Correlation Effects in Real Material

Transcription

1 . p.1/55 Correlation Effects in Real Material Tanusri Saha-Dasgupta S.N. Bose National Centre for Basic Sciences Salt Lake, Calcutta, INDIA

2 . p.2/55 Outline Introduction: why strong correlations? - Failure of one-electron theories - Hesitant electrons: delocalized waves or localized particles? - Examples of strongly correlated materials - Different energy scales and MIT in TMO Methods to deal with correlations in realistic ways - Concepts (LDA+U, LDA+DMFT) - Practical details - Examples Spin-physics out of Correlation - t-j and Heisenberg models - Super-exchange

3 . p.3/55 Electronic Structure Calculations: Good description of many microscopic properties are obtained in terms of - Born-Oppenheimer Approximation Nuclei and the electrons to a good approximation may be treated separately. One-electron Approximation Each electron behaves as an independent particle moving in the mean field of the other electrons plus the field of the nuclei.

4 . p.4/55 LDA Most satisfactory foundation of the one electron picture is provided by the local approximation to the Hohenberg-Kohn-Sham density functional formalism LDA LDA leads to an effective one electron potential which is a function of local electron density. Leads to Self consistent solution to an one electron Schrödinger Eqn.

5 . p.5/55 Strongly correlated electron materials The conventional band-structure calculations within the framework of LDA is surprising successful for many materials. However, they fail for materials with strong e-e correlation! correlation effect necessarily arise, and the consideration of electron correlation effects provides the natural way to understand the phenomena like the insulating nature of CoO.

6 Strongly correlated electron materials Predictions from LDA (Bandstructure) C Ca, Sr Na, K Energy Energy Energy Ef Ef Even No. of e s per unitcell ρ (ε F ) = 0 k k Even No. of e s per unitcell + band overlap ρ (εf) = 0 ρ (ε F ) = 0 k Odd No. of e s per unitcell Accordingly to LDA, odd no. of e s per unit cell always give rise to Metal!. p.6/55

7 Strongly correlated electron materials Failure of Band Theory Total No. of electrons = 9 +6 = 15 Band theory predicts CoO to be metal, while it is the toughest insulator known Failure of LDA ) Failure of single particle picture ) Importance of e e interaction effects (Correlation). p.7/55

8 Strongly correlated electron materials 3s energy 2p 2s a0 a (lattice constant) H_3s = H_band + H_columb e energy/atom itinerant localized Na 0 Na 0 Na + Na ε 3s ε 3s 0 2 ε 3s + U 3s ε 3s At ε 3s 3s U 3s / t 3s. p.8/55

9 . p.9/55 Hesitant e-s: delocalized or localized picture? broad energy bands- associated with large KE, highly itinerant well described using wave-like picture narrow energy bands- e-s spend larger time around a given atom - tendency towards localization, effects of statistical correlations between the motion of individual electrons become important particle-like picture may be more appropriate Intermediate situation - localized character on short time-scales and itinerant character on the long time-scale co-exist! e-s hesitate between being itinerant and localized.

10 . p.10/55 Examples of strongly correlated materials Transition metals: - d-orbitals extend much further from the nucleus than the core electrons. - throughout the 3d series (and even more in 4d series), d-electrons do have an itinerant character, giving rise to quasiparticle bands! - electron correlations do have important physical effects, but not extreme ones like localization.

11 . p.11/55

12 . p.12/55 Examples of strongly correlated materials f-electrons: rare earths, actinides and their compounds: - rare-earth 4f-electrons tend to be localized than itinerant, contribute little to cohesive energy, other e- bands cross E F, hence the metallic character. - actinide (5f) display behavior intermediate between TM and rare earths - e- correln becomes more apparent in compounds involving rare-earth or actinides. - extremely large effective mass heavy fermion behavior. - At high temp local mag. mom and Curie law, low-temp screening of the local moment and Pauli form Kondo effect

13 . p.13/55 Examples of strongly correlated materials - TMO - direct overlap between d-orbitals small, can only move through hybridization! Crystal Field Splitting 10 4 eg 6 t2g d x2 y2 d 3z2 r2 d yz d zx d xy Free Atom Cubic Tetragonal Orthorhombic

