Crystal field and spin-orbit interaction

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Helena Stokes
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1 Crstal field and spin-orbit interaction In the solid state, there are two iportant contributions doinating the spin properties of the individual atos/ions: 1. Crstal field The crstal field describes the influence of the surrounding atos/ions on the electronic energ. This ethod assues that the atoic/ionic quantu echanical proble has been solved first. One then calculates the electrostatic interactions of the surrounding sites b perturbation theor. The ethod was put on a quantitative basis b Bethe (1929) and Van Vleck (1932). It is particularl useful for paraagnetic ions in inorganic crstals. 2. Spin orbit interaction Orbital otion and the agnetic spin oent are coupled via the agnetic interaction of the spin agnetic oent with the agnetic field of the orbital otion. In atos this interaction has the for: ˆSO ˆ ˆ = λ L S 15/11/2005 GKMR ESR lecture WS2005/2006 Denninger 1

2 The ailton operator of an electron with orbital oentu L and spin oentu S in a agnetic field B 0 is: 2 ˆ 2 = + V ˆ ˆ ˆ ˆ 0( r) + V1( r) + L S + gl B B0 L + g B S S B 0 λ µ µ 2 Kinetic energ potential energ of the free ato Crstal field Spin orbit interaction Orbital Zeean interaction Spin Zeean interaction Ver iportant for the solution of this ailtonian are the relevant energ scales: Kinetic energ: 1eV 10 ev 8000 c c -1 Potential energ V 0 (r): 1 ev 10 ev 8000 c c -1 Crstal field V 1 (r): 12.5 ev 1.25 ev 100 c c -1 Spin orbit interaction: 1.25 ev 250 ev 10 c c -1 Zeean interaction: < 1.25 ev < 10 c -1 E = h f = hc = hcν λ E = 1eV = h 241 T = hc 8000 c -1 1 Wave nubers 15/11/2005 GKMR ESR lecture WS2005/2006 Denninger 2

3 The Zeean interactions are usuall sall. Even for B = 10 T: c 1 µ B = hc B 0 The spin orbit interaction in atos increases with the nuclear Z. Tpical atoic values are: ˆ SO = λ LS ˆ ˆ λ: spin orbit coupling constant ato B C F Cl Br λ 10 c c c c c -1 Fro atoic phsics: For hdrogen-like wave functions, quantu nuber n: λ Z n 4 6 The crstal field V 1 (r) and the spin orbit interaction λ L S are nearl alwas (uch) larger than the Zeean interactions µ B B 0 L and 2 µ B B 0 L. Thus the agnetic oent is alwas a sall perturbation on the electronic proble. The agnetic properties have to be solved b perturbation ethods after the solution of the electronic structure. With respect to the influence of the crstal field and the spin orbit interaction, we have to distinguish between the following cases: 15/11/2005 GKMR ESR lecture WS2005/2006 Denninger 3

4 1. Spin orbit interaction «crstal field λ L ˆ Sˆ V ( r ) 1 This is the case for light transition eleents like the 3-d eleents. The crstal field proble is solved first, and the spin orbit interaction is treated as a perturbation. 2. Spin orbit interaction» crstal field λ L ˆ Sˆ V ( r ) 1 This is the case for the heav rare earth eleents 4f, 5f. ^ ^ ^ The states corresponding to the total angular oentu J = L + S are taken as basis states and the crstal field V 1 (r) is treated as a perturbation. 3. Spin orbit interaction crstal field λ L ˆ Sˆ V ( r ) 1 ^ ^ V 1 (r) and λ L S have to be diagonalised together. 15/11/2005 GKMR ESR lecture WS2005/2006 Denninger 4

5 Case 1: λ L ˆ Sˆ V ( r ) 1 We regard an electronic wave function of p-setr surrounded b charged ions. The 4 charges along ±, ± set up the crstal field for the electron in the p-functions. +q +q The p orbital will be lowered in energ, due to the vicinit of the positive charges +q along the ± aes. +q +q + q p : f () r 15/11/2005 GKMR ESR lecture WS2005/2006 Denninger 5

6 +q +q + q p : f () r The p orbital will be shifted up in energ, due to the vicinit of the negative charges along the ± aes. +q +q + q p : f () r The p orbital will be unaffected in energ. Without spin-orbit interaction, the total wave functions would be product wave functions: Orbital wave function * spin function u The energ level diagra is depicted on the right. Each level is doubl degenerate without spin-orbit interaction. E 0 f () r u f () r u f () r u 15/11/2005 GKMR ESR lecture WS2005/2006 Denninger 6

