Talk M.Alouani

Home , Identical particles

1/107

ourg, France, July 01–07, 2012 ourg,

ysics at the nanoscale, Strasb Mebarek Alouani

From atoms to solids nanostructures

IPCMS, 23 rue du Loess, UMR 7504, Strasbourg, France

European Summer Campus 2012:European Ph 2/107

ourg, France, July 01–07, 2012 ourg, t

Course conten ysics at the nanoscale, Strasb

European Summer Campus 2012:European Ph I. Magnetic moment and magnetic field II. No magnetism in classical mechanics III. Where does magnetism come from? IV. Crystal field, superexchange, double exchange V. Free electron model: Spontaneous magnetization VI. The local spin density approximation of the DFT VI. Beyond the DFT: LDA+U VII. Spin-orbit effects: Magnetic anisotropy, XMCD VIII. Bibliography 3/107

ourg, France, July 01–07, 2012 ourg, t

Course conten ysics at the nanoscale, Strasb

ourg, France, July 01–07, 2012 ourg, neton) g potential, in the magnetic dipole ˆ r 2 × r μ is defined as:

0 ysics at the nanoscale, Strasb π μ eh m μ antification axis z gives 4 = Be μ = (/2Bohrma the gyromagnetic factor) rl (' ) γ () ( ∫ Id Ar 2 1 μ =× gm L μ − γ = =

zB Summer Campus 2012:European Ph

μ μ Magnetic moments In classical electrodynamics, the vector approximation, is given by: where the magneticmoment It is shown that the magnetic moment is the angular momentum proportional to The projection along the qu 5/107 H. χ . M= 0 μ / a (H +M) 0 =B μ a is defined as B = France, July 01–07, 2012 ourg, χ H =H . so that M to find 0 μ + a

= B ysics at the nanoscale, Strasb i relative permeability B ). H H because there is no net magnetization. r 0 μ μ 0 μ demagnetizing field )H= χ For a cylinder (1 + 0 European Summer Campus 2012:European Ph μ In free space B = B = Inside a magnet, the relation is more complicated Example: For linear materials, the magnetic susceptibility Magnetization and Field In this case there is still a linear( relationship between B and H In this case it is the free current (coil) that controls the magnetization and hence controls the magnetic field inside the cylinder. In other more complex geometry, one has to subtract out the H. The magnetization M is the magnetic moment per unit volume 6/107

ourg, France, July 01–07, 2012 ourg, t

Course conten ysics at the nanoscale, Strasb

amounts in France, July 01–07, 2012 ourg, 3333 ln 2 ) e m B etic sample where the orbital current that 2 − Bohr-van Leeuwen theorem pA magnetic field ( = NN N N ,, K TV TV W

ysics at the nanoscale, Strasb BB FZ ∂∂ ∂∂ ⎛⎞ ⎛ ⎞ ⎜⎟⎝⎠ ⎜ ⎝ ⎟ ⎠ 11 1 1 −= = = Ep p r r dp dpdr dr MNkT β − to : exp( ( ,..., , ,..., )) ...... ∫ ∼

Z Summer Campus 2012:European Ph Since the integral goes all over phase space, effect of a magnetic field is just to shift the momentum zero. The partition function therefore independent of the magnetic field ( scatter at the surface cancels out orbital volume current. The magnetization is zero in classical mechanics The magnetization is zero This can be easily seen in a finite magn This is because the application of a kinetic energy 8/107

ourg, France, July 01–07, 2012 ourg, t

Course conten ysics at the nanoscale, Strasb

9/107 12 rrt (, ,) Ψ

ψ 12 12 rrt rre (, ,) (, ) Ψ=

ourg, France, July 01–07, 2012 ourg, = (, ,)

ψ 33 A Two-particle wave function: n relativistic Schrödinger equation (SE): n relativistic Schrödinger equation 2 =Ψ = rt Ψ (,) t Ψ ∂ ∂ 12 12 12 rrt drdr the wave function is given by the wave function 1212 (, ,) ' 1 (, ) (, ) iH ysics at the nanoscale, Strasb Ψ of the system given by ψ 22 mm HrrErr 22 ∫∫ =− Δ − Δ + HVrrt time-independent potential time-independent

European Summer Campus 2012:European Ph where E is the total energy total where E is the A single particle wave function: For The Hamiltonian is given by The Hamiltonian The time evolution is determined by the no Two particle system 10/107 , for + sign Pauli exclusion , and particles with we use ] bosons bosons . The state of the 2 particles is: l. We construct a wave function construct a wave l. We

2 are ψ

ourg, France, July 01–07, 2012 ourg, ± integer spin ψψ n be proved in relativistic QM. = and particle 2 in 1 ψ not occupy the same state. ψψ ψψ (W. Pauli, PR 1940). [ 12 11 22 rr r r

(, ) () () ysics at the nanoscale, Strasb h particle is in which state:

ψ = . : particles with : In QM all particles are identica particles : In QM all : particle 1 in 12 11 22 21 12 fermions rr C r r r r (, ) () () () () ±

ψ minus sign ) is a consequence of this Theorem. spin are the PEP ( This spin statistics Theorem ca

European Summer Campus 2012:European Ph : Two identical fermions can half-integer Spin statistics Theorem QM accommodates 2 kinds of identical particles, for fermions Indistinguishable particles whic as to that is noncommittal PEP Note: principle Symmetrization of the wave function Distinguishable particles BOSONS AND FERMIONS .