14 Examples of strongly correlated materials- TMO d xy d xz d yz d 3z2 r2 d x2 y2 Hybridization via the Ligands (orbitals p/o) d x2 y2 p σ d x2 y2 e g t 2g d xy p π d xy. p.14/55

15 . p.15/55 Examples of strongly correlated materials - TMO Three crucial Energies t pd Metal-ligand Hybridization = ǫ d ǫ p Charge Transfer Energy U On-site Coulomb Repulsion Band-width is controlled by: t eff = t 2 pd/

16 . p.16/55 Examples of strongly correlated materials - TMO Minimal model for a TMO H = ǫ p p jσ p jσ + ǫ d d iσ d iσ jσ t pd (d iσ p i±δσ + h.c.) + U d iσ iσ δ i n d i n d i And may be add..... t pp, U pp, U pd..... orbital degeneracy, etc.....

17 Examples of strongly correlated materials - TMO The infamous Hubbard U Naively: φ i φ i 1 r r φ i φ i But this is HUGE (10-20 ev)! SCREENING plays a key role, in particular by 4s electrons - Light TMOs (left of V): p-level much below d-level; 4s close by : U not so big U < - Heavy TMOs (right of V): p-level much closer; 4s much above d-level : U is very big U >. p.17/55

18 . p.18/55 Examples of strongly correlated materials - TMO The Mott phenomenon: turning a half-filled band into an insulator Consider the simpler case first: U < Moving an electron requires creating a hole and a double occupancy: ENERGY COST U This object, once created, can move with a kinetic energy of order of the bandwidth W! U < W: A METALLIC STATE IS POSSIBLE U > W: AN INSULATING STATE IS PREFERRED

19 . p.19/55 Hubbard bands d band U Energy Interaction U = ε d ε p p band The composite excitation hole+double occupancy forms a band (cf excitonic band)

20 Charge transfer insulators Heavy TMOs d band charge gap Fermi level Energy Interaction U p band = ε d ε p U Gain: t t eff ~ pd / 2 Cost: = ε ε d p Transition for > t pd Zaanen, Sawatzky, Allen; Fujimori and Minami. p.20/55

21 Methods Weakly correlated Metal Strongly correlated Metal Mott insulator Intermediate regime Hubbard bands + U < W QS peak (reminder of LDA metal) U >> W Can be described LDA gives correct answer by "LDA+U" method? courtesy: K. Held. p.21/55

22 . p.22/55 Basic Idea of LDA+U PRB 44 (1991) 943, PRB 48 (1993) 169 Delocalized s and p electrons: LDA Localized d or f-electrons: + U using on-site d-d Coulomb interaction (Hubbard-like term) U i j n in j instead of averaged Coulomb energy U N(N-1)/2

23 Hubbard U for localized d orbital: n+1 n 1 U = E(d ) + E(d ) 2 E(d ) n e n+1 n 1 n n U. p.23/55

24 . p.24/55 LDA+U energy functional (Static Mean Field Theory): E LDA+U local = E LDA UN(N 1)/ U i j n i n j LDA+U potential : V i (ˆr) = δe δn i (ˆr) = V LDA (ˆr) + U( 1 2 n i)

25 . p.25/55 LDA+U eigenvalue : ǫ i = δe δn i = ǫ LDA i + U( 1 2 n i) For occupied state n i = 1 ǫ i = ǫ LDA U/2 For unoccupied state n i = 0 ǫ i = ǫ LDA + U/2 ǫ i = U MOTT-HUBBARD GAP ε LDA U = δ εlda δn d

26 . p.26/55 Rotationally Invariant LDA+U LDA+U functional: E LSDA+U [ρ σ (r), {n σ }] = E LSDA [ρ σ (r)] + E U [{n σ }] E dc [{n σ }] Screened Coulomb Correlations: E U [{n σ }] = 1 2 {m},σ { m, m V e,e m, m n σ mm n σ m m + LDA-double counting term: ( m, m V e,e m, m m, m V e,e m, m n σ mm nσ m m E dc [{n σ }] = 1 2 Un(n 1) 1 2 J[n (n 1) + n (n 1)]

