Section 3.4 - 1

3.4 Making a Metal: Conductivity and Distortions • Given what we’ve seen so far, it appears that the obvious way of “creating” a metallic

conductor is to arrange atomic or molecular sites, with partially filled A.O.s / M.O.s, in a regular array such that orbital overlap occurs and a partially filled band arises.

• A metallic conductor is illustrated as a partially filled band. • In a metal, conductivity decreases with an increase in temperature. • The decrease in conductivity observed with an increase in temperature is a result of

electron-phonon interactions (lattice vibrations).

σ conductivity n number of charge carriers e charge of one charge carrier (i.e., one electron) µ carrier mobility

• In a metal, n is large and effectively constant. • µ is affected by the interaction of the charge carriers with the lattice vibrations.

µσ en=

Section 3.4 - 2

Conductivity in Metals: The band structure picture • As noted previously, in a free electron model, for every filled state with momentum k,

there is a filled state with equal and opposite momentum -k. • The application of an electric field changes the relative energies of these momenta.

For example, the effect may be the raising the energy of the k states and lower the energy of -k states.

• In the case that the band is partially filled, a net current is generated by the flow of

charge carriers from the higher lying k states to the lower -k energy levels. • If the band is filled, no net flow of carriers is possible since there are no empty energy

levels to receive the electrons from the higher lying k states.

Section 3.4 - 3

There are some potential pitfalls to our simple design, however... Instabilities in Large Rings and Chains • Second-order Jahn-Teller distortions (SOJT) Consider a long chain (CH)4n+2

β1

β2

β1 and β2 are resonance integralsfor non-equivalent bonds

λ1

Eα λ2

βo = β1 = β2

no distortion

βo = β1 β2

distortion

∑−=occupied

o njE π

βπ

2cos22 ∑ +++−=

occupied njE12

2cos22 2122

21

πββββπ

Section 3.4 - 4

Evidence for this distortion ... • The trend in π* ← π transitions for large rings and chains deviates from the simple

pattern expected for a “normal” Hückel system.

• The band structure for conjugated polymers (e.g., polyacetylene) also shows this

distortion.

This instability sets in at n = 34. The resonance stabilization collapses!

Ideal (CH)n Real (CH)n

Section 3.4 - 5

INTERPRETATION: ETotal = Eσ + Eπ In order for there to be a distortion, Edistorted < Euniform

Edistorted < Euniform Edistorted > Euniform • For E to be a minimum at Q = 0

02

2

2

2

2

2

>⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛

!"!#$!"!#$ %%%

A

dQEd

B

dQEd

dQEd σπ

• A is positive and directly proportional to the number of σ-bonds.

• B is negative ⎟⎟⎠

⎞⎜⎜⎝

⎛

+= ∑

−

=

n

nj nj12

sec4 πβ!

• B gets more negative (with increasing n) faster than A gets more positive. At some

large value of n (C34H34), A + B goes negative, and the bond alternation sets in. • This type of instability pervades many discussions of solid state chemistry. • It is the PEIERLS

instability.

Section 3.4 - 6

A chain of H atoms ... a thought experiment • Consider a chain of evenly spaced hydrogen atoms. • The atoms are close enough in proximity that overlap of the 1s orbitals occurs

generating a half-filled band → metallic conductor

• We know, however, that this system is more stable as discrete molecules of H2

(at STP conditions) • However, at high pressure ( > 4 Mbar), hydrogen does, in fact, form a solid 3D

network of evenly spaced atoms with metallic conducting properties! Peierls Distortion (or Charge Density Wave Driven Distortion) ... General Notes • Every partially filled 1D system is unstable as a regular array and can distort under

certain conditions. • This effect is known as Peierls distortion or a charge density wave (CDW) driven

distortion. • For a system in which there is one unpaired electron per site (i.e., band filling = 0.5)

this distortion is manifested as alternating bond distances (DIMERIZATION!) • Any degree of partial band filling in a 1D system has an associated CDW and is

therefore subject to a Peierls distortion. • A CDW is a repeating pattern of alternating regions of higher and lower charge

density within a lattice.

H H H H 1D Metal

H H H H Molecular H2

Section 3.4 - 7

CDW-driven Instability ... ½ filled band model • Consider a regular infinite chain of periodic potentials

• Redefine the lattice constant (use 2a instead of a) • Unit cell length a (giving k = -π/a ... π/a) becomes

unit cell length 2a (giving k = -π/2a ... π/2a) • Since k is a vector in reciprocal space, a doubling of the lattice constant (from a to

2a) causes a halving of k. • Graphically, this appears as a “folding” of the dispersion curve.

Section 3.4 - 8

• NOTE: All we have done is redefined our lattice constant as 2a. • The band model now describes two bands that have no band gap between them. They

are still a continuous band.

• Now that we have redefined the lattice constant as 2a, the next step is to introduce the

Peierls instability. • The Peierls distortion that is associated with a band filling level of exactly 0.5 (i.e.,

one unpaired electron per site) is a dimerization (see above). • The Peierls instability is introduced into our model by displacing each atom by some

amount x. • NOTE: x is properly a vector, so it is more correct to say that one atom in the unit cell

is displaced by x and the other by -x. • Alternatively, this distortion can be achieved by displacing only one of the two atoms

in each unit cell by 2x. • This distortion introduces two different resonance integrals, β1 and β2, that are

dependent upon the magnitude of x.

