IV. Electronic Structure and Chemical Bonding J.K. Burdett, Chemical Bonding in Solids Experimental...

IV. Electronic Structure and Chemical Bonding J.K. Burdett, Chemical Bonding in Solids

Experimental Aspects

(a) Electrical Conductivity – (thermal or optical) band gaps; (b) Magnetic Susceptibility – localized or itinerant; para- or diamagnetic; (c) Heat Capacity – specific heat due to conduction electrons; lattice; (d) Cohesive Energy – energy required to convert M(s) to M(g); (e) Spectroscopy – XPS, UPS (for example); (f) Phase Changes – under temperature or pressure variations

Experimental Aspects

(a) Electrical Conductivity – (thermal or optical) band gaps; (b) Magnetic Susceptibility – localized or itinerant; para- or diamagnetic; (c) Heat Capacity – specific heat due to conduction electrons; lattice; (d) Cohesive Energy – energy required to convert M(s) to M(g); (e) Spectroscopy – XPS, UPS (for example); (f) Phase Changes – under temperature or pressure variations

Theoretical Aspects

(a) Electronic Density of States (DOS curves) – occupied and unoccupied states; (b) Electron Density – where does electronic charge “build up” in a solid? (c) Analysis of DOS – overlap (bonding) populations, charge partitioning,… (d) Band structure – energy dispersion relations; (e) Equations of State – E(V) curves for various structures; (f) Phonon DOS – vibrational states of crystals; stability of structures ( < 0 ??) (g) “Molecular Dynamics” – phase transitions; crystallization models;

Metallic

Covalent, Molecular

Zintl Phases

Metalloids

"Polymeric"

van Arkel-Ketelaar Triangle

Average Electronegativity

ElectronegativityDifference

L.C. Allen, J. Am. Chem. Soc. 1992, 114, 1510

Hand-Outs: 1

Metallic

Covalent, Molecular

Zintl Phases

Metalloids

"Polymeric"

= “Configuration Energy”

L.C. Allen et al., JACS, 2000, 122, 2780, 5132

Hand-Outs: 1

Metallic

Covalent, Molecular

Zintl Phases

Metalloids

"Polymeric"

Low valence e/orbital ratioLow IP(I)Small

High valence e/orbital ratioHigh IP(I)Small

Large Charge transfer from cation to anion

Hand-Outs: 1

Metallic

Covalent, Molecular

Zintl Phases

Metalloids

"Polymeric"

Electrical ConductorsParamagnetic; Itinerant magnetism

Soft – malleable, ductile

Electrical InsulatorsDiamagneticLow boiling points

Electrical Insulators; Conducting liquidsDiamagnetic; Localized magnetismBrittle

Hand-Outs: 1

Metallic

Covalent, Molecular

Zintl Phases

Metalloids

"Polymeric"

Elect. Semiconductors / SemimetalsDiamagnetic

“Hard” – Brittle

Electrical SemiconductorsDiamagnetic

Elect. Semiconductors / SemimetalsDiamagnetic

Hand-Outs: 1

Schrödinger’s Equation: {n} = E{n}{n}

: “Hamiltonian” = Energy operatorKinetic + Potential energy expressions; external fields (electric, magnetic)

{n}: Electronic wavefunctions (complex)(r) = *{n}{n} dV: Charge density (real)

E{n}: Electronic energies

Temperature: How electronic states are occupied –

Maxwell-Boltzmann Distribution: f(E) = exp[(EEF)/kT]

Fermi-Dirac Distribution: f(E) = [1+exp((EEF)/kT)]1

“A solid is a molecule with an infinite number (ca. 1023) of atoms.”

• Molecular Solids: on molecular entities (as in gas phase); packing effects?

• Extended Solids: how to make the problem tractable?

(a) Amorphous (glasses): silicates, phosphates – molecular fragments, tieoff ends with simple atoms, e.g., “H”;

(b) Quasiperiodic: fragments based on building units, tie off ends withsimple atoms, e.g., “H”;

(c) Crystalline: unit cells (translational symmetry) – elegant simplification!

L X W L K

Electronic Structure of Si:

Fermi Level

Electronic Band Structure Electronic Density of States

What can we learn from this information?