27 . p.27/55 Slater parametrization of U Multipole expansion: 1 r r = kq 4π 2k + 1 r< k r> k+1 Ykq(ˆr)Y kq (ˆr ) Coulomb Matrix Elements in Y lm basis: mm m m = k a k (m, m, m, m )F k F k Slater integrals Average interaction: U and J U = F 0 ; J (for d electrons) = 1 14 (F 2 + F 4 )

28 . p.28/55 How to calculate U and J PRB 39 (1989) 9028 Constrained DFT + Super-cell calculation Calculate the energy surface as a function of local charge fluctuations. Mapped onto a self-consistent mean-filed solution of the Hubbard model. Extract U and J from band structure results.

29 . p.29/55 Notes on calculation of U Constrained DFT works in the fully localized limit. Therefore often overestimates the magnitude of U. For the same element, U depends also on the ionicity in different compounds higher the ionicity, larger the U. One thus varies U in the reasonable range (Comparison with photoemission..).

30 . p.30/55 Where to find U and J PRB 44 (1991) 943 : 3d atoms PRB 50 (1994) : 3d, 4d, 5d atoms PRB 58 (1998) 1201 : 3d atoms PRB 44 (1991) : Fe(3d) PRB 54 (1996) 4387 : Fe(3d) PRL 80 (1998) 4305 : Cr(3d) PRB 58 (1998) 9752 : Yb(4f)

31 . p.31/55 CO in CaFeO 3 tilt LDA LDA+U O2 Fe rotation FeB O 6 breathing IC RT Fe e g RTB FeA eg RTB FeB e g O1 FeA O 6 b c a a δ 2 Fe > Fe + Fe 4 δ Phys. Rev. B 72, (2005)

32 . p.32/55 CO in CaFeO 3 4 spin up Fe(A) LSDA spin dn e g 4 spin up LDA+U spin dn Fe(A)+Fe(B)e g Energy (ev) 0 E f Fe(A)e g Fe(B)e g t Fe(A)+Fe(B) 2g Fe(B) e g Fe(A)+Fe(B)t 2g Energy (ev) 0 E f Fe(A) Fe(B) e g e g Fe(A)+Fe(B)t 2g 4 O(p) O(p) 4 O(p) O(p) Fe(A)+Fe(B) t 2g

33 . p.33/55 Strongly correlated electron materials LDA + Materials specific Fails for strong correln. + Model Approaches Input parameters unknown Account for correln. effects

34 Strongly correlated electron materials LDA + Materials specific Fails for strong correln. + Model Approaches Input parameters unknown Account for correln. effects Improve the description starting from LDA, by combining ab-initio calculations with many-body methods. A major break-through in this respect is (DMFT). - Impurity model takes into account of the local dynamics. - SCF captures the translational invariance and and coherence invariance of the lattice.. p.33/55

35 . p.34/55 Marrying DMFT and DFT-LDA DMFT emphasizes local correlation we need a localized basis set, i.e. basis functions which are centered on the atomic positions R in the crystal lattice Use Wannier function basis!

36 . p.35/55 NMTO: truly minimal set and Wannier functions A basis set of localized orbitals is constructed from the exact scattering solutions of a superposition of short-ranged, spherically-symmetric potential wells (the so-called muffin-tin approximation to the potential) at a mesh of energies, ǫ 0, ǫ 1,...,ǫ N The number of energy points, N, defines the order of such muffin-tin orbitals, the NMTO s. Each NMTO satisfies a specific boundary condition which provides it with an orbital character and makes it localized. The NMTO s being energy-selective in nature are flexible and may be chosen to span selected bands Downfolding If these bands are isolated, the NMTO set spans the Hilbert space of the Wannier functions. In other words, the orthonormalized NMTO s are the localized Wannier functions. O. K. Andersen and T. Saha-Dasgupta Phys. Rev. B 62, R16219 (2000)

37 Phase diagram of V 2 O 3 * undergoes 1st order M I transition can be induced by temp., pressure, alloying * PM PI : same crystal (corundum) & magn. structure * only known example among transition metal oxides to show a PM PI transition. p.36/55

