IV. Electronic Structure and Chemical Bonding J.K. Burdett, Chemical Bonding in Solids Experimental...
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Transcript of 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
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
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;
IV. Electronic Structure and Chemical Bonding J.K. Burdett, Chemical Bonding in Solids
Cs F
Metallic
Ionic
Covalent, Molecular
Zintl Phases
Metalloids
"Polymeric"
CsF
HF
Si Al
van Arkel-Ketelaar Triangle
Average Electronegativity
ElectronegativityDifference
L.C. Allen, J. Am. Chem. Soc. 1992, 114, 1510
Hand-Outs: 1
IV. Electronic Structure and Chemical Bonding J.K. Burdett, Chemical Bonding in Solids
Cs F
Metallic
Ionic
Covalent, Molecular
Zintl Phases
Metalloids
"Polymeric"
CsF
HF
Si Al
van Arkel-Ketelaar Triangle
Average Electronegativity
ElectronegativityDifference
L.C. Allen, J. Am. Chem. Soc. 1992, 114, 1510
= “Configuration Energy”
L.C. Allen et al., JACS, 2000, 122, 2780, 5132
Hand-Outs: 1
IV. Electronic Structure and Chemical Bonding J.K. Burdett, Chemical Bonding in Solids
Cs F
Metallic
Ionic
Covalent, Molecular
Zintl Phases
Metalloids
"Polymeric"
CsF
HF
Si Al
van Arkel-Ketelaar Triangle
Low valence e/orbital ratioLow IP(I)Small
High valence e/orbital ratioHigh IP(I)Small
Large Charge transfer from cation to anion
Average Electronegativity
ElectronegativityDifference
L.C. Allen, J. Am. Chem. Soc. 1992, 114, 1510
Hand-Outs: 1
IV. Electronic Structure and Chemical Bonding J.K. Burdett, Chemical Bonding in Solids
Cs F
Metallic
Ionic
Covalent, Molecular
Zintl Phases
Metalloids
"Polymeric"
CsF
HF
Si Al
van Arkel-Ketelaar Triangle
Electrical ConductorsParamagnetic; Itinerant magnetism
Soft – malleable, ductile
Electrical InsulatorsDiamagneticLow boiling points
Electrical Insulators; Conducting liquidsDiamagnetic; Localized magnetismBrittle
Average Electronegativity
ElectronegativityDifference
L.C. Allen, J. Am. Chem. Soc. 1992, 114, 1510
Hand-Outs: 1
IV. Electronic Structure and Chemical Bonding J.K. Burdett, Chemical Bonding in Solids
Cs F
Metallic
Ionic
Covalent, Molecular
Zintl Phases
Metalloids
"Polymeric"
CsF
HF
Si Al
van Arkel-Ketelaar Triangle
Elect. Semiconductors / SemimetalsDiamagnetic
“Hard” – Brittle
Electrical SemiconductorsDiamagnetic
“Hard” – Brittle
Elect. Semiconductors / SemimetalsDiamagnetic
“Hard” – Brittle
L.C. Allen, J. Am. Chem. Soc. 1992, 114, 1510
Hand-Outs: 1
IV. Electronic Structure and Chemical Bonding J.K. Burdett, Chemical Bonding in Solids
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
IV. Electronic Structure and Chemical Bonding J.K. Burdett, Chemical Bonding in Solids
Schrödinger’s Equation: {n} = E{n}{n}
“A solid is a molecule with an infinite number (ca. 1023) of atoms.”
IV. Electronic Structure and Chemical Bonding J.K. Burdett, Chemical Bonding in Solids
Schrödinger’s Equation: {n} = E{n}{n}
“A solid is a molecule with an infinite number (ca. 1023) of atoms.”
• Molecular Solids: on molecular entities (as in gas phase); packing effects?
IV. Electronic Structure and Chemical Bonding J.K. Burdett, Chemical Bonding in Solids
Schrödinger’s Equation: {n} = E{n}{n}
“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!
IV. Electronic Structure and Chemical Bonding J.K. Burdett, Chemical Bonding in Solids
(eV)
-14
-12
-10
-8
-6
-4
-2
0
2
4
6
8
10
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
1 2 3
; 1m i i
mm mm N
N N 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
a1
a2
Hand-Outs: 3
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
a1
a2
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
IV. Electronic Structure and Chemical Bonding Periodic Boundary Conditions: Reciprocal (“k”) Space
Real Space (2D)
N1 = 3; N2 = 3
4a2 = 3a2 + a2
4a2 a2
n1 = 1, 2, 3; n2 = 1, 2, 3
a1
a2
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
IV. Electronic Structure and Chemical Bonding Periodic Boundary Conditions: Reciprocal (“k”) Space
Real Space (2D)
N1 = 3; N2 = 3
Lattice Group (9 members) = {a1+a2, 2a1+a2, 3a1+a2,a1+2a2, 2a1+2a2, 3a1+2a2, a1+3a2, 2a1+3a2, 3a1+3a2}
4a2 = 3a2 + a2
4a2 a2
n1 = 1, 2, 3; n2 = 1, 2, 3
Reciprocal Space
a1
a2 a1*
a2*
Hand-Outs: 3
IV. Electronic Structure and Chemical Bonding Periodic Boundary Conditions: Reciprocal (“k”) Space
Real Space (2D)
N1 = 3; N2 = 3
4a2 = 3a2 + a2
4a2 a2
n1 = 1, 2, 3; n2 = 1, 2, 3
Reciprocal Space
a1
a2 a1*
a2*
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*
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
IV. Electronic Structure and Chemical Bonding Group of the Lattice: IRs
a
t1 = axt2 = 2ax
t3 = 3axt4 = 4ax = Identity
Lattice: {t1, t2, t3, t4 = identity}
Real Space
Hand-Outs: 4
IV. Electronic Structure and Chemical Bonding Group of the Lattice: IRs
a
t1 = axt2 = 2ax
t3 = 3axt4 = 4ax = Identity
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
IV. Electronic Structure and Chemical Bonding Group of the Lattice: IRs
a
t1 = axt2 = 2ax
t3 = 3axt4 = 4ax = Identity
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
4,3,2,1;;1
)()4(
)()(
4/24
4
ne
xftxf
xftxf
ninn
nn
nn
2 / 4
( ) ( )
; , 1,2,3,4
m
m
k n n
m mnin
f x t f x
e m n
Real Space Reciprocal Space
Hand-Outs: 4
f (x) = General functionon 1D Lattice
= Basis functionof 1D Lattice
( )mkf x
IV. Electronic Structure and Chemical Bonding Group of the Lattice: IRs
a
t1 = axt2 = 2ax
t3 = 3axt4 = 4ax = Identity
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
4,3,2,1;;1
)()4(
)()(
4/24
4
ne
xftxf
xftxf
ninn
nn
nn
Basis Function for IR km: xmk
xetxmm k
imnnk 4/2
Real Space Reciprocal Space
Hand-Outs: 4
2 / 4
( ) ( )
; , 1,2,3,4
m
m
k n n
m mnin
f x t f x
e m n
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
Real:
Imag:
ei x/2a
(Complex conjugate of k3)
k2 1 1 1 1 Real
ei x/a
(Real Representation)
k3 i 1 i 1
Real:
Imag:
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
1023
1023
{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
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)!
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
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)!
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
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)!
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*
a1*
a2*
Hand-Outs: 6
Consider a 2DOrthorhombic Lattice:
a1
a2
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” =
a1*
a2*
Hand-Outs: 6
a1 < a2, a1* > a2*