DEpartment of physics

Learning Goals:  What is Classical Theory of electron gas - Introduction to Drude's Model Successful Explaination of electronic Properties using Drude's Model Electronic Conductivity of Metals Qualitative behavior of Widemann-Franz Law -Ratio of Thermal Conductivity to the Electrical Conductivity of any metal at a given temperature is a constant Magneto-resistance behavior of Metals - Positive Hall Coefficient Limitations of Drude's Model Quantitative explanation of Wiedemann-Franz Law Drude's model cannot explain why some materials show negative Hall coefficient Please refer to: Text : Oxford Series Band Theory of Solids

In this section, we discuss the classical theory of electron Gas.

A little bit of history:

Electron was discovered by J. J. Thompson in 1897. Based on the esperienetal results, it was known at that time that the ratio of thermal conductivity to the electrical conductivity at a given temperature in a constant for all metals, which is known as the Wiedemann-Franz Law. In 1900, Drude was able to come up with an explanation to that using the classical theory of electron gas. In this chapter, we will discuss the important results from Drude's model; specificalyl, we will get an expression for electrical conductivity and thermal conductivity. And get a number for Lorentz Constant $$ L=\frac{\kappa}{\sigma T} $$

 Basic assumptions of the Drude Model:

•  Electrons in a solid behaves like a classical ideal gas. They do not collide. That is they do not interact with each other. That is also called the "independent electron approximation".

•  The positively charge ion cores are at rest. Electrons can collide with the ion cores. These collisions instantaneously change the velocity of electrons. However, electrons do not interact with ion cores in between collisions. That is also called "free electron approximation" 

•  The electron reach thermal equilibrium with the lattice through collisions with ions. $$\frac{1}{2} m_e v_t^2 = \frac{3}{2} k_B T $$
  

That tells the average speed of electron at room temperature is $v_t = 10^5 ms^{-1}$.

• Electrons move freely in between collisions. The mean length of the free motion is called mean free path. Knowing the packing density, we can get an estimation for this as $\lambda -\sim 1 nm$$.
  

Mean time between collisions is known as the relaxation time. $$\tau=\frac{\lambda}{v_t}$$

With the above information, we can get an estimation for the relaxation time as $$\tau \sim 1 \times 10^{-14} s $$.

We can get an idea of the particle density in a classical ideal gas, which will be in the order $10^28 m^{-3}$ .

Knowing the mass density, and the atomic number, we cal also calculate the free electron density in a metal which comes in the order of $\sim 10^25 m^{-3}$.

Obviously, electrons in a metal has a higher density than the ideal gas particles.

Let's use the Drude Model for understanding some electronic and thermal properties of metals. We will specifically consider electron conductivity, thermal conductivity, Wiedemann-Franz Law and Hall Coefficient.

DC Electrical Conductivity of a Metal with Drude's Model:

We are going to consider the behavior of electrons in the presence of applied electric field, E. Knowing the force acting on an electron in an electric field, let's apply Newton's Law.

$$m_e \frac{d\vec{v}}{dt}=-e \vec{\epsilon}$$

which gives: $$\vec{v}(t) = \frac{-e \epsilon t}{m_e} $$

This tells electron is accelerated in the direction opposite to the electric field. The above equation also suggests that the velocity keeps increasing indefinitely, which is not true. The motion is destroyed when electrons collide with ions. Average time electron accelerates is equal to the relaxation time.

$$ \vec{v}(t) = \frac{-e \epsilon \tau}{m_e} $$

This is called the drift velocity of electrons. Knowing the drift velocity, we can calculate the electron current.

 Let's consider a conducting material with a cross section A. We can find the number of electron pass through the materials per unit area as:

$$n |\vec{v}| A $$

So the current passing through the material: $$I=-e n v A $$

From that we can get an expression for the current density as:

$$\vec{J} = -e n \vec{v}$$

We have already got an expression for the drift velocity.

By plugging that in, we get:

$$J= \frac{n e^2 \tau }{m_e} \vec{\epsilon} $$

 That is, we have proved that the current density is proportional to the electric field. The proportionality constant is the electrical conductivity of the material.

$$ \vec{J}=\sigma \ec{\epsilon}$$

where the conductivity $$\sigma =\frac{n e^2 \tau}{m_e}$$

 This is the Ohm's Law.

We learnt $$ V=IR$$ where $$R=\frac{\rho L}{A}$$

$$V=T \frac{\rho L}{A}$$ $$\frac{V}{L}=\frac{I}{A} \rho $$ $$\frac{E} { J}= \rho$$ or $$J=\sigma E$$

That is, we were able to relate to the expression we got to the familiar expression for the Ohm's Law.

Electron Mobility:

Another important definition we can get from the above equations is the mobility of electrons.
$$\sigma =\frac{n e^2 \tau}{m_e}$$ $$\sigma =n \frac{ e \tau}{m_e} e $$ $$\sigma=n \mu e$$

where we defined $$\mu \frac{e \tau }{m_e}$$

Let's think about the electron mobility. We can also go back to the previously proved equation. $$J=-e n v =\frac{n e^2 \tau }{m_e} E $$

which can be re-arranged to get $$\frac{v}{E} =\frac{e \tau}{m_e} $$

That means, electron mobility can also be defined as the ratio between the velocity and the electron field. Or more specifically, it is a measure of how easy/hard to give a velocity to the electrons. Electron mobility is tunable.