38 V2O3: Corundum Structure O p V t2g V eg V s E2 3 E2 3 E2 10 E1-4 E1 2 E1 2 E1 8-6 E0 1 0 E0 1 0 E0 6 4 E0-8 L Z Γ F -1 L Z Γ F -1 L Z Γ F 2 L Z Γ F V t2g xy V eg x2 y2 V eg 3z2 1 pd π antibonding pd σ antibonding pd σ antibonding. p.37/55

39 NMTO+DMFT Input Hamiltonian- multi-band Hubbard Hamiltonian: H = H lda Umm n imσ n im σ m m [U mm J mm ]n imσ n im σ The many-body Hamiltonian depends on the choice of the basis func. NMTO s are the ideal candidates! The low-energy Hamiltonian defined above, involves ONLY correlated, localized Wannier orbitals and no other orbitals achieved via NMTO-downfolding Assume dc. corrections to be orbital-independent within d-manifold results into simple shift of the chemical potential!. p.38/55

40 . p.39/55 NMTO+DMFT Many-body Hamiltonian solve by DMFT. 1 2 Σσ lm (ω) Single site problem Dyson eqn. Georges Kotliar 92 Multi-orbital quantum impurity problem solve by QMC. - Maps the interacting electron problem onto a sum of non-interacting problems (single particle moves in a fluctuating, time-dependent field) - Evaluates this sum by Monte Carlo sampling

41 . p.40/55 DMFT results Energy [ev] L Z Γ F

42 PM spectra - comparison with PES Expt: Mo et.al.. p.41/55

43 PI spectra - comparison with PES DOS [states/(ev spin V atom)] Total e g a 1g Intensity 1 Theory Exp 0-3 Energy [ev] Energy [ev]. p.42/55

44 . p.43/55 Hubbard model = t-j model How spin physics arises from Strong Electron Correlations?

45 Large U limit of Hubbard Model (t/u 1) ε at ε at +U Atomic limit (t=0) has huge degeneracy. For L sites occupied by N e s, singly occupied sites (which is or spin) L can be picked out in C 2 N ways! N Large degeneracy makes the standard perturbation theory inapplicable. Treat consecutive orders of t/u systematically > Accomplished by a suitable cannonical transformation Motion of e s are constrained by having to avoid the creation of double occupancy Hopping mixes the states! Rotate to a basis whose states are not mixed in order t Rotate to a basis whose states are not mixed in order t p.44/55

46 Projected Hopping Local basis: O > j = c j > = j O > c j > = j O > c j d > = j c j O > PROJECTION OPERATORS: ^ ^ ^ P = O > < O = (1 n )(1 n ) jo j j j j ^ P j ^ P j ^ P jd = > < j j = > < j j = d > < d j j ^ ^ = n (1 n ) j j ^ ^ = (1 n )n j j ^ ^ = n n j j Ensure there is at i Ensure there is at j Perform the hopping Ensure site j is d> Ensure site i is 0> = ^ P io ^ = n ^ P jd j ^ P jd c j c j c j projected hopping i j i i B = c c c = (1 c c)c = c c cc = c c i c i c i ^ P i ^ n ^ P i i ^ P i O d. p.45/55

47 Correlated Hopping H = t t H t = + H t = H t + H t + H 0 t ^ ^ ^ ^ n jσ c j σ c (1 n i σ i )+ n i σ c iσ c jσ (1 n jσ ) <i,j>,σ + H = t t σ H = t t D > D +1 (1 ^ n )c c ^ n +(1 ^ n ) c c n iσ iσ jσ jσ jσ jσ iσ iσ <i,j>,σ D > D 1 O d i j i j d O i j i j 0 ^ ^ ^ O (1 n iσ )c iσ c jσ (1 j n ) + n σ iσ c O iσ c jσ n jσ i j i j <i,j>,σ D > D ( c iσ c jσ + c jσ c iσ ) + H 0 t = H t + H t + H t H t compared to H t Do not care the correlations! Mind the local correlations!! has accquired a complicated many body character! d d i j i j. p.46/55

