Download Inroduction, Drude model

Condensed matter physics I: 2015-2016
Introduction, Drude model
R. Ganesh
1
Introduction
Condensed matter physics (CMP) is the study of systems with many interacting particles. The adjective ‘condensed’ should be understood as the opposite of
‘dilute’ – the constituent particles are so close to each other that their interactions cannot be ignored. Some typical examples of ‘condensed’ systems include
electrons in a metal, spins in an insulating magnet, neutrons in a neutron star,
Na atoms in an ultracold atomic gas, He atoms in liquid He, etc. Historically,
the field grew out solid state physics. It used to be called ’solid state physics’
until 1967 when Phil Anderson and Volker Heine coined the term ‘condensed
matter physics’.
CMP serves as a counterpoint to the ‘reductionism’. In the reductionist approach, complex systems are studied by breaking them down into smaller, fundamental units. This is exemplified by particle accelerator physics in which atoms
are broken down into nucleons, which are further broken down into quarks, etc.
In the condensed matter point of view, the whole is greater than the sum of its
parts. For example, we may understand how a single He atom behaves according
to the laws of quantum mechanics. However, that does not tell us how a beaker
with liquid He will behave. Completely unforeseen behaviour (superfluidity in
this case) can emerge from the interactions of many constituent particles. The
new physics that emerges from collective behaviour at different length scales is
the domain of condensed matter physics.
There are two broad (not necessarily mutually exclusive) divisions within
CMP. ‘Soft’ CMP deals with classical systems – the constituent entities obey
classical equations of motion, e.g., grains of sand which obey Newton’s laws. In
‘hard’ CMP, the quantum nature of constituents is important and can not be
ignored, e.g., La2 CuO4 , a cuprate, is an antiferromagnet composed of spin-1/2
spins.
Among contemporary research areas in physics, CMP stands out for its
particularly rich interplay of theory and experiment. Theoretical ideas have
suggested new experimental pathways and vice versa. By and large, new discoveries in this field have come from experiments. Some prominent examples
include superfluidity (Kapitza and Allen, 1930’s), high-temperature superconductivity (Bednorz and Müller, 1986), quasicrystals (Shechtman, 1982), etc.
1
Theoretical studies have also lead to new discoveries. A prominent topical example is topological insulators (Kane and Mele, 2005).
CMP is a truly complex field. It encompasses a wide spectrum of phenomena
that cannot meaningfully be reduced to a set of basic equations. This can be
compared with:
• Electromagnetism ↔ Maxwell’s equations
• Fluid mechanics ↔ Navier-Stokes equations
• Classical mechanics ↔ Newton’s equations.
Unlike these areas, there is no underlying ‘basic’ law of CMP. Of course, any
many particle system is ultimately described by a many body Schrödinger equation. However, in practice, it is impossible to deduce the system’s macroscopic
behaviour from this. In part due to this complex nature, CMP remains an active
area of research despite one century of experimental and theoretical work.
CMP is an excellent window into the scientific process. Models are constructed for complex systems, based on the knowledge available at a given point
in time. They are continually continually tested for theoretical consistency and
experimental validity and updated accordingly. The classic example is the Drude
model for metals that was proposed in 1900; it continues to be used, albeit with
several modifications and caveats.
In this course, we will focus on solid state systems to develop several basic
ideas. Within solids, we will mostly concern ourselves with crystalline materials.
The translational symmetry of crystals allows us to make great progress.
2
Drude model
What makes metals metallic? Why does copper conduct electricity, while common salt does not? The simplest model to explain conduction in metals was
put forward by Paul Drude, a German, in 1900. Some of the crucial discoveries
that paved the way for the Drude model include
• Ohm’s law, 1827
• Joule heating of metals, 1841
• Equipartition theorem, 1845
• Discovery of the electron, J. J. Thomson, 1897
J. J. Thomson showed that cathode rays emitted by different atoms all contain
the same negatively charged particle. This subatomic particle was called this
the electron. Drude identified these as the charge carriers responsible for conduction in metals. His model pictures a metal as a gas of electrons. Positive
charge centres, which are required for charge neutrality, are taken to be very
heavy and immobile – thereby playing no role in conduction. This remarkably
2
simple-minded model continues to be invoked to this day, despite the presence
of sophisticated numerical techniques and complex quantum analyses. This is
an example of the tremendous utility of simple physically-motivated models in
CMP.