11/107

ψ = fermions 12 21 for (, ) (,) – ψ Prr rr and

ourg, France, July 01–07, 2012 ourg, bosons . for -1 + ized state, it remains in such state and is required to be symmetric +1 and H commute (compatible observables). P are P ψ ysics at the nanoscale, Strasb ± =1 and = interchanges the two particles: 2 P 12 21 rr rr (, ) (,)

ψ The eigenvalues of . P

exchange operator P Summer Campus 2012:European Ph It is then obvious that The wave function of a QM system or antisymmetric. The If the system starts in a given symmetr during its evolution. The exchange operator P We can find a complete set of functions that are simultaneous eigenstates of H and 2 2 1,2 12 12/107 2 is 1 || 2 2| | 2 ∓ x D two identical 2 22 = 12 1,2 12 22 two bosons 12 2 xx x =± () of between identical 2 −=+− =− 12 1 2 12 12 xx x x xx xxxxx () xx x x x −=+− 12 x ()

ourg, France, July 01–07, 2012 ourg, and and || , ⎤ ⎦ 22 ∓ xdxxdx exchange force separation value so that () ()

ψψ 21 2 111 1 22 2 ishable particles, and that of 12 1,2 ψψ x 12 ∫∫

= ψψ ψψ [] = ⎡ ⎣ 2 1 22 2 1 = 12 11 22 xx x x =± =+

(, ) () () ysics at the nanoscale, Strasb D 222 2 * 12 ψ xxx 12 11 22 21 12 xx x x rr r r r r (, ) () () () () 2 ± () () of the square 12 , . It is like there an ψ ⎤ ⎦ 12

and ψψ xx xdx 2 22 2 ∫ (, ) D = = ψ 12 increased 1,2 ± 22 2 xx x 2 is ⎡ ⎣ compared to that of distingu ∫∫ 2 1 = =+ European Summer Campus 2012:European Ph expectation value −=+− 22 222 1 111212 12 xxx x x x x dx dx xx x x xx x where Identical particles: () Exchange force fermions particles. The reduced Proof (one dimension case) Similarly, Distinguishable particles: 13/107 e- + symmetrization of the +

ourg, France, July 01–07, 2012 ourg, e- repulsive exchange force ψχ no energy is spent in moving the particles. Anitisymmetric configuration produces at a singlet (antisymmetic spin state) les it is the antisymmetric configuration = the spin part is antisymmetric. because rr rr ss (, ) (, )(, ) ysics at the nanoscale, Strasb ,12 12 12 12 ss purely geometrical consequence ψ + is a is not a real force e- e- attractive exchange force + In molecules, the bonding state is

European Summer Campus 2012:European Ph exchange force Symmetric configuration produces The spatial part can be symmetric if Note: and the anti-bonding a triplet state (antisymmetric spatial wave function, but symmetric spin function). Exchange force (continued) Because electrons are fermions, in molecu that should happen. This is true but one has to take into account the spin of the electrons. The requirement. It 14/107 . . . b E singlet state different from the atomic s level. e-e interaction 22 for simplification. = H 11 , ij δ

() France, July 01–07, 2012 ourg, = H 0 〉= E | ij 〈

ψ and the eigenvectors are: because of the with β 2 has the lowest energy - > and 0 b 〉= 〉 of H ψ |

ψ = E || a E HE () et

22 ysics at the nanoscale, Strasb 11 β and assume that += + += 〉= 〉+ 〉 〉= 〉− 〉 0 ba

21 1 0 2ψψ 2 ||1|2||1|2 01 122 1 = E Ec H cHc Ec Ec Ec b ⎧ ⎨ ⎩ E The hydrogen molecule the molecular state . β = 21 12 cc =H denote the state of an electron in molecule, it can be written as a linear 12 difficult to get the correct GS 〉= 〉+ 〉 European Summer Campus 2012:European Ph ψ ψ ||1|2 < 0 combination of the state of the isolated atoms: Let It is Instead, a simple molecular orbital approach captures most of the physics. Molecules: The SE for the molecule is : To solve it we project into the atomic basis and obtain: The fact that the Hamiltonian is hermitian the s orbital is real implies and that H The solutions are: β The spin state is therefore antisymmetric and correspond to a 15/107 12