48 Canonical Transformation Intuitive notion: Low energy excitations are propagarted in the lower Hubbard subband. Pure band motion mixes states from two subbands via H + t and H t. Unmixing can be achieved via rotation to a new basis. Define H eff : H eff = e is He is = H + i[s, H] + i2 2 [S, [S, H]] = H U + H + t + H t + H o t + i[s, H U ] + i[s, H U + H + t + H t + H o t ] + Choose S such that: H + t + H t gets cancelled by i[s, H U ] S = i U [H+ t H t ] H eff = H U + H o t + 1 U [H+ t, H t ] to order t 2. p.47/55

49 Hubbard Operators b < a X j = b > < a [ Projection Operator: j j ^ a < a P a = a > < a = X ] C < 0 d < j = X j + X j Convention: d > = C j C j 0 > < 0 C d < j = X j X j C jσ σ < 0 d < σ = X j + η (σ) X j η ( σ ) = + 1 if σ 1 if σ = = H t 0 = t <i,j> σ σ < 0 0 < σ d < σ d < σ X i X + X X + H.c. j i j 0 0 d d H t + = t η ( σ ) d < σ 0 < σ d < σ 0 < σ X X + X X i j j i. p.48/55

50 t-j model Consider action of H eff on the low energy sub space σ σ H t 0 = = t <ij>σ + H t, H t = σ < 0 0 < σ σ < 0 0 < σ X X + X X i j j i <ij> + H t ij +, H t ij + H H t, ij t jk <ijk> [ disjoint pairs commute] + H σ σ 0 d + 1 H, H j i t ij t j i ij U H σ σ + 1 H H t ij t U ij j i 2 t η(σ)η(σ ) U σ : original spin arrang. σ σ σ < σ σ < σ 2 X X 2t z z ^ ^ i j σ : interchange of spin ( S i S j n i n j ) U 4. p.49/55

51 Heisenberg model H eff H t J = t ij σ (1 ˆn i σ )c iσ C jσ(1 ˆn i σ ) + h.c. Exact half-filling (n=1): + 4t2 U [S is j ˆn iˆn j 4 ] + 3 siteterms.. H = J ij S i S j AF Heisenberg model [ J = 4t2 U ] if neighboring sites are, a virtual hopping process can create an intermediate 0d pair state with energy U [ associated energy gain t 2 U. If spins are or, hopping is prohibited by Pauli principle.. p.50/55

52 Super-exchange or cation A 3z Anion cation B 2 2 p z 3z pz > ~ pz > + b d A > < pz H d A > 1 + b 2 [ b ~ ] ε p ε d For configuration: E ~ b 2 ( < pz + b < d + b < d ) A B < pz H pz > ( pz > + b d > + b d >) For configuration: E ~ 2 ε ε p ε p + 2b 2 A B d b 2 ( ) Exchange: E E ~ 2 b 4 ( ε ε d p ) SUPER EXCHANGE. p.51/55

53 La 2 NiMnO 6 (PRL, 100, ). p.52/55

54 . p.53/55 Mn Ni Ni eg Mn t2g Superexchange e g e g t 2g ~ 2.5 ev e g E = 0 2 E = 4t /(U+ ) Ni eg Mn eg Superexchange ~ 2 ev e g t 2g t 2g ~ 1 ev ~ 1.5 ev t 2g 2 E = 4t /(U J + H ) 2 E = 4t /(U+ ) Use NMTO downfolding to estimate s and t s

55 La 2 NiMnO 6 - Magnetism NMTO-dowfolding study (Effective Ni-Mn model) NMTO downfolding Study: t e t = 0.02 ev o 155 e t = 0.25 ev t e e= 0.20 ev e e= 1.90 ev J Ni Mn = J AF + J FM (te,t ) 2 = 4 (U + e,t ) 4 (te,e ) 2 J H (U + e,e J H )(U + e,e ) J Ni Mn 4-7 mev (U 4-5 ev,j H =0.9 ev). p.54/55

56 . p.55/55 Conclusion Recent technological developments allow for the realistic description of strongly correlated electron system taking into account both the material specific chemical knowledge and the strong correlation aspect.