Section 3.4 - 9

A Peierls Distortion Creates a Bandgap

Energy equation for a distorted system: akE !!2cos2 2122

21 ββββα ++±=

Therefore: At k = 0, E = α ± (β1 + β2) k = π/2a, E = α ± (β1 - β2) so the bands no longer intersect at k = π/2a !

• A CDW-driven distortion of the originally evenly-spaced 1D lattice with one

unpaired electron per site has given rise to dimerization causing the formation of a band gap.

• The previously metallic band structure is now an insulator (or semiconductor).

Section 3.4 - 10

Jahn-Teller Distortion vs. Peierls Distortion • Note the creation of a band gap.

Peierls Distortion ... non ½ filled bands • In principle, all 1D systems are subject to a CDW-driven distortion. • However, there do exist 1D systems that appear to be unaffected by this type of

distortion. • In reality, although a CDW can be determined for any partially filled 1D array, onset

of a Peierls distortion is dependent on temperature and pressure. • 1D systems that do not undergo a Peierls distortion under STP conditions are often

systems in which the fraction of band filling is not equal to ½. • i.e., there is not exactly one unpaired electron per lattice site. Examples:

1/4-filled band

1/6-filled band

Section 3.4 - 11

Fermi Surfaces and Nesting • Whether or not a Peierls distortion is observed can be explained in terms of the extent

of “nesting” in the Fermi surface. • The Fermi surface can be described as the junction between filled and empty energy

levels in a partially filled band. • For a system with filled bands, there is no Fermi surface, so the concept only applies

to metals. • The Fermi surface is a constant energy plot, in k-space, of the highest occupied

energy levels at T = 0 K. • If large parts of the Fermi surface may be translated by a unique vector q such that

they are superimposable on other parts of the surface, the Fermi surface is said to be strongly nested.

i.e., Many levels in the k-space continuum will be stabilized by the same distortion!

• The unique vector by which these two points can be connected is q = 2 kF • The value of kF is determined by the extent of band-filling.

Therefore: kF (Max.) = π/a and q (Max.) = 2 π/a BUT this is a wave vector system, so q = 2 π/a = 0/a ! Therefore, in reality: q (Max.) = π/a which makes kF (Max.) = π/2a

Section 3.4 - 12

• This can be understood in the following way:

1. kF = π/2a describes an exactly ½ filled band

2. A band that is ½ filled with negative charge carriers (electrons) is also ½ filled with positive charge carriers (holes).

A band that is more than half-filled with electrons (e.g., ¾ filled) is less than half filled with holes (e.g., ¼ filled) Thus a band filling level B is equivalent to a band filling level 1-B. The CDW associated with B = ¾ is the same as the CDW for B = ¼

3. Therefore kF = π/2a is the maximum kF necessary to describe a system.

General Formula for CDW-driven Distortions ... 1D only • Since kF (Max.) = π/2a and q = 2 kF ... q (Max.) = 2 kF (Max.) = π/a ... This is for a ½ filled band (i.e., exactly one charge carrier per lattice site) • If we now define a reciprocal lattice unit a* = 2π/a

... and define the average number of charge carriers per site as ρ ... we can redefine q (Max.) = π/a = (1/2) a* and ρ (Max.) = 1 ... therefore the general equation relating q to ρ is: q = (ρ/2) a*

• So for a ½ filled band, ρ = 1 and q = (1/2) a* for a 3/8 filled band, ρ = 0.75 and q = (3/8) a* • With the onset of a distortion, the 1D translation vector of a lattice point is t = la

where l = (1, 2, 3, ...) • In the presence of a harmonic modulation in the a direction, with a wavevector q, the

displacement of the lattice points (Δt) caused by a CDW-driven distortion are given by:

where A = amplitude (weighting factor) and φ = phase

)sin( φ+⋅=Δ tqAt!!!

Section 3.4 - 13

• The periodic lattice displacements are given by Δt/A, where the value of A (a weighting factor) is a function of the CDW.

• For the drawing below, the phase factor φ has been set to π/2 for q = (1/2)a*

and to 0 for q = (3/8)a*

**HOMEWORK: verify that this drawing is correct!** The Temperature and Pressure Dependence of a Peierls Instability • Driving forces that favour a distortion arise from interactions of the highest occupied

energy levels. • “Elastic forces” from the underlying electronic structure resist this distortion.

e.g., σ-bond compression ← minimum energy is at Δa = 0

Section 3.4 - 14

• A CDW must overcome an underlying “stiffness” of the lattice to introduce a

distortion. • Furthermore, interatomic (or intermolecular) repulsion is stronger in asymmetric

(distorted) arrays because it is proportional to the square of the interatomic separation.

1. Which is the stable form at low temperature?

high temperature?

2. Which is the stable form at low pressure? high pressure?