Drude’s original model assumed that electrons wander around and scatter
of the positive-charge centres. We now know this to be incorrect. We will discuss some scattering mechanisms later on. In a modern re-interpretation of the
Drude model, valence electrons in metals delocalise and scatter off defects. For
future reference, we note that the typical length scale in metals is 1-3Å. The
typical density of electrons in (1-30) ×1016 m−3 (or 1022 cm−3 ).
Assumptions of the Drude model:
• Electrons move freely inside the metal, except for collisions with ‘ions’.
Between collisions, electrons move in a straight line with fixed velocity
– Independent electron approximation: no interaction between electrons
– Free electron approximation: no interaction between electrons and
ions
• Collisions occur with some scattering centres. Collisions are instantaneous.
• Probability of a given electron undergoing a collision in an infintesimal
interval dt is given by dt/τ . This is independent of any details such as the
electron’s position or velocity. τ is a system parameter called ‘relaxation
time’ or ‘mean free time’.
• Collisions immediately lead to thermal equilibrium. After a collision, the
electron comes out with a randomly oriented velocity. Its speed is related
to T , the ‘local’ temperature (assuming that a temperature can be assigned to a region within the metal – more on this later). This is major
assumption – the outgoing velocity is taken to be independent of details
such as velocity before the collision, presence of neighbouring ions, etc.
Exercise 1: Show that for a given electron, the average until the next collision
is given by τ . What is the average time between two collisions?
Exercise 2: Show that the average time elapsed since the immediately preceding collision is also given by τ .
Note that τ encodes a lot of information, including number of defect structures, temperature, etc. See the book for a table showing how τ decreases with
increasing temperature – we will try and interpret this at a later point in this
course.
DC electrical conductivity:
The Drude model immediately implies Ohm’s law V = IR. Resistance,
an extrinsic quantity, can be described using resistivity, an intrinsic quantity
R = ρL/A. The conductivity σ = 1/ρ can be deduced from the Drude picture
as follows.
3
Consider a metal slab with an imposed voltage V . An electron in the bulk
of this slab travels freely and independently between collisions. In addition, it
now experiences an electric field E due to the imposed voltage. The velocity of
a given electron, at time t since the immediately preceding collsion, is now given
by ve = vcollision − teE/m. vcollision is the velocity with which it emerged after
the immediately preceding collision – a randomly oriented vector. Therefore,
the average velocity of an electron is given by vavg = −teE/m. The mass and
charge of an electron are denoted by m and −e respectively.
If we now take an ensemble of electrons in one region of the metal (assuming
that the local temperature is the same everywhere), the average velocity is given
by vavg = −τ eE/m. We have used hti = τ , i.e., the average time since the last
collision for a given electron is τ . Apart from τ , we need one more piece of
information about the material – the electron density n.
Consider an imaginary square of cross-sectional area A within the metal,
oriented normal to the electric field. How many electrons pass through this in a
time interval dt? We can geometrically argue that all the electrons in a cuboid
of cross sectional area A and length dt × vavg will pass through this square.
Thus, the current through the square is given by the amount of charge passing
in time dt divided by dt: −neA × dt × vavg /dt = nAτ e2 E/m. This is a vector
parallel to the direction of electric field. The current density is this quantity
divided by A.
This can be rewritten as
J = σ0 E,
(1)
where
ne2 τ
.
(2)
m
One can easily show that this leads to Ohm’s law with R = l/σ0 A for a bulk
conductor.
Drude estimated the mean free time τ from the experimentally known conductivities of metals (∼ µ ohm-cm). He used equipartition theorem to obtain
vavg , and estimated that the ‘mean free path’ (vavg τ ) is of the same order as
the lattice spacing. This is consistent with Drude’s picture that electrons scatter of ions. However, with materials available today, we find mean free paths
ranging from ∼ 1 − 108 times the lattice constant. Clearly, we need a more
sophisticated understanding of the scattering mechanism.
Frictional damping in the Drude model:
Let us denote the average momentum of an electron at time t as p(t). How
does this momentum change in an infinitesimal interval dt? During this interval,
a fraction of the electrons (number of electrons ×dt/τ ) will undergo a collision.
