Thermal Conductivity in Solid State Physics : Formula, Derivation, Applications & Notes

 

Thermal Conductivity in Solid State Physics Explained Easily | Formula, Notes & Applications

Introduction

Thermal conductivity is one of the most important physical properties of solids. It tells us how easily heat flows through a material. In solid state physics, we study how atoms, electrons, and crystal structures behave inside solids. Thermal conductivity helps us understand how heat energy moves through these particles.

When one end of a solid is heated, the temperature of that end increases. The particles there gain energy and begin to transfer it to nearby particles. Slowly or quickly, heat moves from the hot region to the cold region.

This property is very useful in:

•  Engineering
•  Electronics
•  Construction
•  Metal industries
•  Semiconductor devices
• Daily household objects

What is Thermal Conductivity?

Thermal conductivity is the ability of a material to conduct heat.

Good conductors

Some materials allow heat to pass quickly. These are called good conductors.

Examples:

•  Silver
•  Copper
•  Aluminium
• Iron

Bad conductors

Some materials do not allow heat to pass easily. These are called bad conductors or insulators.

Examples:

• Wood
•  Plastic
•  Rubber
•  Glass
•  Air

 Simple Example

This image AI illustration

Imagine a metal spoon placed in hot tea.

After some time, the handle becomes hot.

 Heat traveled through the spoon.

Now place a wooden spoon in hot tea.

 Handle remains cool for longer time.

This means:

Metal has high thermal conductivity.

Wood has low thermal conductivity.

 Definition

Thermal conductivity is defined as:

The amount of heat flowing per second through unit area of a material of unit thickness when the temperature difference between opposite faces is one kelvin.

It is represented by:

k or K

SI Unit:

W m⁻¹ K⁻¹

Fourier’s Law of Heat Conduction

French scientist Fourier gave the law of heat conduction.

It states:

 The rate of flow of heat through a material is directly proportional to area and temperature gradient.

Q/t = - kA dT/dx

Where:

 Q = quantity of heat

 t = time

 A = area of cross section

 k = thermal conductivity

dT/dx = temperature gradient

Negative sign means heat flows from high temperature to low temperature.

Meaning of the Formula

If area increases → more heat flows.

If temperature difference increases → more heat flows.

If material has high k → heat flows faster.

The basic law of heat conduction is called Fourier’s Law.

It states:

Rate of heat flow is directly proportional to area and temperature gradient.

$\frac{\mathrm{<span\; style="font-family:\; georgia;\; font-size:\; large;">Heat\; Transfer\; in\; Solids\; <p\; class="MsoNormal"><span\; style="font-weight:\; normal;">In\; solids,\; heat\; is\; transferred\; mainly\; in\; two\; ways:</span></p>\; <p\; class="MsoNormal"><span\; style="font-weight:\; normal;">1.\; By\; Free\; Electrons</span></p>\; <p\; class="MsoNormal"><span\; style="font-weight:\; normal;">This\; occurs\; in\; metals.</span></p>\; <p\; class="MsoNormal"><span\; style="font-weight:\; normal;">Metals\; have\; many\; free\; electrons.\; These\; electrons\; move\; throughout\; the\; solid.\; When\; heated,\; electrons\; gain\; kinetic\; energy\; and\; move\; toward\; cooler\; regions\; carrying\; heat.</span></p>\; <p\; class="MsoNormal"><span\; style="font-weight:\; normal;">Thus\; metals\; conduct\; heat\; quickly.</span></p>\; </span>}}{}$

 2. By Lattice Vibrations

$ Atoms in solids are arranged in a crystal lattice. They vibrate about fixed positions. When one atom gains energy, it vibrates more strongly and transfers energy to neighboring atoms. This vibration transfer carries heat. These vibration energy packets are called phonons.  Thermal Conductivity in Metals Metals are excellent conductors because: Large number of free electrons  Fast movement of electrons  Easy transfer of energy Examples: Metal Thermal Conductivity Silver Very High            Copper  High  Aluminium High  Iron  Moderate That is why: Cooking vessels use aluminium  Electric irons use metal plates Heat sinks use copper $

 Why Silver is Best but Copper is Common?

$\frac{\mathrm{<span\; style="font-family:\; georgia;\; font-size:\; large;">\; <p\; class="MsoNormal">Silver\; has\; highest\; conductivity,\; but\; it\; is\; expensive.</p>\; <p\; class="MsoNormal">Copper\; has\; slightly\; lower\; conductivity\; and\; lower\; cost.</p>\; <p\; class="MsoNormal">So\; copper\; is\; widely\; used.</p>\; </span>}}{}$