ψψ = rrr () 2 () () a b ρ ρ QM interference bond ρ Charge density . There is no analogue in

ourg, France, July 01–07, 2012 ourg, where s is a direct consequence of ysics at the nanoscale, Strasb 12 b a ψ ψ 2 wave function ==++

European Summer Campus 2012:European Ph rrrrr () 2| ()| () () () b b bond a. Bonding b. Antibonding Representation of bonding and antibonding states ρψρρρ classical physics. between the wave functions of constituent atoms The chemical bond in molecule 21

16/107 and H and 12 (0)=1 1 c ba ba hh

hh β ωω

ωω so that it it it it −− −− : sin |1> Ψ are then given by:

ourg, France, July 01–07, 2012 ourg, 2 =−

β =+ c 102 () () 01 2 21 2 11 2 and (

β ct Aect Ae Ae Ae 1 ⎧ ⎪ ⎨ ⎪ ⎩ =+ c =+ d 2 1 β dt dt − dc dc ih Eih c c c E c ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

β hh ⎛⎞⎜⎟⎝⎠ ⎛⎞ ⎜⎟ ⎝⎠ cos an 00

hh ysics at the nanoscale, Strasb are given by are given EE it it and −− e molecule is in the state 2 et et and insert them into these coupled equations, we obtain: into these them and insert β == oscillate harmonically with time. ) . The coefficients et c + hh 2 () 1 12

ω 00 c it ct ct EE − =1/2 2 == and ba are arbitrary constants to be determined by the initial conditions. 1 ωω =A 2 1 c A A = ii cAe and (,) (,) 1 , then Ψ=Ψ A t ∂

∂ Summer Campus 2012:European Ph (0)=0 ih r t H r t 2 If we set where and the coefficients c If we know that at t=0s, th c We start from the time-dependent SE for state The amplitudes Importance of the off-diagonal Hamiltonian matrix elements H 17/107 . and back again again and back , and returns to β |2> h/4 to state

ourg, France, July 01–07, 2012 ourg, |1> h β r orbital is vibrating between 3 4 that the molecule is in state |2> ⎠ 2

β h ⎛⎞ ⎜⎟ ⎝ (t)| 2 2 corresponding to the ionization energy the off-diagonal elements of Hamiltonian β = h 2 2 passing from state

ysics at the nanoscale, Strasb et |c () sin 13.6 eV 2 2 ct t (t)| 1 ⎠

β of about h . An electron in this molecula 4

β h . For this reason /h ⎛⎞ ⎜⎟ ⎝ β 2 /h 2 β 2 is . At the same time probability β = 2 |1> h/2 () cos and 1

ct t Summer Campus 2012:European Ph energy barrier 1 0 Probabilities |c one in the other is The frequency of the molecule to state the two atoms. The electron tunnel from atom 1 to 2 back and forth despite the is exactly out of phase. The probability starts at 1 and then decreases to 0 in time The probability per unit time that the electron tunnels or “hops” from one atom to are called hopping integrals. )

ABS 18/107 . is now 11 ) results in it 2 − 12 EE E1 > E2 is to increase the 2 Δ= et E 1 ) and antibonding ( state

, respectively. France, July 01–07, 2012 ourg, charge density to be 2 BS where reverse direction for the ABS and E 1

β molecule except that the H for some 2 += β β += BS 1222 ccEc and is in 22 cEcEc 22 11 2 1

E β ⎧ ⎨ ⎩

ysics at the nanoscale, Strasb 2 2 + + 12 12 . Let us call them E EE EE () () 22 =+Δ+ =−Δ+ a b are equal. E E ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ABS : and

European Summer Campus 2012:European Ph BS splitting between the bonding and antibonding states. Notice that the difference in on-site energies E For the heteronuclear molecule, the fact that (for example being energetically favorable in the transferred from atom 1 to 2 different from the H The problem is similar to that of the H In the homomuclear the charge density associated with each atom case in both the Heteronuclear diatomic molecule The SE of the molecule’s gives: eigenvalues The secular determinant leads to a bonding state ( The secular determinant leads to a bonding 19/107 because because . β These simple ideas hold / in the secular equation Δ = є є ourg, France, July 01–07, 2012 ourg, ABS nsferred to atom 2 for the BS and all or the BS bond becomes partially ionic BS the respectively. Here ,

ABS ysics at the nanoscale, Strasb kes place in the molecule.

