Solid State Physics – Lecture 10:
Oulton (2020/21)
Lecture 10 Prelude:
In lecture 8 and 9, we have found that the electronic states of a crystal, predicted by quantum mechanics, are
like plane waves that extend over an entire crystal. Bloch’s quantum theory of electrons predicts lossless
propagating electronic states. In this way we introduced the idea of crystal momentum in the previous lecture.
It is quite remarkable that the periodic potential of a crystal only affects the dispersion (momentum) of electrons
– it does not scatter them! In this lecture, we explore how to calculate the momentum of electrons within solids
so we can evaluate the effect of external forces?
The quantum states of the crystal must be viewed as basis states. The wavefunction of a real electron, which
we presume has a finite size, should be described by a sum over these quantum states such that the
uncertainties in momentum and position are set by Heisenberg’s uncertainty principle. (Currently, we assume
an exact value of k so it is no surprise our states are delocalised across the entire crystal!) This interpretation
allows us to reconcile the quantum model with our classical view of particle-like electronic charges.
Since an electron must be represented as wave-packet of a crystal’s electronic states, it is not the phase
velocity that describes the motion of electrons, but the group velocity. This seems to make good sense: e.g.
consider the free electron model’s wave solutions, 𝜓(𝑥, 𝑡) ∝ exp{𝑖(𝑘𝑥 − 𝐸𝑡/ℏ)}, to the 1D time dependent
Schrodinger equation,
4𝜓(𝑥, 𝑡) = −
𝐻
ℏ! ""
𝜓 (𝑥, 𝑡) = 𝑖ℏ𝜓̇(𝑥, 𝑡)
2𝑚
(10.1)
Substitution of the harmonic wave solution shows states of energy, 𝐸 = ℏ! 𝑘 ! /2𝑚, where 𝑚 is the electron
mass and 𝑘 is the wavenumber of the electronic wave. The phase (𝑣# ) and group (𝑣$ ) velocities are:
𝑣# =
𝐸
ℏ𝑘
1 𝑑𝐸 ℏ𝑘
=
, 𝑣$ =
=
ℏ𝑘 2𝑚
ℏ 𝑑𝑘
𝑚
(10.2)
Only the group velocity generates an acceptable correspondence between the classical electron momentum
(𝑚𝑣$ ) and the quantum mechanical electron momentum (ℏ𝑘).
In this lecture, we will explore how to calculate velocity and momentum of electrons from the Bloch
wavefunctions of a crystal, 𝜓%,' (𝒓) = 𝑒 (𝒌.𝒓,(-./ℏ 𝜙%,' (𝒓), where 𝜙%,' (𝒓), are the periodic lattice functions for a
particular band, labelled 𝑛. (See Lectures 9.) We will find that classical equations of motion with quantum
mechanically defined velocity and mass capture all essential physics. This semi-classical description will allow
us to calculate the effect of external forces on electrons in solids simply by applying Newton’s 2nd law of motion.
This will simplify greatly our understanding of the behaviour of electrons in crystal.
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Solid State Physics – Lecture 10:
Oulton (2020/21)
Lecture 10: semi-classical theory of Bloch electrons
Electron momentum in a crystal
Let us continue to investigate the correspondence between our quantum theory of electronic states in a crystal
and the classical view of electrons as charged particles moving through a uniform medium. This is a truly
remarkable phenomenon from a classical standpoint: it is not clear how electrons as particles can negotiate
the periodic array of atoms within the crystal. Even Felix Bloch was delighted by his solution to this problem:
“When I started to think about it, I felt that the main problem was to explain how the electrons could sneak by
all the ions in a metal… By straightforward Fourier analysis I found to my delight that the wave differed from
the plane wave of free electrons only by a periodic modulation”.
Felix Bloch
Bloch’s theorem is a very surprising result! An atom presents a substantial scattering cross section to a passing
electron yet within a solid, atoms spaced by mere angstroms, do not seem to scatter electrons at all!
Apparently, the effect of the crystal only modifies the velocity at which electrons can move in certain directions
and at certain energies. This phenomenon is reliant on the perfect periodicity of the crystal lattice, which
enables the electron wavefunction to “fit” precisely between atoms. In effect electrons in a crystal can move
as if they are in a vacuum!
However, the Bloch wavefunction is complicated and it is not immediately clear how we can calculate the
momentum or velocity of an electron in a crystal. Consider therefore Schrodinger’s equation for one electron
in a periodic potential, 𝑉(𝒓), of a crystal. (The following derivation is not examinable.) The Hamiltonian is
4=
𝐻
𝒑
C!