Immediately after this time interval, they will have randomly oriented velocities
as they have just undergone a collision. Their average momentum will thus be
zero. The remaining electrons (number of electrons ×{1 − dt/τ }) have experienced a force f (t) during this time interval – their momentum changes due to
the impulse. Below, we will examine several examples with different f (t).
σ0 =
4
B
z
y
x
Figure 1: Configuration for Hall effect measurement.
The average momentum at time t + dt becomes
dt p(t + dt) ≈ 1 −
p(t) + f (t)dt + O(dt2 )
τ
p
dp(t)
= − + f (t).
dt
τ
(3)
(4)
We see that a frictional damping term emerges, which is inversely proportional
at τ . Thus, the assumptions of the Drude model amount to damped motion of
electrons in an external field.
As a quick check, let us apply this formula to the case of a static electric field
in a metal. The force on an electron f (t) is then given by −eE. We demand that
the system be in equilibrium, i.e., dp
dt = 0. It is easily shown that this reduces
to Ohm’s law j = σ0 E. We now proceed to apply this formula to several more
interesting cases.
Hall effect: The Hall effect was discovered by Edwin Hall in 1879, when he
was a doctoral student. Much before Drude, Hall was working with a notion
of mobile charge carriers transporting current within a metal. To understand
the context, we go back to 1820 when Oersted noticed that a tcurrent carrying
wire exerts a force on a nearby magnet (compass needle). Soon afterwards,
in 1821, Faraday discovered that a current carrying wire is deflected by an
external magnetic field. Edwin Hall wanted to determine if this force acts on
the whole conductor, or if it only acts on some free-moving mobile carriers inside
the metal. Suppose the conductor itself is kept fixed, then the putative mobile
carriers would be deflected to one side of the conductor. The cross sectional area
of the conductor would be effectively reduced; thereby the resistance would go
up. His experimental setup is as shown in Fig. 2. A magnetic field is applied
along the Z axis and an electric field is applied along the Y direction. Hall did
not find a change in resistance as he had so cleverly envisioned, instead he found
a transverse voltage. Let us now analyse this setup within the Drude model.
Applying Eq. 4 to the case shown in Fig.2, we have
p
dp(t)
p
= −e E +
×H − .
(5)
dt
mc
τ
5
We work with cgs units, following the notation of Ashcroft and Mermin. Note
that we have used the modern equation for Lorentz force. This was known in
Drude’s time, having been derived by Oliver Heaviside in 1889 and refined by
Henrik Lorentz in 1892. We assume that the externally applied electric and
magnetic fields penetrate into the metal and are experienced by the charge
carriers. We resolving this vector equation into its x and y components and
demand a steady state solution dp/dt = 0:
0 = −eEy + ωc px − py /τ
(6)
0 = −eEx − ωc py − px /τ ;
(7)
ωc = eH/mc
(8)
We have defined the cyclotron frequency ωc . Multiplying the above two equations by −neτ /m, we obtain
σ0 Ey = −ωc τ jx + jy
(9)
σ0 Ex = ωc τ jy + jx
(10)
(11)
In equilibrium, we can have a current along y since the external leads transport electrons and avoid any buildup of charge. However, we cannot have any
current along the x direction, i.e., jx = 0. Therefore,
Ex =
ωc τ
jy .
σ0
(12)
Thus, a transverse voltage Ex develops. It is proportional to both the longitudinal current jy and the magnetic field strength. The Hall resistivity is given
by
1
H.
(13)
ρH =
nec
Surprisingly, the Hall resistivity does not depend on the relaxation time τ .
Naı̈vely, the only material dependent parameter in this expression is n. There
is one more crucial qualitative input – the sign of the charge carrier. It can
easily be checked that the sign of the charge carrier, positive or negative, does
not matter in the derivation of σ0 . This can be seen from the fact that σ0 ∼ e2 ,
unlike the Hall resistivity.
AC conductivity: Let us assume that an oscillating electric field exists within
a metal. We denote
E(t) = Re{E(ω)eiωt }.
AC conductivity
Plasma oscillations
Thermal conductivity
Wiedemann Franz law
Seebeck effect
6
(14)