εε is not zero 1 Δ 22 ε +± + , when 12 2 1 = increases all the charge density is tra 2 2 2 1 European Summer Campus 2012:European Ph c c є 0 Heteronuclear diatomic molecule 1 charge is transferred to atom 1 for the ABS. Indeed if we insert the eigenvalues of we find: As Conclusion some charge transfer ta also for alloys. )

) 20/107 triplet state triplet singlet state 3 4 4 1 − ⎪ ⎩ ⎧ ⎪ ⎪ ⎨ ⎪ ⋅= 12 ˆˆ SS ferromagnetic coupling ( antiferromagnetic coupling ( ourg, France, July 01–07, 2012 ourg, this is because ˆˆ ij i j JS S 12 s to the Heisenberg Hamiltonian ˆˆ > ij ∑ J is the exchange parameter ˆ −⋅

ysics at the nanoscale, Strasb = ˆ H

ψψ ψψ χ ψψ ψψ χ [] [] ** 33 11 22 12 21 1 2 2 2 1 1 ψψ ψψ 12 and the triplet state is favored ∫ and the singlet state is favored ˆˆ 2()()()() =− =+ , the spin dependent term of effective Hamiltonian is: T T T +−−⋅ ST ST -E EE EESS 12 11 22 21 12 S 12 11 22 21 12 rr r r r r rr r r r r (3)() −= (, ) () () () () (, ) () () () () =− ⋅ 4 1 SS TT ST EE r rdrdr rHr

ψ ψ = J =E ˆ HJSS ˆ H if J>0 Es>E if J<0 Es

European Summer Campus 2012:European Ph The wave functions of the singlet and triplet states are: Heisenberg Hamiltonian The effective spin Hamiltonian is therefore given by: Defining The generalization to N spins lead Notice 21/107

ourg, France, July 01–07, 2012 ourg,

ysics at the nanoscale, Strasb

European Summer Campus 2012:European Ph 22/107 L J ourg, France, July 01–07, 2012 ourg, maximize M more than half filled, then L tal spin has the lowest energy, and 2S+1 tal angular momentum and L is the orbital l angular momentum J=|L-S|; if it 0 0 0 4 9/2 4 2 P S S F D P F

ysics at the nanoscale, Strasb 23 6 5 74 8 3 43 10 1 3p 3d 3d 2p 3d 3d 2 : If a subshell: If (n,l) is no 2 2 1 2 2 2s 2 21 : The state with the highest to He 1s O 1s Si (Ne)3s Fe (Ar)4s Co (Ar)4s Ni (Ar)4s Cu (Ar)4s

European Summer Campus 2012:European Ph Examples: angular momentum written S, P, D, F, G, H, I, K, L, M, N, etc. Hund’s second rule the lowest energy level has tota is more than half filled, then J=|L+S| has the lowest energy. S is the total spin momentum, J is the to Hund’s rules The ground state of an atom is then denoted: for the largest multiplicity of S, it has lowest energy largest L value. the Hund’s first rule + 23/107 + (1) SS (Brillouin) gJJ eff

μ 2(1) B 0 = kT 3d orbital quenching

= μ 3 because the crystal field splitting is much larger than the SOC. ′ eff B n is the Landéis the factor eff B J

J μμ g μμ =

χ Expt 1.70 5.36 5.82 1.83 B

μ France, July 01–07, 2012 ourg, / eff ’ Expt. 2.51 7.98 9.77 9.5 μ 1.73 4.90 5.92 1.73 B μ B μ eff/ / μ 2.54 7.94 9.72 9.59 eff 1.55 6.70 5.92 3.55 μ 5/2 7/2 6 15/2 F S F I term 2 8 7 4 3/2 4 5/2 5/2 D D S D

term 2 5 6 2 ysics at the nanoscale, Strasb 5/2 7/2 6 15/2 J J 3/2 4 5/2 5/2 L 2 2 0 2 3 0 3 6 L S 1/2 2 5/2 1/2 ½ 7/2 3 3/2 S 1 6 5 9 1 7 8 11 shell 3d 3d 3d 3d 4f 4f 4f 4f shell

3+ Summer Campus 2012:European Ph 3+ 3+ 3+ 2+ 2+ 3+ 3+ Ce Gd Tb Er ion ion Ti Fe Fe Cu Magnetic ground states of 4f and 3d ions using Hund’s rules 24/107

ourg, France, July 01–07, 2012 ourg, t

Course conten ysics at the nanoscale, Strasb

ourg, France, July 01–07, 2012 ourg, ) in cubic O 2g t The d orbitals form two types of irreducible representations (e

ysics at the nanoscale, Strasb

European Summer Campus 2012:European Ph spherical negative field Crystal field

26/107

0 Δ

ourg, France, July 01–07, 2012 ourg, > 3 eV Strong crystal field 0 Δ

ysics at the nanoscale, Strasb orbitals will lie along these axes are affected more with the ionic < 3 eV: Weak< 3 eV: crystal field and if g depends on:

0 Summer Campus 2012:European Ph Δ 0 Octahedral crystal field In octahedral field the degenerecy of the d-orbitals is lifted. Δ 1.2. ligands Nature of the 3. ion metal on the The charge The nature of the orbital (3d, 4d, 5d) If The e electrostatic interaction and move higher in energy. The d-orbitals interact differently with the point charges located at +x, -x, +y,-y, +z and -z 27/107 : 2 and therefore the

ourg, France, July 01–07, 2012 ourg, 0 ughly 2/3 from its octahedron complex 4 9 =Δ high spin due to the reduced crystal t d is therefore reduced by (2/3) Δ

ysics at the nanoscale, Strasb 4 ligands the tetrahedral complex in

European Summer Campus 2012:European Ph Tetrahedral crystal field electrostatic potential is reduced by ro value. The tetrahedron crystal fiel There are only This make all tetrahedral complexes field splitting compared to the pairing energy (U). 28/107 energy 2g and t g Δ U>

ourg, France, July 01–07, 2012 ourg, splitting 2g -t g Δ

ysics at the nanoscale, Strasb U< Hund’s rule

st Summer Campus 2012:European Ph 1 Pairing energy (U) versus e The colors of metallic ions is a consequence the e splitting .

29/107 . z compression 2 short and 4 long bonds octahedron compressed along the z direction z elongation 2 long and 4 short bonds ourg, France, July 01–07, 2012 ourg, 2 -y configuration 2 1 x ) x2-y2 (d ysics at the nanoscale, Strasb 0 system in a degenerate ) 2 Z d ( : ligands along z, -z will be repelled more and bonds elongated 0 ) The octahedron will be elongated along the z direction x2-y2 (d 1 ) 2 Z

European Summer Campus 2012:European Ph Jahn-Teller theorem, 1937: Any non-linear molecular electronic state will be unstable and will undergo distortion to form a system of lower energy thereby removing the degeneracy. For (d Jahn-Teller distorsion The opposite happen for 30/107

ourg, France, July 01–07, 2012 ourg,

ysics at the nanoscale, Strasb

European Summer Campus 2012:European Ph Examples of Jahn-Teller distortions 31/107 M. Verdaguer, Paris VI overlap

ourg, France, July 01–07, 2012 ourg, orthogonality

ysics at the nanoscale, Strasb ↓↑↓ CrNi CrMn ↑ ↑↑↓↓ ⎩ ⎧ ⎪ ⎨ ⎪ ↑↑↓↑ =↑↓↑↓ { =

European Summer Campus 2012:European Ph Ferro AntiFerro The superexchange reduces the kinetic energy of electrons and stabilizes the antiferromagnetism Magnetic order Superexchange mechanism 32/107

ourg, France, July 01–07, 2012 ourg, e most stable because of

ysics at the nanoscale, Strasb structure is energetically th AF2

European Summer Campus 2012:European Ph Magnetic order in NiO the superexchange mediated by the oxygen atoms. The 3 ) 3 MnO

x 33/107 Sr MnO x 1-x Sr 1-x 4+ in La 4+ Mn and Mn

ourg, France, July 01–07, 2012 ourg, g g 2g 2g 3+ e t e t oduces ferromagnetic coupling in La No yes

ysics at the nanoscale, Strasb 3+ Mn g 2g g 2g e t e t

European Summer Campus 2012:European Ph The double exchange mechanism pr Double exchnage In some mixed valence oxydes (example: Mn 34/107

ourg, France, July 01–07, 2012 ourg, t

Course conten ysics at the nanoscale, Strasb

ourg, France, July 01–07, 2012 ourg, spin split DOS and appearance of spin split energy reduction of the magnetic moment 22 2 2 2 Spontaneous magnetism. E/2 Δ 22 2 11 1 ysics at the nanoscale, Strasb ) split ()()(1())0 є 2 B 2 1

for the μλ 0

. μ = (')' ( ) (()) U 000 μλ μλ μμλ 0 ∫ M KMFF F two subbands MF B F into 2 spin directions Paramagnetic DOS g( Summer Campus 2012:European Ph WMdMMnnUgEE WW W UgE gEE where Δ=− =− =− −=− Δ Δ=Δ+Δ= Δ − < Question: Can a material save energy by becoming ferromagnetic? This energy cost is compensated by an with the molecular field: The total kinetic energy cost due to spin split is: Free-electron model: magnetism of metals (Stoner Criterion) 36/107 > < () gE F 2 BF F ()1 0 ()1 EMB Ug E = = g Ug E P

χμμ F ()1 2 gE M Ug E 2 BF where μ )

ourg, France, July 01–07, 2012 ourg, E. U U Δ < > EMB U E E magnetic field then we can determine g Δ Δ 2 Stoner enhancement (

ysics at the nanoscale, Strasb EE U F ()()(1()) (1()) g P 22() 1 χ W withleads to: respect to M Δ − 1() KMFF F F BUgE 0