Thermal Conductivity in Non-Metals

$\frac{\mathrm{<span\; style="font-family:\; georgia;\; font-size:\; large;">\; <p\; class="MsoNormal">Non-metals\; do\; not\; have\; free\; electrons.\; Heat\; transfer\; happens\; only\; through\; atomic\; vibrations.</p>\; <p\; class="MsoNormal">This\; is\; slow.</p>\; <p\; class="MsoNormal">Examples:</p>\; <p\; class="MsoNormal"></p><ul\; style="text-align:\; left;"><li>Wood</li><li>Plastic</li><li>Rubber</li><li>Glass</li></ul><p></p>\; <p\; class="MsoNormal">Hence\; they\; are\; insulators.</p>\; <p\; class="MsoNormal">Used\; for:</p>\; <p\; class="MsoNormal">Vessel\; handles</p>\; <p\; class="MsoNormal"> Electrical\; insulation</p>\; <p\; class="MsoNormal"> Refrigerator\; walls</p>\; </span>}}{}$

 Thermal Conductivity in Semiconductors

$\frac{\mathrm{<span\; style="font-family:\; georgia;\; font-size:\; large;">\; <p\; class="MsoNormal">Examples:</p>\; <p\; class="MsoNormal"></p><ul\; style="text-align:\; left;"><li> Silicon</li><li>Germanium</li></ul><p></p>\; </span>}}{}$

Heat conduction occurs by:

$\frac{\mathrm{<span\; style="font-family:\; georgia;\; font-size:\; large;">\; <ul\; style="margin-top:\; 0cm;"\; type="disc">\; <li\; class="MsoNormal"\; style="mso-list:\; l0\; level1\; lfo5;\; text-indent:\; -18pt;"><span\; style="color:\; \#e06666;">Electrons</span></li>\; <li\; class="MsoNormal"\; style="mso-list:\; l0\; level1\; lfo5;\; text-indent:\; -18pt;"><span\; style="color:\; \#e06666;">Phonons</span></li>\; </ul>\; </span>}}{}$

Phonon and Electron

$ In solids, heat and electricity are transferred mainly by electrons and phonon.  1. Electron An electron is a tiny negatively charged particle present in atoms.(e-)  In Solids In metals, some electrons are free to move inside the material. These are called free electrons. They help in  Electrical conduction Thermal conduction  Example When copper wire is connected to battery:  Free electrons move  Electric current flows When one end of copper rod is heated:  Electrons gain energy  Move and transfer heat So electrons carry charge + heat in metals.  2. Phonon A phonon is not a particle like electron. It is a quantized vibration energy of atoms in a crystal lattice. Simple meaning: When atoms in a solid vibrate, that vibration energy travels like a wave. This energy packet is called a phonon.  Easy Example Imagine people standing in line holding hands. If first person shakes, vibration passes to next person. Same way:  One atom vibrates  Neighbor atom vibrates  Energy moves through crystal That moving vibration energy = phonon  Phonons help in: Heat conduction in insulators  Lattice vibration study Specific heat of solids Difference Between Electron and Phonon Property Electron  Phonon    Nature Real particle      Vibration energy packet Charge    Negative charge  No charge               Mass  Has mass           No rest mass            Carries    Electricity + heat Heat only               Important in Metals   Insulators / Crystals   $

In Metals

$\frac{\mathrm{<span\; style="font-family:\; georgia;\; font-size:\; large;">\; <p\; class="MsoNormal">Heat\; transfer\; by:</p>\; <p\; class="MsoNormal"> Electrons\; (mainly)</p>\; <p\; class="MsoNormal"> Phonons\; (small\; contribution)</p>\; </span>}}{}$

 In Insulators

$ Heat transfer mainly by:  Phonons only Because no free electrons. Electron:  Negatively charged particle responsible for electrical and thermal conduction in metals. Phonon: Quantum of lattice vibrational energy responsible for heat conduction in solids.  Easy Memory Trick Electron = charge carrier Phonon = vibration heat carrier Semiconductors are used in mobile phones, laptops, computers. Heat removal is important because excess heat damages circuits. So cooling systems are needed. $

 Microscopic Explanation

This image AI illustration

$\frac{\mathrm{<span\; style="font-family:\; georgia;\; font-size:\; large;">\; <p\; class="MsoNormal">Suppose\; left\; end\; of\; metal\; rod\; is\; heated.</p>\; <p\; class="MsoNormal">At\; hot\; end:</p>\; <p\; class="MsoNormal">Electrons\; move\; faster</p>\; <p\; class="MsoNormal"> Collide\; with\; other\; electrons\; and\; atoms</p>\; <p\; class="MsoNormal"> Transfer\; energy\; toward\; cold\; end</p>\; <p\; class="MsoNormal">Thus\; temperature\; rises\; gradually\; along\; rod.</p>\; </span>}}{}$

 Temperature Gradient

$\frac{\mathrm{<span\; style="font-family:\; georgia;\; font-size:\; large;"><span\; style="font-weight:\; normal;">\; </span><p\; class="MsoNormal"><span\; style="font-weight:\; normal;">Temperature\; changes\; from\; one\; point\; to\; another\; inside\; a\; rod.</span></p>\; <p\; class="MsoNormal"><span\; style="font-weight:\; normal;">This\; change\; per\; unit\; length\; is\; called\; temperature\; gradient.</span></p>\; <p\; class="MsoNormal"><span\; style="font-weight:\; normal;">dT/dx</span></p>\; <p\; class="MsoNormal"><span\; style="font-weight:\; normal;">If\; temperature\; difference\; is\; large,\; heat\; flow\; is\; faster.</span></p>\; </span>}}{}$