By increasing the relaxation time, mobility can be increased. Also in semiconductor systems, we will later learn that the electron mass can be tuned.

OK..... At this point, what we have is, we are able to explain the Ohm's law using the Drude's model.

Hall Effect: (using the Drude;s Model):

Drude's model can also be found to describe the Classical Hall effect.

Let's start with the equation of motion. Earlier we wrote:

$$m\frac{dv}{dt}=-e\epsilon$$ It basically says the electron experiences a constant force, which means the electron velocity keeps increasing (in the direction opposite to electrons). Earlier, we introduced the relaxation time at the end to fix that problem. Another way of thinking that is: there is a retarding force which is proportional to the velocity.

 $$m \frac{d\vec{v}}{dt}=\vec{F} -F_{\text{retarded}} $$

and we take

$$F_{\text{retarded}}=\frac{m\vec{v}}{\tau}$$

Now, since we are going to consider the Hall effect, which is an observation made on the electron dynamics in the presence of magnetic field: We take that the force on the electron is coming from the electric field and magnetic field. Specifically, we consider the device geometry as shown. An electric field is applied on the specimen in x direction. and magnetic field is applied in the z direction.

Figure Here

$$ m\frac{d\vec{v}}{dt}=-e \left(\vec{E}+\frac{1}{c} \vec{v} \times \vec{B} \right) - \frac{m}{\tau} \vec{v} $$

$$m \frac{d\vec{v}}{dt} + \frac{m}{\tau} \vec{v}=-e \left(\vec{E}+\frac{1}{c} \vec{v} \times \vec{B} \right) $$

We have

$$\vec{B}=B \hat{z}$$

So,

$$\vec{v}\times \vec{B}=\left(v_x \hat{x}+v_y \hat{y} + v+z \hat{z} \right) \times B\hat{z} $$

$$\vec{v}\times \vec{B}= -v_x B \hat{y}+v_y B \hat{x} + 0 \hat{z} $$

That is:

$$m \frac{d\vec{v}}{dt} + \frac{m}{\tau} \vec{v}=-e \left(E_x \hat{x}+ E_y \hat{y} + E_z \hat{z} \right) - \frac{e}{c} \left( -v_x B \hat{y}+v_y B \hat{x} + 0 \hat{z} \right) $$

Now we can consider the x, y, and z components of the above equations:

$$m \frac{dv_x}{dt} + \frac{m}{\tau} v_x = -e \left(E_x + \frac{v_y}{c} B \right) $$

$$m \frac{dv_y}{dt} + \frac{m}{\tau} v_y = -e \left(E_y - \frac{v_x}{c} B \right) $$

$$m \frac{dv_z}{dt} + \frac{m}{\tau} v_z=-eE_z$$

In the steady state, the time derivative of the velocity will be zero.

So the above equaitons reduce to:

$$\frac{m}{\tau} v_x = -e \left(E_x + \frac{v_y}{c} B \right) $$

$$\frac{m}{\tau} v_y = -e \left(E_y - \frac{v_x}{c} B \right) $$

$$ \frac{m}{\tau} v_z=-eE_z$$

From this, we get:
$$ v_x = -\frac{e \tau}{m} E_x - \frac{e \tau v_y}{cm} B $$

$$ v_y = -\frac{e \tau}{m} E_y + \frac{e \tau v_x}{cm} B $$

Let's use the definition of the cuyclotron frequency to simplify these equations:

$$\omega =\frac{eB}{mc} $$

$$ v_x = -\frac{e \tau}{m} E_x - \omega \tau v_y $$

$$ v_y = -\frac{e \tau}{m} E_y + \omega \tau v_x $$

Figure here.....

At the equilibrium state, an electric field is developped in the negative y direction, which stops further motion of electron in y drection.

Which says
$$v_y=0$$. $$ \frac{e \tau}{m} E_y = \omega \tau v_x $$ $$v_x = \frac{e}{m \onega} E_y$$

From the other equation,

we get: $$ v_x = -\frac{e \tau}{m} E_x $$

By submstiuting for vx:
 $$\frac{e}{m \omega} E_y=-\frac{e \tau}{m} E_x $$

 We get: $$E_y = -\frac{eB\tau}{mc} E_x $$ And the Hall coefficient is defined as: $$R_H=\frac{E_y}{j_x B}$$ By using $$J_x=\sigma E_x$$ We get $$R_H=-\frac{1}{nec}$$

Hall coefficient is used to measure the electron concentration of materials. So according to this equation, Hall coefficient is a positive quantity (e-being negative). For some materials like Al, Cu, the measured positive Hall coefficient is in agreement with the experiement. However, there are some materials which shows a negative Hall coefficient, which cant be explained with the Drude's model. It can be explained with the band structure of solids. We will discuss that in a later lesson.

Thermal Conductivity by Electrons:

 It is important to point out the heat can be transferred through a material by electrons and lattice vibrations. But for metals, the heat conductivity by lattice vibrations is not as significant compared to that of lattice vibrations.

Here we only discuss the electron heat conductivity (since we are considering metals).