IV. Electronic Structure and Chemical Bonding Periodic Functions J.K. Burdett, Chemical Bonding in Solids

General, single-valued function, f (r), with total symmetry of Bravais lattice:

)()()(:)( rrtrr in Aefff Plane waves: ei = cos + i sin

(r) = K r, K: units of 1/distance

( )( ) ( )n ni iinf Ae Ae e f K r t K tK rr t r

1,Therefore nie tK K tn = 2N

{Km} = Reciprocal Lattice: Km = m1a1* + m2a2* + m3a3*(m1, m2, m3 integers)

Therefore, for r = ua1 + va2 + wa3, the general periodic function of the lattice is

wmvmumii AeAef m 3212)( rKr

Hand-Outs: 2

IV. Electronic Structure and Chemical Bonding Group of the Lattice J.K. Burdett, Chemical Bonding in Solids

Bravais Lattice: {tn = n1a1 + n2a2 + n3a3; n1, n2, n3 integers}

(1) Closed under vector addition: tn + tm = tn+m lattice

(2) Identity: t0 = 0 lattice

(3) Vector addition is associative: (tn + tm) + tp = tn + (tm + tp)

(4) Inverse: tn = tn, tn + tn = 0

ALSO: (5) Vector addition is commutative: tn + tm = tm + tn

The (Bravais) Lattice is an “Abelian group”:(a) # classes = # members of the group(b) # members of the group = # irreducible representations (IRs)(c) each IR is one-dimensional (a 11 matrix; a complex number, ei )(d) Periodic (Born-von Karman) Boundary Conditions: Set N1a1 = identity (like 0), N2a2 = identity, and N3a3 = identity

1 n1 N1, 1 n2 N2, 1 n3 N3, Order of {tn} = N = N1N2N3

Therefore, N IRs, each labeled km: * * *31 21 2 3

; 1m i i

mm mm N

k a a a

Hand-Outs: 2

IV. Electronic Structure and Chemical Bonding Periodic Boundary Conditions: Reciprocal (“k”) Space

Real Space (2D)

N1 = 3; N2 = 3

n1 = 1, 2, 3; n2 = 1, 2, 3

Hand-Outs: 3

Real Space (2D)

N1 = 3; N2 = 3

n1 = 1, 2, 3; n2 = 1, 2, 3

Lattice Group (9 members) = {a1+a2, 2a1+a2, 3a1+a2,a1+2a2, 2a1+2a2, 3a1+2a2, a1+3a2, 2a1+3a2, 3a1+3a2}

Hand-Outs: 3

Real Space (2D)

N1 = 3; N2 = 3

4a2 = 3a2 + a2

4a2 a2

n1 = 1, 2, 3; n2 = 1, 2, 3

Hand-Outs: 3

Real Space (2D)

N1 = 3; N2 = 3

4a2 = 3a2 + a2

4a2 a2

n1 = 1, 2, 3; n2 = 1, 2, 3

Reciprocal Space

a2 a1*

Hand-Outs: 3

Real Space (2D)

N1 = 3; N2 = 3

4a2 = 3a2 + a2

4a2 a2

n1 = 1, 2, 3; n2 = 1, 2, 3

Reciprocal Space

a2 a1*

Allowed IRs (9 k-points)

k11 = (1/3)a1*+ (1/3)a2*; k12 = (1/3)a1*+ (2/3)a2*;k13 = (1/3)a1*+ (3/3)a2*; k21 = (2/3)a1*+ (1/3)a2*;k22 = (2/3)a1*+ (2/3)a2*; k23 = (2/3)a1*+ (3/3)a2*;k31 = (3/3)a1*+ (3/3)a2*; k32 = (3/3)a1*+ (2/3)a2*;k33 = (3/3)a1*+ (3/3)a2*

Hand-Outs: 3

IV. Electronic Structure and Chemical Bonding Group of the Lattice: IRs

t1 = axt2 = 2ax

t3 = 3axt4 = 4ax = Identity

Lattice: {t1, t2, t3, t4 = identity}

Real Space

Hand-Outs: 4

t1 = axt2 = 2ax

K = 0x K = a* = (2/a)x

k1 = (1/4)a*

k2 = (2/4)a*

k3 = (3/4)a*

k4 = (4/4)a*k4 = (0/4)a*Lattice: {t1, t2, t3, t4 = identity}

IRs: k1, k2, k3, k4

Real Space Reciprocal Space

Hand-Outs: 4

t1 = axt2 = 2ax

K = 0x K = a* = (2/a)x

k1 = (1/4)a*

k2 = (2/4)a*

k3 = (3/4)a*

IRs: k1, k2, k3, k4

4,3,2,1;;1

( ) ( )

; , 1,2,3,4

m mnin

f x t f x

Hand-Outs: 4

f (x) = General functionon 1D Lattice

= Basis functionof 1D Lattice

( )mkf x

t1 = axt2 = 2ax

K = 0x K = a* = (2/a)x

k1 = (1/4)a*

k2 = (2/4)a*

k3 = (3/4)a*

IRs: k1, k2, k3, k4

4,3,2,1;;1

Basis Function for IR km: xmk

xetxmm k

imnnk 4/2

Hand-Outs: 4

( ) ( )

