iv. electronic structure and chemical bonding j.k. burdett, chemical bonding in solids experimental...
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![Page 1: IV. Electronic Structure and Chemical Bonding J.K. Burdett, Chemical Bonding in Solids Experimental Aspects (a) Electrical Conductivity – (thermal or optical)](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649d785503460f94a5b2cc/html5/thumbnails/1.jpg)
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
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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;
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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
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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
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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
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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
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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
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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
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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.”
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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?
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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!
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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?
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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*