 2, 4, 5 M U=4 n=.58 4 χ 1 F F n T N(ω).8.4 U=4 n=.58 T=.4 M=.4 Ulmke et al ω µ

16 FM in binary alloy itinerant electrons Anderson Hubbard Hamiltonian H = X ij,σ t ij ĉ iσĉjσ + X iσ ɛ iˆn iσ + U X i ˆn i ˆn i where ɛ is random variable with bimodal PDF P(ɛ) = xδ ɛ + «+ (1 x)δ 2 ɛ «2 Physical observable averaged arithmetically Z dis = dɛp(ɛ)( ) d = FCC DOS stabilizes FM N (ω) = exp[ 1+ 2ω 2 ] q π(1 + 2ω)

17 Dynamical Mean Field Theory Local Green function - Hilbert transform of DOS with self energy G σn = Z N (ɛ) dɛ iω n + µ Σ σn ɛ expressed by path integral, which is calculated with Hubbard Stratonovich and QMC over auxiliary Ising spins G σn = *R D [cσ, c σ ] c σnc σn ea i {c σ,c σ,g 1 R D [cσ, c σ ] ea i {c σ,c σ,g 1 σ } single impurity action for each ɛ i = ± /2 σ } + dis A i {c σ, c σ, G 1 σ } = X n,σ c σn G 1 σn c σn ɛ i X σ Z β dτn σ (τ) U 2 X Z β σ dτc σ (τ)c σ(τ)c σ (τ)c σ(τ) k integrated Dyson equation for Weiss function G 1 σn = G 1 σn + Σ σn

18 Curie temperature =1 =4 Mott n=.3 U=2 U= concentration, x (b) (x) increases!!! at some cases U=2 (a) U=4 n>x U=6 n<x DISORDER n =.7 x =.5 n =.3 x = n=.7 U=2 =1 =4.6.5 x= U=6 Mott U=2 U=4 U=6 n = concentration, x disorder,

19 Is there an alloy band splitting at U >? U = 4, n =.3, n =.5, T =.71, MEM LHB =. E 1 x SPECTRAL DENSITY UHB =.4 =1. =1.4 =1.8 x ω µ DOS Subtle interplay between and U increases!

20 Why is Curie temperature enhanced? T.6.4 P n < 2x n eff = n x T p ( c n eff ).2 F = xt p c U n E p Tc(n) x (n) U 1 1 x (x) increases!!! at some cases n n eff n Good for large U x DOS 1 x U 2 n > 2x n eff = n 2x 1 x = (1 x)t p c p (n eff ) U 1 (n) T p c (n) n eff n n Good for small U

21 Magnetization and Curie-Weiss law =1 =4 Mott n=.3 U=2 M(T), χ 1 (T) x=.6, U=6, n=.3 x=.1, U=2, n= U=6.1 = concentration, x kt If W and n < 2x M s = n but n > 2x M s = n 2x.3.2 n=.7 U=2 =1 =4 M(T) Ms = tanh[ M(T) TMs ].1.8 χ(t) = C T Tc, where C M s.6 U=6 Mott.4.2 C 1 C 2 =.623 close to concentration, x

22 Mott Hubbard metal insulator transition =1 =4 n=.3 U=2 A( ω) µ 2N L ω LAB 2xNL µ U UAB 2(1 x)nl Mott U= concentration, x µ LHB UHB xn L µ µ U c METAL µ xnl INSULATOR U c If n = x (or 1 + x) Mott-Hubbard MIT occures for > x and U > 6 x (or 1 x and 6 1 x) U = 6, x =.5, n =.5, T =.71, MEM.3 n= U=2 U=6 =1 =4 Mott SPECTRAL DENSITY TEMPERATURE T PM FM 1 2 DISORDER PI concentration, x ω µ

23 Correlated insulators alloy Mott insulator alloy charge transfer insulator U< U> ε +U UHB ε+ UAB ε + UAB ε+u UHB ε LHB ε LHB alloy Mott insulator alloy charge transfer insulator

24 Quantum critical points =1 =4 n=.3 U=2.2.1 n=.7 U=2 =1 = Mott U= U=6 Mott concentration, x concentration, x At T = quantum phase transitions: FM met PM ins or FM met PM ins (Mott). Correlation (band-width) controlled, Filling controlled, Alloy concentration controlled Mott MITs. Is non-fermi liquid in d <? Role of correlations in space.

25 Summary New collective effects induced by correlation and disorder Possibilities of increase in binary alloy ferromagnet New Mott Hubbard metal insulator transition at n 1 Alloy Mott insulator vs. Alloy charge transfer insulator Alloy concentration controlled Mott MIT Outlook (x) - QPT? 2nd vs 1st order PT? Multi-band Hubbard model, role of Hund and exchange coupling, which from our findings are generic for many orbitals? Material specific models?? LDA+DMFT+disorder???