Paramagnetic μ Ferromagnetic WW W Δ=Δ+Δ= Δ − − = − − H MM

European Summer Campus 2012:European Ph =≈ =

Stoner Criterion (continued) χ In the ferromagnetic ground state, the DOS at Fermi level are split by the exchange splitting If we add the effect of an applied If we add the the magnetic susceptibility The minimization of 37/107

ourg, France, July 01–07, 2012 ourg, t

Course conten ysics at the nanoscale, Strasb

ourg, France, July 01–07, 2012 ourg,

ysics at the nanoscale, Strasb Nobel prize in Chemistry 1998

European Summer Campus 2012:European Ph Nobel lectures by W. Kohn and J. A. Pople 39/107 . magnetic Quantum transport

ourg, France, July 01–07, 2012 ourg, tter from ab initio methods optical electronic

ysics at the nanoscale, Strasb PROPERTIES of materials structural vibrational Goal: Describe properties of ma European Summer Campus 2012:European Ph 40/107

ourg, France, July 01–07, 2012 ourg, valence. ─ als possible (Bloch theorem) electron interactions. electron interactions. ─ ─

ysics at the nanoscale, Strasb Decouple the movement of electrons and nuclei. Treatment of the electron Treatment of the electron Treatment of the (nuclei + core) To expand the eigenstates hamiltonian. of the Efficient and self-consistent computations of H S. Makes the calculation of materi

European Summer Campus 2012:European Ph Born-Oppenhaimer Hartree-Fock Approximation Density Functional Theory Pseudopotentials all electron potentials or Basis set Numerical evaluation of matrix elements Supercells Main approximations: -

41/107 p >> v e v

ourg, France, July 01–07, 2012 ourg, (2) e ation decouple the electronic and >> M in the potential generated by e Move the nuclei as classical particles y instantaneous ionic configuration. p M Nuclei much slower than the electrons ⇒

ysics at the nanoscale, Strasb nuclear degrees of freedom Born-Oppenheimer Approximation Electrons in their ground state for an ground state Electrons in their fixed positions for nuclei fixed positions for

European Summer Campus 2012:European Ph (1) Solve electronic equations assuming Adiabatic or Born-Oppenheimer approxim -

42/107 Fixed potential “external” to e ourg, France, July 01–07, 2012 ourg,

ysics at the nanoscale, Strasb Born-Oppenheimer Approximation Classical displacement Nuclei

European Summer Campus 2012:European Ph Electrons Ifthenuclear positionsare fixed, thewavefunctioncan be decoupled be thewavefunctioncan fixed, positionsare Ifthenuclear 43/107 One electron density ourg, France, July 01–07, 2012 ourg, Integrates out this information All properties of the system can All properties of the be considered as unique functionals of the ground state density One equation for the density is simpler than the full many-body Schrödinger equation ion of electronic structure

ysics at the nanoscale, Strasb Density Functional Theory in condensed matter ground-state of a quantum many-body system

Many electron wave function Summer Campus 2012:European Ph A special role can be assigned to the density of particles in 3N degrees of freedom for N electrons Contains all information about the system Satisfies the many-electron Schrödinger equation DFT: primary tool for calculat 44/107

ourg, France, July 01–07, 2012 ourg, icles in an external potential, xcept for a constant, by the ground-

ysics at the nanoscale, Strasb Density Functional Theory . : For any system of interacting part First theorem of Hohenberg-Kohn: European Summer Campus 2012:European Ph Theorem I this potential is determined uniquely, e particle density 0 ) ) 45/107 for (r) (r) t t '' ' ex (r) ex VE V ext −+

). ) V’ r

( ’, and

0 t ψ ext n ex ( ( and (r) (r) (r) (r (r) and 0 00 ψ ext 3 3 drn V drn V E ∫ ∫ −−+ = =

ourg, France, July 01–07, 2012 ourg, ψ ’> respectively. ψ ' ψ | H H’ ' ’| be two different external potentials ψ HH E : H e wave function for this ground state =< 0 ' ' H E’ E '| | ' H || <+

ψ ' 00 〈−+〉 ψ E 〈−+〉 = > and (nondegenerate state) ysics at the nanoscale, Strasb + = ψ 00 E |H| Density Functional Theory ψ =< 0

ψψ ψ ψψ ψ |' | | '' : || '||' '|

this ground state density. ψ ψ Summer Campus 2012:European Ph =〈 〉<〈 〉 =〈 〉<〈 〉 These 2 potentials produce different wave functions 2 total energies E iii. Let’s suppose that we have 2 external potentials V Proof by reductio absurdum: i. Let’s suppose that we have an exact ground state density ii. Let’s assume that there is only on The outcome is absurd, so there cannot that produce the same density for their ground state. ' 0 By adding these 2 equations we obtain 0 Similarly The variational insures that theorem EH H EH H 46/107 For any . ext V in terms of E[n]

ourg, France, July 01–07, 2012 ourg, that minimizes the functional is econd n r any external potential .