; , 1,2,3,4

m mnin

f x t f x

f (x) = General functionon 1D Lattice

= Basis functionof 1D Lattice

( )mkf x

IV. Electronic Structure and Chemical Bonding Group of the Lattice: IRs (Character Table)

t1 t2 t3 t4 Basis Functions (Real / Imaginary) Most General

k1 i 1 i 1

ei x/2a

(Complex conjugate of k3)

k2 1 1 1 1 Real

ei x/a

(Real Representation)

k3 i 1 i 1

e3i x/2a = eix/2a

(Complex conjugate of k1)

k4 1 1 1 1 Real

e2i x/a = 1

(Totally symmetric rep)

Hand-Outs: 4

IV. Electronic Structure and Chemical Bonding Group of the Lattice: Reciprocal Space

As the size of the real space lattice increases, N large (ca. 108 in each direction)

Reciprocal space becomes continuous set of k-points: …

t1 t2 … tN

k1 : 1

k2 : 1

: : : : :

kN 1 1 1 1

11 tk ie 1 2ie k t

2 1ie k t 2 2ie k t

{km} is a “quasi”-continuous space; “km” = “k-point” or “wavevector”

IdentityOperation

TotallySymmetricRepresentation

Hand-Outs: 4

IV. Electronic Structure and Chemical Bonding Bloch’s Theorem

The wavefunctions for electrons, phonons (= lattice vibrations) subjected to periodic potential, i.e., U(r + t) = U(r), take the form

nk(r) = eikr un(r)

where un(r) has the full periodicity of the lattice, i.e., un(r + t) = un(r).

Note that nk(r + t) = eikt nk(r)

Therefore, for a determination of electronic states or vibrational modes in crystals, we only need to treat the contents of the unit cell (primitive cell)!

Hand-Outs: 5

The wavefunctions for electrons, phonons (= lattice vibrations) subjected to periodic potential, i.e., U(r + t) = U(r), take the form

nk(r) = eikr un(r)

Corollary #1If K = reciprocal lattice vector, then nk(r) and nk+K(r) have the samesymmetry properties with respect to translations (same IR!)…

nk(r + t) = eikt nk(r);

nk+K(r + t) = ei(k+K)t nk+K(r) = eikt nk+K(r)

Hand-Outs: 5

The wavefunctions for electrons, phonons (= lattice vibrations) subjected to periodic potential , i.e., U(r + t) = U(r) take the form

nk(r) = eikr un(r)

Corollary #1If K = reciprocal lattice vector, then nk(r) and nk+K(r) have the samesymmetry properties with respect to translations (same IR!)…

nk(r + t) = eikt nk(r);

nk+K(r + t) = ei(k+K)t nk+K(r) = eikt nk+K(r)

Hand-Outs: 5

The wavefunctions for electrons, phonons (= lattice vibrations) subjected to periodic potential , i.e., U(r + t) = U(r) take the form

nk(r) = eikr un(r)

Corollary #2n,k(r) is the complex conjugate of nk(r)…

n,k(r) = eikr un(r) = nk*(r)

Hand-Outs: 5

IV. Electronic Structure and Chemical Bonding Brillouin Zones

Allowed IRs for the set of lattice translations are confined to one primitive cell inreciprocal space: (first) Brillouin zone

(0,0) (0,1)

(1,1)(1,0)(1, 1)

(0, 1)

( 1, 1) ( 1,0) ( 1,1)

{km: km = m1a1* + m2a2*; 0 < mi 1}

a1 < a2, a1* > a2*

Hand-Outs: 6

Consider a 2DOrthorhombic Lattice:

IV. Electronic Structure and Chemical Bonding Brillouin Zones

Allowed IRs for the set of lattice translations are confined to one primitive cell inreciprocal space: (first) Brillouin zone

(0,0) (0,1)

(1,1)(1,0)(1, 1)

(0, 1)

( 1, 1) ( 1,0) ( 1,1)

{km: km = m1a1* + m2a2*; 0 < mi 1}

(0,0) (0,1)

(1,1)(1,0)(1, 1)

(0, 1)

( 1, 1) ( 1,0) ( 1,1)

{km: km = 1a1* + 2a2*; 1/2 < i 1/2}

(First)Brillouin Zone (FBZ)(Wigner-Seitz cell)

“Zone Boundary”“Zone Edge”

“Zone Center” =

Hand-Outs: 6

a1 < a2, a1* > a2*

IV. Electronic Structure and Chemical Bonding J.K. Burdett, Chemical Bonding in Solids Experimental...

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