electrical conductivity - babeș-bolyai university
TRANSCRIPT
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•Electrical resistivity due to electron-phonon interaction.
•Bloch-Gruneisen law
•Residual resistivity of metals
•Impurity effect in a metal
Electrical conductivity
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Relaxation time aproximation:
The rate of return to equilibrium is proportional to
The proportionality coeficient τ/1 is a function of only k
ετ
ddfvEef
fff
01
10
⋅⋅⋅⋅=
+=
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For steady state: 0=
∂∂
tf
For a homogeneous Sommerfeld metal: 0=∇ fr
00)( feEkff k∇⋅+=
τ
mk
k
mk
kf
kf
2
22
00
2
=∂∂
=
∂∂⋅
∂∂
=∂∂
ε
ε
εε
mk
mpv ==
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Small oscillations ααα uRR += 0
The ions oscillations create changes in the crystalline periodic potential
Changes in the electron wave functions
Electron-phonon interaction
phpheephephe HHHHHH −−− ++++=
Hamiltonian of the solid: electrons and phonons
In harmonic approximation 0=− phphH
Electron-electron interaction Screening Electron-phonon interaction
effphephe HHHH −++=
σσσ
ε ,k,k,k
ke ccH
+∑=
λλλ
λω ,q,q,q
,qph caH
+∑=
Fröhlich
(See Mizutani)
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σσ ,k,qkq cca ++
k
q
qk'k +=
σσ ,k,qkq cca +−
+
•An electron in state disappears
•A phonon in state
appears
•An electron in state apears
k
q
qk'k −=
∑ +−
+++− +=
σσσσσ
,q,k,k,qkq
*effq,k,qkq
effq
effphe ccaAccaAH
qk'kk ±=→electron
emission and absorption of phonons
•An electron in state disappears
•A phonon in state
disappears
•An electron in state appears
( )qUiDqAq
effq ∝
−=
ρω2
0
32
FZD ε=
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Solid State Course by Mark Jarrel (Louisana Univ.),
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We presume the electron-phonon interaction is coulombian
2
24qZeUq Ω
=π
The Fourier transform of the interaction potential
DC electrical conductivity due to inelastic electron–phonon interaction
the Boltzmann transport equation for an isotropic metal consisting of only a single element:
)R(U α
Ω: volume per atom
see Mizutani
Electrical resistivity due to electron-phonon interaction
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We get:
If the relaxation time approximation is employed again, the Boltzmann transport equation is simplified as:
where
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To simplify further, the relaxation time τk is assumed to be independent of the wave vector k and is denoted as τ.
Then, the Boltzmann transport equation is reduced to:
If a spherical Fermi surface is assumed for an isotropic metal, the summation over the wave vector k’ can be replaced by the integration over the whole of k’-space.
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The displacement of atoms from their equilibrium positions disrupts the periodicity of the lattice and contributes to the resistivity through inelastic scattering of electrons with the absorption or emission of phonons.
we need to evaluate the dynamical structure factor
(See Mizutani)
Phonon absorption depends on the phonon density n(q), while phonon emission depends on the factor 1+n(q).
Bloch-Grüneisen law
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phonon energies are small compared to the Fermi energy only electrons close to the Fermi surface can participate in scattering events.
consideration of the inelastic electron–phonon interaction is essential in treating electron transport phenomena in both periodic and non-periodic metals at finite temperatures.
from Mizutani
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Normal (N) and Umklapp (U) processes
Large angle scattering dominates at high temperature
Small angle scattering is important at low temperature
The structure factor remains finite when
The process with:
g Is a reciprocal lattice vector
is a normal process
qk'k ±=−
gUmklapp process
from Kittel
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•The umklapp process becomes important when the temperature dependence of the resistivity of a metal needs to be quantitatively evaluated.
In the present discussion, we consider only the normal process and discuss qualitatively the temperature dependence of the electrical resistivity of a crystal metal.
•the resistivity due to the inelastic one-phonon normal process is deduced to be
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•we take the ratio of resistivity at temperature T over that at the Debye temperature ΘD.
where
This indicates that the temperature-dependent resistivity can be normalized with respect to that at the Debye temperature.
the maximum value of x at any temperature T is replaced by ΘD/T, since qmax=qD.
Bloch-Grüneisen law
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where the upper limit is replaced by the Debye radius corresponding to the available maximum wave number in the Debye model
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For example, pure Cu with ΘD=333 K exhibits an almost linear temperature dependence in the temperature range from 80 K up to about 800 K.
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5T∝
from Mizutani
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Only lattice waves with long wavelengths or shorter wave vectors can
survive in a perfect crystal at low temperatures.
•the scattering angle becomes smaller and smaller with decreasing
temperature.
•The Bloch–Grüneisen law, particularly the T5-law, is a phenomenon
characteristic of a crystalline metal, where the mean free path of
the conduction electron is much longer than the atomic distance.
•The Bloch–Grüneisen law breaks down in non-periodic solids like
amorphous alloys and quasicrystals, where the mean free path of the
conduction electron is shortened because of the failure of the Bloch
theorem and often becomes comparable to an average atomic distance.
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Linde law no longer holds when the solute concentration exceeds about 5 at.%.
the impurity–impurity interaction cannot be ignored.
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All these compounds form a completely periodic structure, since the lattice sites which Au and Cu atoms occupy are uniquely assigned.
superlattice or superstructure.
the resistivity drops sharply when an intermetallic compound is formed. •This is due to the restoration of the periodic potential, resulting in a substantial reduction in the scattering of conduction electrons
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from UvA-VU Master Course: Advanced Solid State Physics by Anne de Visser (University of Amsterdam)
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from UvA-VU Master Course: Advanced Solid State Physics by Anne de Visser (University of Amsterdam)