0 ysics at the nanoscale, Strasb n Density Functional Theory can be defined, valid fo , the exact ground state of system is global minimum A universal functional for the energy n ext V

European Summer Campus 2012:European Ph Second theorem of Hohenberg-Kohn: the density particular Theorem II: exact ground state density value of this functional, and the density theorem of Hohenberg-Kohn S , ext V’ 47/107 0 ’(r) cannot n ’]. n [

E ≤

0 E ,

ourg, France, July 01–07, 2012 ourg, ’(r)= and an external potential

n then: ’ ψ r 3 ext d V ∫ 3 ondence between the wave function and the between the wave function and ondence 3 . ∫

0 and 0

≥ E ∫ r)

n’(

ysics at the nanoscale, Strasb HK ext Density Functional Theory =+ HK ext [] [] (r) (r) En F n drn V corresponds a wave function n’ HFndrnVEnEn '| | ' [ '] '(r) (r) [ '] [ ]

European Summer Campus 2012:European Ph ψψ 〉+=≥ 〈〉=+ be lower that the exact energy For a trial electron density The ground-state total energy calculated from the trial density This is because there one to corresp electronic density. This second theorem can be rewritten as: This theorem is easily proved since: to the density and to n correspond the external potential 48/107 ansatz Kohn-Sham (never proven in general) n

ourg, France, July 01–07, 2012 ourg, independent particle system independent particle Density of an auxiliary non-interacting =

ysics at the nanoscale, Strasb only the ground state density only the Density Functional Theory an independent-particle problem All the properties of system are completely determined given Ground state density of the density of Ground state

many-body interacting system Summer Campus 2012:European Ph The Kohn-Sham ansatz replaces the many-body problem with But no prescription to solve the difficult interacting many-body hamiltonian 49/107

ourg, France, July 01–07, 2012 ourg, e auxiliary independent-particle system

ysics at the nanoscale, Strasb Density Functional Theory We assume that each electron moves independently in a potential created by the nuclei and rest of electrons.

European Summer Campus 2012:European Ph One electron or independent particle model Actual calculations performed on th 50/107

δ [] n

En δ δ = [] En n } δ

ourg, France, July 01–07, 2012 ourg, −= rticle kinetic energy which is given that determines the density, 3

ψεψ ψ me we take a variation with respect to ∫ ⎡⎤ ⎣⎦ and knowing that and knowing ext is known then the problem solved, since − equations, we can use the variational theorem Vrr

HK +=

[] (r) 0 ysics at the nanoscale, Strasb

HF F En drn N { al of the orbitals:

δε * Density Functional Theory (r) [(r)] ψψ n = HK δ Fn

(r) (r) (r) δ ((r))()() n

European Summer Campus 2012:European Ph We then obtain from the variationalWe then obtain the KS SE: theorem the SE will produce a selfconsistent and hence ground the state total energy. The trick is to use the independent-pa explicitly as a function If we write If the exact form of the Like the derivation of The Kohn-Sham (KS) equations to find out the KS equations. This ti the electron density: 51/107 The rest: Exchange- correlation .

ourg, France, July 01–07, 2012 ourg, XC orrelation hole has to fulfill the sum rule:

ysics at the nanoscale, Strasb Density Functional Theory can be written in terms of the exchange-correlation XC : xc

European Summer Campus 2012:European Ph hole n The variational gives then: theorem The exchange-correlation energy has to obey certain rules: Then one has to rewrite the functional as This last sum rule help constrain the search for E 1. Like in the HFA, E 2. Again, like in the HFA, exchange-c 52/107

ourg, France, July 01–07, 2012 ourg,

ysics at the nanoscale, Strasb Density Functional Theory approach proposed by Kohn and Sham Density functional theory is the most widely used method today for electronic structure calculations because of the

European Summer Campus 2012:European Ph 53/107 3 sum xc

ε 3 ∫ =

ourg, France, July 01–07, 2012 ourg, nnrnrdr [] () (()) ous electron gas, it depends only on

ε XC XC E 3 ∫ = nnrnrdr [] () [,]

ysics at the nanoscale, Strasb XC XC E : Density Functional Theory XC

ε ∫ =∇ nnrnrnrdr [] () ((), (),...) at r depends on the density and its Summer Campus 2012:European Ph at r depends only on the density (exact for a homogeneous electron gas). at r depends on the density shape n in the whole space. XC XC XC XC XC The success of the LDA is certainly due to fact that it satisfy the n rule, and being constructed from a homogene the spherical average of this hole density. E є gradient at r. It is a function of the density and not functional! є Generalized Gradient Approximation (GGA) The general form of E є Local density approximation (LDA): 54/107 m(r).

