In the [previous chapter](https://steemit.com/physics/@mystifact/let-low-temperature-electron-transport-or-chapter-1-abstract), the Abstract very briefly summarised the **experiment** and its **outcomes**. This chapter will introduce the relevant Physics behind this experiment, as well as **low-temperature electron transport**. 

https://cdn.pixabay.com/photo/2017/03/02/15/13/molecules-2111539_960_720.jpg

# 2 - Introduction
This section will be split into **2 main categories**; the objectives of the experiment and the Physics behind resistivity.
## 2.1 - Objectives
By looking at how the resistivity of a material *changes* with varying temperature, some basic properties can be determined. In terms of a material’s electrical properties, they can be split into three groups: **metals**, **semiconductors** and **insulators**. Only metals and insulators have been analysed in this study. 
### 2.1.1 - Metals
By taking resistance measurements of multiple Copper samples of varying thickness; **the bulk resistivity**, **mean free path** and **Debye temperature** could be determined.
### 2.1.2 - Alloys
The resistivity values of the alloy samples were experimentally calculated and studied, to understand how the different concentrations of Copper and Nickel within the alloy *affected* the resistivity. 
### 2.1.3 - Kondo
Multiple Kondo samples, made up of Copper and Iron of **unknown concentrations** were also analysed for their resistivity-temperature dependence. From there, the concentrations were determined.
### 2.1.4 - Superconductors
For the Superconductors, multiple Niobium samples were tested in order to observe the special effect of **Superconductivity**. This was also achieved by looking at resistivity-temperature dependence. 
## 2.2 - Resistivity
In order to measure the resistivity through a wire, the relation, **ρ=RA/L** can be used. To calculate a resistance, a current must be passed through a sample and its voltage must be recorded, as shown in figure 1. 

<center>https://i.gyazo.com/f6004e1bc4aaa2f16db342c245953839.png</center>
<center>*Figure 1: Circuit set up to measure resistivity*</center>

The issue here, is that the measured potential difference also contains that of the **lead** and **contact resistance**. To measure the potential difference of just the wire, the **4 point probe method** [1] is used, as shown in figure 2. 
 
<center>https://i.gyazo.com/310d7a1a7db51616995b7b34e29a4de3.png</center>
<center>*Figure 2: Circuit using the 4 point probe method*</center>

Any unwanted contact or lead resistance, in this method, is **eliminated** due to the driven current measuring the wire directly. If a DC current is used, thermal EMF becomes an issue. This can *easily* be eliminated by switching the current direction and finding the **difference** in the voltages; where this difference is the thermal, unwanted EMF. This 4 point probe method works well for wires, but when a sample is irregular-shaped, another method must be used. This method is called the **Van der Pauw method** and it is used to calculate the resistivity of objects that are not in the shape of a wire. Using this technique, 4 contact points are chosen around the edge of the sample; from this, two points are used to run a current through the sample and the other two measure the voltage, allowing for a resistance to be calculated. Figure 3, below, shows the **contact points** around the irregular-shaped sample.
 
<center>https://i.gyazo.com/6038bbab86a3b3aa5416e431652f7c00.png</center>
<center>*Figure 3: Four points located around the irregular shaped sample*</center>

Once a resistance is calculated, the current is driven through the two points **90°** to the previous configuration, for another resistance to be acquired. The following equation is then used to calculate the resistivity: 

https://i.gyazo.com/0086e02f3f4cc847a72398e2468cdd07.png
*Where d= thickness of the sample and  f= correction factor*

Resistivity is a measure of how well a passing current is ’**resisted**’ from flowing through a material. Its origins arise from this flowing current interacting with **phonons** and **impurities** in the lattice of the specified material. **Matthiesen’s rule** explains this in terms of **collision rates**:

https://i.gyazo.com/4951635562a880a1b0fa3dcfa3397db6.png

Electron-phonon scattering can be explained by the **Bloch Gruneisen function** [2]. Using the resistivity-temperature relation of a sample, a Bloch Gruneisen function can be fitted to give a *range* of **useful values** including **Debye temperature** and the **mean free path** of the material. The equation of the Bloch Gruneisen function is as follows: 

https://i.gyazo.com/eefd1a8e59120787fbba6f1af0a9edfa.png

Where ρ(0)= the impurity contribution to the resistivity and ρ_phonon= the phonon contribution to the resistivity. 

ρ_phonon has an equation for itself, as shown below.

https://i.gyazo.com/f2aa9f1cd8b182c561daaf0f7ced2d94.png

As can be seen from the integral, varying T_D in ρ_phonon gives relations **ρ∝T** and **ρ∝T^5**. The Fuchs Sondheimer relation is used to *further* study the effect of resistivity of a sample of **varying thickness**. The equation for the *simplified* **Fuchs Sondheimer** relation is [3]:

https://i.gyazo.com/6ae74e3320bd5e64832985b9ecd09633.png
*Where ρ_B= the bulk resistivity and p= the reflectivity ratio.*

**If you have any questions, leave them below and until next time, take care.**

**~ Mystifact**
***
**References:**
[1]: http://iopscience.iop.org/article/10.1088/0953-8984/27/22/223201
[2]: https://link.springer.com/article/10.1007%2Fs11232-011-0003-4
[3]: [http://www.qucosa.de/fileadmin/data/qucosa/](http://www.qucosa.de/fileadmin/data/qucosa/documents/7092/Dissertation_Martin_Philipp.pdf)

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**Previous LET chapters:**
[Chapter 1 - Abstract](https://steemit.com/physics/@mystifact/let-low-temperature-electron-transport-or-chapter-1-abstract)

**Next chapter:**
[Chapter 3 - Experimental Techniques](https://steemit.com/physics/@mystifact/let-low-temperature-electron-transport-or-chapter-3-experimental-techniques)

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