Advanced Materials Science - Lab Intermediate Physics

University of Ulm

Solid State Physics Department

Electrical Conductivity

Translated by Michael-Stefan Rill

January 20, 2003

CONTENTS 1

Contents

1 Introduction 2

2 Basic concepts 22.1 Crystalline metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.4 Amorphous metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Experimental setup 53.1 The cryostat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2 Liquid helium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.3 Temperature controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4 Tasks 7

5 Important Note 7

1 INTRODUCTION 2

1 Introduction

The electrical conductivity of solids shows – among other properties depending on temperature– very interesting effects, which can be used technically, too. In this course you will get anoverview over various materials revealing different temperature dependencies of their electricalconductivitiy.Practically, you will determine the electrical resistivity of various samples as function of tem-perature. The experimental setup enables you to measure all samples simultaneously over alarge temperature range. This seems to be very efficient, because the coolant (liquid helium)used to achieve low temperatures (≈ 4K) requires a complex equipment.Temperature control and data aquisition are handled by means of a computer system. Thisallows you to write all data to a hard drive giving you the opportunity to easily perform dataprocessing on any computer (you can use, e.g., the system provided by the practical).

2 Basic concepts

This section only gives a short outline of some basic models for the electrical conductivity. Moredetailed aspects as well as an accurate quantum-mechanical derivation for crystalline metals,semiconductors, superconductors, and disordered metals can be found in the literature (e.g.[1][2], [3].

2.1 Crystalline metals

The conductivity of crystalline simple metals can be described using the model originally pro-posed by Drude and Lorentz. This model assumes that each atom allows its valence electrons totravel through the solid (the so called conduction electrons). This leads to a high charge-carrierdensity (1023 electrons per cm3)which can be treated as a free electron gas.Within the Drude theory the electrical resistance results from scattering processes of the elec-trons while travelling through the solid. Here the distance, over which an electron can move onaverage without any collision, will be denoted as “inelastic mean free path”. In an ideal crystalthese collisions are caused by thermally induced collective vibrations of the atoms (phonons).This leads to an increase of the electrical resistance with rising temperature, caused by anincreasing amplitude of the lattice vibrations (phonon scattering).Due to these considerations the resistivity of an ideal crystal will disappear at zero temperature(T = 0K). Because, in a real crystal, we always find imperfections like defects, impurities, grainboundaries, the resistivity at zero temperature does not vanish but reaches a finite (non-zero)value the so-called . This value is often found to be temperature independent allowing todescribe the total resistance as the superposition of two terms: ρ = ρ0 + ρ(T ). At highertemperatures the temperature-dependent part ρ(T ) approaches a linear behaviour.The platinum sample used in the experiment is a polycrystalline specimen. This means, thatcrystallites ranging in size from typically 10nm to 100nm are present. The individual crystallitesare randomly oriented, separated from each other by so called grain boundaries. The latter act

2 BASIC CONCEPTS 3

as additional scattering centers for electrons and therefore contribute to the total resistivity(grain boundary scattering).In the case of samples whose dimensions are large as compared to the inelastic mean free pathof the conduction electrons, the resistivity mainly results from scattering at phonons, grainboundaries and impurities. For very thin films (thickness of the order of the inelastic mean freepath within the corresponding bulk material) additional contributions arise from an increasedscattering at the sample surface (“size”-effects).Quantum-mechanics allows to accurately calculate the behaviour of the electrons by means ofthe Schrodinger equation. Taking into account that electrons are described by means of theFermi-Dirac distribution function, the temperature dependent contribution to the resistivity canbe determined. This again leads to a linear behaviour of the resistivity at higher temperaturesbut to a T 5 dependence at low temperatures described by the Bloch-Gruneisen equation:

ρ(T ) = A

(T

ΘD

)5 ∫ ΘT

0

x5

(ex − 1)(1− e−x)dx (1)

where A is a material parameter and ΘD the Debye temperature. The following relations canbe used

ρ(T ) ∼ T for T >Θ

2(2)

ρ(T ) ∼ T 5 for T <Θ

10(3)

Hence for platinum (ΘD,Pt = 229K) a linear resistivity can be expected at room temperature

and a T 5-behaviour can be expected for temperatures T<∼ 20K.

2.2 Semiconductors

The conductivity of semiconductors can be understood within the energy band model. AtT = 0K the valence band is completely filled and the conduction band is left empty. Hence,no free charge carriers are available, which can be accelerated by an applied electrical field.Thus, at zero temperature, semiconductors are insulators. At finite temperature, electrons canbe excited from the valence band to the conduction band by absorption of thermal energy, nowbeing able to be accelerated by an external electrical field. At the same time, electrons missingin the valence band by excitation of electrons to the conduction band can act a positive charges(hole states) which also contribute to the total conductivity.As compared to metals, for semiconductors the density N of free charge carriers shows adistinct dependence on temperature: with rising T the density of electrons in the conductionband (correspondingly: holes in the valence band) increases exponentially (why?). In order todetermine the total conductivity the mobility of different types of charge carriers (electrons,holes) must be considered, which can be different from each other.In the case of polycrystalline carbon one finally obtains for the conductivity

2 BASIC CONCEPTS 4

σ(T ) ∼ e− Egap

2 kB T (4)

with Egab representing the activation energy and kB the Boltzmann constant.