7 The basis functions with spin u are: f () r u f() r f () r u f () r u =1,2 u = 1 u 2 = f() r f () r f () r f () r f () r 6 basis functions for the p-orbitals Spin up, spin down One now calculates in a perturbation theor the influence of the operators: Tpical atri eleents are: f () r u ( µ B Lˆ ) f () r u dr dτ * B f () r u ( λ Lˆ Sˆ ) f () r u dr dτ * s s µ BL ˆ B ˆ ˆLS λ dr : integration over the spatial coordinates dτ s : integration over the spin degree of freedo f() r Lˆ f()d r r 0 f() r Lˆ f()d r r 0 f() r Lˆ f()d r r 0 All diagonal atri eleents of the orbital operators vanish! This is tered: quenching of orbital angular oentu. Proof: e.g. Slichter p.89 ff 15/11/2005 GKMR ESR lecture WS2005/2006 Denninger 7

8 After a rather length calculation (see details e.g. in Slichter p ) one arrives at an effective spin ailtonian operator: ˆ = µ ( g B Sˆ + g B Sˆ + g B Sˆ ) B g = 2 g g λ = 2 1 ( E E) λ = 2 1 ( E E ) ( ) ( ) Electron is in the state f(r)!! circulation via p, p circulation via p, p The ain effect is a g-shift proportional to the spin-orbit interaction λ and inversel proportional to the energ difference of the higher ling state. Let E E c -1, λ 100 c -1 g = 2 ( ) : 1% effect! Note: The electron with this g-tensor is in the lowest energ eigenstate, which is the p -state! 15/11/2005 GKMR ESR lecture WS2005/2006 Denninger 8

9 B : field is along the occupied p state. g = 2 B B circulation via the ecited state p at E = E -E : p + p + p - p - g λ = 2 1 ( E E ) ( ) B circulation via the ecited state p at E = E -E : p + p + p - p - g λ = 2 1 ( E E ) ( ) 15/11/2005 GKMR ESR lecture WS2005/2006 Denninger 9

10 General structure of the g-shift: correction ter E is the energ distance to a state near the occupied state. λ E This schee can even be applied to etended electron states in a seiconductor band structure. The g factor g c of the conduction electrons in III-V seiconductors like GaAs, InSb, InP, GaN etc. can be calculated b a odel tered k p theor. The essential paraeters are again the spin-orbit coupling 0 and the energ difference E g. Since band electrons have an effective ass eff, this is another paraeter. g c = Energ (rel. units) ( 1) eff 3 Eg [ 1 ] Conduction band Energ Valence bands -1,0-0,5 0,0 0,5 1,0 wavevector k GaAs: g c InSb: g c -50 gap E g E n (k) 0 E j (k) bandstructurediagra.opj 15/11/2005 GKMR ESR lecture WS2005/2006 Denninger 10

11 Crstal field for d-electrons, case of transition eleents on lattice sites of e.g. seiconductors, insulators. The eaple with the p-electron is not often encountered in real solids, since the p-electrons participate ver strongl in cheical bonds. What is ver coon, however, is the case of d- electrons in solids. Man transition eleent ions can be build into solids on lattice sites (with the appropriate ionic charge). Or these transition eleents for coplees with high setr. We eaine the case of octahedral setr and the case with a sall tetragonal distortion fro octahedral setr. Positive ion like Fe 3+, Cr 2+ water olecule 15/11/2005 GKMR ESR lecture WS2005/2006 Denninger 11

12 The 5 d orbitals (d, d, d, d 2-2,d 32-r2 ) have different interactions with the dipole fields fro the 2 O olecules. This is due to the directions into which the lobes of these orbitals point. ψ 2,2 ψ ψ ψ ψ + ψ 2,-2 21, 2,- 1 2,1 2,-1 dγ ψ 2,2 ψ + ψ 2,-2 20, dε Energ splitting b the cubic field 15/11/2005 GKMR ESR lecture WS2005/2006 Denninger 12

13 Ver often, one has a tetragonal distortion of the cople, e.g. a slight oveent of the 2 O olecules along the ± ais. This leads to an additional splitting of the dγ and dε states. dγ dε Free ion Cubic field Tetragonal field δ The ground state is a single level, and thus a spin doublet. It can be split b a agnetic field. 15/11/2005 GKMR ESR lecture WS2005/2006 Denninger 13

14 The g-tensor values for the ground state are: g = g = ge 2 λ δ This is an aial tensor. g = g λ e 8 For KTi(C 2 O 4 ) O the relevant ion is Ti 3+ : g = g = 1.96 g = 1.86 λ = 154 c -1 δ = 7700 c -1 = 8800 c -1 15/11/2005 GKMR ESR lecture WS2005/2006 Denninger 14

Eigenvalues of the Angular Momentum Operators

Eigenvalues of the Angular Momentum Operators Eigenvalues of the Angular Moentu Operators Toda, we are talking about the eigenvalues of the angular oentu operators. J is used to denote angular oentu in general, L is used specificall to denote orbital