ourg, France, July 01–07, 2012 ourg, and the magnetization n(r) Von Barth and Hedin, J. Phys. C 5, 1629 (1972).

ysics at the nanoscale, Strasb Density Functional Theory

European Summer Campus 2012:European Ph depend on the position. We can always use z-direction for convenience. The wave function for each spin direction (+ or -) is given by: each spin direction (+ or -) is given The wave function for Local spin density approximation Here the functional is of density In the collinear case, we assume that direction of magnetization does not The spin-up (+) and down (-) electron densities are given by: The total charge density and the magnetization are given by: LSDA was first introduced by 55/107 The rest: Exchange- correlation

ourg, France, July 01–07, 2012 ourg, icle equation for each spin direction:

ysics at the nanoscale, Strasb yields the number of valence electrons in the unit cell: Density Functional Theory are determined so that the total density of states integrated i w Fermi energy

European Summer Campus 2012:European Ph The exchange-correlation per spin is determined by functional derivative: up to the The weights The variational theorem gives the one part 56/107 stresses … Energy, forces, Output quantities Yes

ourg, France, July 01–07, 2012 ourg, No

ysics at the nanoscale, Strasb : Structure, Atomic species Density Functional Theory :

European Summer Campus 2012:European Ph output the tolal energy,forces, etc… The Kohn-Sham equations must be solved self- consistently The potential (input) depends on the density (output) I. Input II. for input Guess III. Compute the potential IV. Solve the Ks equations V. Compute the output density VI. If selfconsistent PAW 57/107 ,

ourg, France, July 01–07, 2012 ourg,

ysics at the nanoscale, Strasb

European Summer Campus 2012:European Ph Used approximations, basis-sets, potentials, etc ˆ

58/107 + (). , GikGr nk , Ce + G ∑ GUrBGUrYr ( ) () ( ) () () ,, ourg, France, July 01–07, 2012 ourg, nk nk m lm l l m l l A ⎡⎤ ⎣⎦ lm ∑ Partial waves in the muffin-tin spheres

ysics at the nanoscale, Strasb Plane waves in the interstitial:

European Summer Campus 2012:European Ph LSDA, GGA, or LDA+U - treatment Different of the muffin-tin spheres and interstitial region - (Full-Potential Linear Augmented Plane-Wave method Methods of calculation 59/107

ourg, France, July 01–07, 2012 ourg, results, but not always better. -5% +1% enough (structural properties, …). enough GGA -50% +10% -20%,

ysics at the nanoscale, Strasb LDA -50% +15% -1% , -3% +10, +40% Density Functional Theory

c a B gap E E usually tends to overcompensate LDA simplest approximation but accurate :

: Summer Campus 2012:European Ph Accuracy of the XC functionals in the structural and electronic properties GGA LDA 60/107 lattice constant –NM –NM –fcc –FM –bcc – Correct –FM –bcc LSDA GGA Experiment

ourg, France, July 01–07, 2012 ourg, : DFT ground state of iron LSDA GGA

ysics at the nanoscale, Strasb Density Functional Theory GGA LSDA Courtesy of Karl H. Schwartz Results obtained with Wien2k. GGA

European Summer Campus 2012:European Ph In some cases, GGA is a must P. Bagno al., PRB 40, 1997 (1987) et 61/107 edges 23

ourg, France, July 01–07, 2012 ourg, (SFMO) 6

XAS and XMCD at Fe Mo L ysics at the nanoscale, Strasb - FeMoO 2 Sr DOS

European Summer Campus 2012:European Ph Kanchana et al., PRB 75, 22040(R), 2007 SFMO crystallizes in a bct, a=b= 5.55Å and c=7.90 Å Spin and orbital moments of Fe Mo 62/107

ourg, France, July 01–07, 2012 ourg, t

Course conten ysics at the nanoscale, Strasb

ourg, France, July 01–07, 2012 ourg,

ysics at the nanoscale, Strasb BIBLIOGRAPHY http://www.ifwdresden.de/institutes/itf/members/helmut C. Kittel, Wiley J. M. Ziman, Wiley S. Blundell, Oxford Univ. Press R.M. Dreizler and E.K.U. Gross, Spinger-Verlag, 1990 H. Eschrig, D. C. Mattis, World Scientific J. Kuebler, Oxford science publications N. W. Aschroft and N. D. Mermin

European Summer Campus 2012:European Ph Solid State Physics Magnetism in Condensed Matter, Principles of the Theory of Solids Density functional Theory,, The Fundamentals of Density functional Theory,, Theory of Itinerant Electron magnetism, The Theory of Magnetism Made Simple Solid State Physics, 9 9 9 9 9 9 9 9