2.3 Superconductors

Zero resistivity below a critical temperature (but still at finite temperatures!) in the super-conducting metals is explained by the occupation of the same quantum mechanical state by amacroscopic amount of charge carriers (BCS theory; Bardeen, Cooper and Schrieffer got theNobel price for this model). Because electrons as fermions have to fulfill the Pauli exclusionprinciple they are not allowed to occupy the same quantum mechanical state. To overcomethis problem, Bardeen, Cooper and Schrieffer proposed that two single electrons (fermions)can couple to a pair (Cooper-pair) below a critical temperature which behaves like a bosonand, therefore, can occupy the ground state together with other Cooper-pairs. The couplingis mediated by phonons (lattice distortions)as illustrated in figures 1 and 2 within a simplemechanical model by two spheres on a membrane: The distortion of the membrane representsa phonon, the sphere reflects an electron. Two electrons can form a Cooper pair, since it canbe energetically more favorable, if two electrons interact over the same lattice distortion, as ifthe electrons distort the crystal lattice by two separated phonons. From the interaction withthe same phonon an attractive interaction between the two contributing electrons may result,which leads to a coupling of these electrons. The formation of Cooper pairs is enabled byelectron phonon interaction).

Figure 1: Attraction of spheres on a flexible membrane. The configuration a is unstable andchanges into b

To repeat: While electrons as spin 12-particles must follow the Fermi Dirac distribution function

fFD(ε, T ) =1

exp(

ε−εF

kBT

)+ 1

(5)

and thus fulfill the Pauli exclusion principle, Cooper pairs as particles with whole-numberedspins are bosons and follow the Bose-Einstein statistics

fBE(ε, T ) =1

exp(

~ωkBT

)− 1

(6)

2 BASIC CONCEPTS 5

Figure 2: Distortion of the lattice of the atomic core by electrons

This distribution function enables the macroscopic occupation of the same quantum state, inwhich transport quantities e.g. the resistivity or the thermoelectric power disappear. Above thetransition temperature TC the Cooper pairs are broken by thermal excitation kBT . Therefore,by exceeding the transition temperature during annealing, the superconducting phase (theCooper pairs) will disappear and ”normal” electrons will carry the electrical current.

2.4 Amorphous metals

Amorphous or glassy metals (typically alloys) generally show resistivities which are about oneto two orders of magnitude larger than the values known for their crystalline counterparts.These metals often show a nearly temperature-independent resistivity, which is dominated byelectron mean free paths of the order of the interatomic distance. Thus, electron scattering atthe disordered atomic structure exceeds the scattering induced by phonons. Amorphous metalscan show both, small positive as well small negative temperature coefficaints for the resistivity,respectively. While a small increase in resistivity with temperature can easily be understoodby means of an increasing scattering induced by phonons, the negative temperature coefficientcan only be understood by means of the Ziman formula [3]

ρ =3π

e2~vF2

(N

V

)1

4kF4

∫ 2kF

0

Q3S(Q, T )|u(Q)|2dQ (7)

Here vF is the velocity of electrons with Fermi momentum, NV

is the charge-carrier density. kF

is the Fermi momentum, Q the momentum transfer caused by scattering processes. S(Q, T ) isthe structure factor and |u(Q)|2 the matrix element of the scattering process.Details of the latter phenomenon will be explained be your adviser!

3 EXPERIMENTAL SETUP 6

3 Experimental setup

The experimental setup (see fig. 3) mainly consists of four components: Cryostat, samples,control unit and recording device. The main components are described in more detail in thefollowing subsections.

Figure 3: Schematic sketch of the experimental setup

3.1 The cryostat

The cryostat (gr. cryo: low temperature; gr. statos: maintain, retain) is a container, which isused for thermal isolation to achieve low-temperature. The setup is divided into the followingparts

• Vacuum isolation

• Liquid nitrogen cooling shield (TS = 77.3K)

4 TASKS 7

• Second vacuum isolation

• Reservoir for liquid helium (the actual refrigerant)

• Sample zone

3.2 Temperature controller

The temperature controlling device offers a comfortable possibility to easily vary the sampletemperature. Using a silicon diode (range: 1.4K to 475K) which is in good thermal contactwith the sample holder, the controller repeatedly measures the actual sample temperature. Incase of any deviation from the desired temperature the controller adjusts the value of a heatercurrent (heat transfer to the sample holder). On the other hand, there is a constant coolingrate provided by the continuous evaporation of liquid Helium (heat transfer from the sampleholder). Since there is constant cooling but variable heating, the controller can adjust anydesired temperature of the sample.

4 Tasks

• Measure the resistivities of Pt, Cu60Ni40, carbon, a-FeB, and MgB2 between 4 and 90K.

• Discuss the different phenomena leading to the observed dependencies of the resistivityon temperature.

• Determine the transition temperature TC of the superconducting sample.

5 Important Note

ATTENTION: ALL MANIPULATIONS OF THE VACUUM SYSTEM/ COOLING SYSTEMHAVE TO BE EXPLAINED BY YOUR ADVISER BEFORE YOU ARE ALLOWED TOCHANGE ANYTHING !!

References

[1] C. Kittel, Introduction to Solid State Physics.

[2] W. Buckel, Superconductivity

[3] J. M. Ziman, Models of disorder.

REFERENCES 8

Figure 4: Scetch of the cryostat.