+ 𝑉(𝒓),
2𝑚
(10.3)
where 𝑚 is the electron mass and 𝒑
C = −𝑖ℏ𝜵 is the momentum operator. We know that Bloch’s solution
(ignoring the time dependence for the moment) have the form:
𝜓%,' (𝒓) = 𝑒 (𝒌.𝒓 𝜙%,' (𝒓)
(10.4)
4𝜓%,' (𝒓). We start with the momentum operator,
Let’s try to evaluate 𝐻
4 𝜙%,' (𝒓) − 𝑖ℏ𝑒 (𝒌.𝒓 𝜵𝜙%,' (𝒓) = 𝑒 (𝒌.𝒓 G𝒑
4 H𝜙%,' (𝒓)
𝒑
C 𝜓%,' (𝒓) = 𝑒 (𝒌.𝒓 ℏ𝒌
C + ℏ𝒌
(10.5)
At first glance, the momentum operator does not return the expected eigenvalue for the electron’s momentum,
i.e. 𝒑
C𝜓%,' (𝒓) ≠ ℏ𝒌𝜓%,' (𝒓)! Therefore, ℏ𝒌 is apparently not the electron’s momentum. To distinguish it, as 𝒌 is
still a good quantum number here, it is called the crystal momentum. Continuing with the evaluation of the
Hamiltonian, it is straightforward to show [exercise]:
4 H! 𝜙%,' (𝒓)
𝒑
C𝟐 𝜓%,' (𝒓) = 𝑒 (𝒌.𝒓 G𝒑
C + ℏ𝒌
(10.6)
The Schrodinger equation for a Bloch state may thus be written as:
!
4H
G𝒑
C + ℏ𝒌
𝜙%,' (𝒓) + 𝑉(𝒓)𝜙%,' (𝒓) = 𝐸% (𝒌)𝜙%,' (𝒓)
2𝑚
(10.7)
To explore the relationship between the momentum operator and the electron momentum, we will consider a
perturbation to this system: 𝒌 ↦ 𝒌 + 𝒒, where 𝒒 is assumed to be small. The Kinetic energy term may be
multiplied out to:
!
!
4 + ℏ𝒒
4H + 2ℏ𝒒
4H + ℏ! 𝒒
G𝒑
C + ℏ𝒌
CH = G𝒑
C + ℏ𝒌
C. G𝒑
C + ℏ𝒌
C𝟐
(10.8)
We will explore the effect of this change in momentum through a Taylor expansion of the electron energy,
𝐸% (𝒌 + 𝒒), for 𝒒 small,
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Solid State Physics – Lecture 10:
Oulton (2020/21)
1
𝜕 ! 𝐸% (𝒌)
𝐸% (𝒌 + 𝒒) ≈ 𝐸% (𝒌) + 𝒒
C. 𝜵' 𝐸% (𝒌) + N
𝑞 𝑞 +⋯
2
𝜕𝑘( 𝜕𝑘2 ( 2
(10.9)
(,2
Substituting back into Schrodinger’s equation (sub. Eqs. 10.8 & 10.9 into 10.7), we find
4H!
C + ℏ𝒌
G𝒑
𝜙%,'34 (𝒓) + 𝑉(𝒓)𝜙%,'34 (𝒓) + ⋯
2𝑚
ℏ
4 H𝜙%,'34 (𝒓) + ⋯
C. G𝒑
C + ℏ𝒌
𝒒
𝑚
ℏ! 𝟐
𝒒
C 𝜙%,'34 (𝒓)
2𝑚
𝐸% (𝒌)𝜙%,'34 (𝒓) + ⋯
C. 𝜵' 𝐸% (𝒌)𝜙%,'34 (𝒓) + ⋯
𝒒
≈
(10.10)
!
1
𝜕 𝐸% (𝒌)
(𝒓)
𝑞𝑞𝜙
N
2
𝜕𝑘( 𝜕𝑘2 ( 2 %,'34
(,2
Equation 10.10 has been written like this to highlight the functionally related rows. The top row represents the
original unperturbed Schrodinger equation. The second row contains terms in 𝒒. For small 𝒒 = 𝜹𝒌, this row
includes the term 𝜹𝒌. 𝜵' 𝐸% (𝒌), which represents the amount of energy required to shift the electron by 𝜹𝒌. We
thus find a new operator relationship:
ℏ
4H = 𝜵' 𝐸% (𝒌)
G𝒑
C + ℏ𝒌
𝑚
(10.11)
Notice that we have now introduced the group velocity, which we already suspected would tell us about an
electron’s momentum in the crystal. In 3D, 𝜵' 𝐸% (𝒌) = ℏ𝒗$ , where,
𝜵' =
𝜕
𝜕
𝜕
𝑘T +
𝑘T +
𝑘T
𝜕𝑘5 5 𝜕𝑘6 6 𝜕𝑘7 7
(10.12)
where the hats imply unit vectors as opposed to operators in this case. Let us find the expectation value of
𝜵' 𝐸% (𝒌) acting on 𝜙%,' (𝒓):
𝜵' 𝐸% (𝒌)𝜙%,' (𝒓) =
ℏ
ℏ
ℏ
4H𝜙%,' (𝒓) = 𝑒 ,(𝒌.𝒓 𝒑
G𝒑
C + ℏ𝒌
C𝑒 (𝒌.𝒓 𝜙%,' (𝒓) = 𝑒 ,(𝒌.𝒓 𝒑
C𝜓%,' (𝒓)
𝑚
𝑚
𝑚
(10.13)
∗
(𝒓) and integrating.
We can evaluate the expectation value by multiplying through with 𝜙%,'
∗
(𝒓)G𝜵' 𝐸% (𝒌)H𝜙%,' (𝒓) =
U 𝑑𝑉 𝜙%,'
ℏ
ℏ
∗
∗
(𝒓)𝑒 ,(𝒌.𝒓 𝒑
(𝒓)𝒑
U 𝑑𝑉 𝜙%,'
C𝜓%,' (𝒓) = U 𝑑𝑉 𝜓%,'
C𝜓%,' (𝒓)
𝑚
𝑚
(10.14)
The integral is the expectation value of the electron momentum! We thus find that ℏ,9 ⟨𝜵' 𝐸% (𝒌)⟩ = ⟨𝒗𝒈 ⟩ =
⟨𝒑⟩/𝑚. This result states that the electron momentum in the crystal is related to the group velocity of the Bloch
states, just like in the free electron theory.
Electron effective mass in a crystal
We now return to the final row of Eqn. 10.10. For simplicity, let us treat this row in 1D. We find,
,9
ℏ! 𝑑! 𝐸% (𝒌)
𝑑! 𝐸% (𝒌)
!
=
⇒
𝑚
=
ℏ
Y
Z
𝑚
𝑑𝑘 !
𝑑𝑘 !
(10.15)
The crystal not only modifies the momentum of electrons, but also their mass! To distinguish the effective mass
of the electron in the crystal from its actual mass, we use an asterisk notation, 𝑚 → 𝑚∗ . This notation will be
used throughout this course. In the case of the free electron theory, the effective mass is simply the free
electron mass. [Consider it an exercise to check that this is indeed the case.]
Semi-classical equations of motion
We can now formulate a semi-classical theory of the dynamics of electrons by re-casting Newton’s 2nd law in
terms of the group velocity of the electronic wavefunction and the electron’s effective mass. While the
mechanics are classical in nature, the parameters of the electron are derived from quantum mechanical theory
– hence the term semi-classical. This proves to be a very useful model. Newton’s 2nd law becomes: (the
following material is now examinable.)
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Solid State Physics – Lecture 10:
Oulton (2020/21)
𝑭 = 𝑚∗
𝑑𝒗$
𝑑𝑡
(10.16)
Let us therefore consider the work done, 𝛿𝑊, by an external force 𝑭, in time 𝛿𝑡 on an electron in a crystal,
𝛿𝑊 = 𝑭. 𝛿𝒙 = 𝑭. 𝒗$ 𝛿𝑡
(10.17)
From our quantum mechanical parameters, we know that 𝜵' 𝐸% (𝒌) = ℏ𝒗𝒈 , so the work done is,
𝛿𝑊 = 𝜵' 𝐸% (𝒌). 𝛿𝒌 = ℏ𝒗$ 𝛿𝒌
(10.18)
We saw this also in Eqn. 10.11. Substituting Eqns. 10.23 into 10.24, we find:
ℏ𝒗$ 𝛿𝒌 = 𝛿𝑊 = 𝑭. 𝒗$ 𝛿𝑡
(10.19)
Taking the limit of 𝛿𝑡 ↦ 0, we find,
lim ℏ
;.↦=
𝛿𝒌
𝑑𝒌
=𝑭=ℏ
𝛿𝑡
𝑑𝑡
(10.20)
This is yet another remarkable result. The effect of an external force on an electron in the crystal is to change
the crystal momentum ℏ𝒌. In the absence of a force, the crystal momentum must be constant or, in other
words, conserved. This result is why we can use the semi-classical theory.
We can use this result also, to derive the form of the effective mass. We will perform this in 1D for simplicity.
(This is the examinable derivation for the effective mass). We return to Newton’s 2nd law (Eqn. 10.22) and
substitute for the group velocity:
𝐹 = 𝑚∗
𝑑𝑣$ 𝑚∗ 𝑑𝐸% (𝑘) 𝑚∗ 𝑑𝐸% (𝑘) 𝑑𝑘
=
=
𝑑𝑡
ℏ 𝑑𝑘𝑑𝑡
ℏ 𝑑𝑘 ! 𝑑𝑡
(10.21)
We can now use Eqn. 10.26 to find,
,9
𝐹 = 𝑚∗
𝑑𝐸% (𝑘) 𝐹
𝑑! 𝐸% (𝒌)
∗
!
⇒
𝑚
=
ℏ
Y
Z
𝑑𝑘 ! ℏ!
𝜕𝑘 !
(10.22)
As was found in Eqn. 10.15.
Summary
•
•
•
•
•
We can use a semi-classical model to describe charge transport in solids.
The semi-classical model uses classical equations of motion with carrier momentum and mass defined
by quantum theory.
The effective mass is isotropic and constant in the parabolic band approximation.
Since effective mass may be negative, it is useful to introduce the concept of holes.
The quantum mechanical mass and momentum parametrize a band. When calculating conduction,
we must account for all partially filled bands, not just electron and holes.
4