Factors Affecting Thermal Conductivity

 1. Nature of Material

$\frac{\mathrm{<span\; style="font-family:\; georgia;\; font-size:\; large;"><span\; style="font-weight:\; normal;">\; </span><p\; class="MsoNormal"><span\; style="font-weight:\; normal;">Every\; material\; has\; different\; atomic\; arrangement.</span></p>\; <p\; class="MsoNormal"><span\; style="font-weight:\; normal;">Metals\; >\; Semiconductors\; >\; Insulators</span></p>\; </span>}}{}$

 2. Temperature

$\frac{\mathrm{<span\; style="font-family:\; georgia;\; font-size:\; large;">\; <p\; class="MsoNormal"><span\; style="font-weight:\; normal;">For\; metals:</span></p>\; <p\; class="MsoNormal"><span\; style="font-weight:\; normal;">As\; temperature\; increases,\; atoms\; vibrate\; more\; and\; obstruct\; electrons.</span></p>\; <p\; class="MsoNormal"><span\; style="font-weight:\; normal;">So\; conductivity\; decreases.</span></p>\; <p\; class="MsoNormal"><span\; style="font-weight:\; normal;">For\; insulators:</span></p>\; <p\; class="MsoNormal"><span\; style="font-weight:\; normal;">Conductivity\; may\; increase\; slightly.</span></p>\; </span>}}{}$

 3. Impurities

$\frac{\mathrm{<span\; style="font-family:\; georgia;\; font-size:\; large;">\; <p\; class="MsoNormal">If\; foreign\; atoms\; are\; present:</p>\; <p\; class="MsoNormal"> Electron\; motion\; disturbed</p>\; <p\; class="MsoNormal">Lattice\; vibration\; scattered</p>\; <p\; class="MsoNormal">So\; conductivity\; decreases.</p>\; </span>}}{}$

 4. Crystal Defects

$\frac{\mathrm{<span\; style="font-family:\; georgia;\; font-size:\; large;">\; <p\; class="MsoNormal">Vacancies,\; cracks,\; dislocations\; reduce\; heat\; flow.</p>\; </span>}}{}$

 5. Moisture / Air Gaps

$\frac{\mathrm{<span\; style="font-family:\; georgia;\; font-size:\; large;">\; <p\; class="MsoNormal">Air\; has\; low\; conductivity.</p>\; <p\; class="MsoNormal">So\; porous\; materials\; act\; as\; insulators.</p>\; </span>}}{}$

Thermal Resistance

$\frac{\mathrm{<span\; style="font-family:\; georgia;\; font-size:\; large;"><span\; style="font-weight:\; normal;">\; </span><p\; class="MsoNormal"><span\; style="font-weight:\; normal;">Opposition\; to\; heat\; flow\; is\; called\; thermal\; resistance.</span></p>\; <p\; class="MsoNormal"><span\; style="font-weight:\; normal;">R=L/kA</span></p>\; <p\; class="MsoNormal"><span\; style="font-weight:\; normal;">Where:</span></p>\; <p\; class="MsoNormal"><span\; style="font-weight:\; normal;"> L\; =\; thickness</span></p>\; <p\; class="MsoNormal"><span\; style="font-weight:\; normal;"> k\; =\; conductivity</span></p>\; <p\; class="MsoNormal"><span\; style="font-weight:\; normal;">A\; =\; area</span></p>\; <p\; class="MsoNormal"><span\; style="font-weight:\; normal;">If\; R\; is\; high\; \to \; heat\; flow\; less.</span></p>\; <p\; class="MsoNormal"><span\; style="font-weight:\; normal;">Used\; in\; insulation\; design</span></p>\; </span>}}{}$

Why Metals Conduct Heat Better?

$ Metals contain a large number of free electrons. These electrons move randomly and transfer kinetic energy rapidly. Therefore: Heat conductivity of metals is high Electrical conductivity is also high Example: Silver has very high thermal conductivity Copper is widely used in wires and heat sinks $

Why Insulators Conduct Poorly?

$ Materials like wood, rubber, plastic, glass do not have free electrons. Heat transfer happens only through lattice vibration, which is slower. Hence thermal conductivity is low. Used in: Handle of cooking vessels Electrical insulation Thermal insulation $

 Conductors vs Insulators

$ Property Conductors Insulators  Heat Flow Easy      Difficult      | Electrons Many free electrons No free electrons Use Cooking vessel      Handle            Example Copper   Wood  $

$ In metals: Good electrical conductors are usually good thermal conductors. Because both depend on free electrons. Example: Copper conducts electricity well.  Copper also conducts heat well.  Importance in Solid State Physics Thermal conductivity gives information about:  Electron motion  Crystal structure Defects in solids Purity of material Lattice vibration Semiconductor performance Scientists use conductivity to study materials. $

Applications

$  1. Cooking Utensils Aluminium and copper bases used.  2. Heat Sink Computer processors generate heat. Copper or aluminium heat sinks remove heat.  3. Thermos Flask Walls prevent heat transfer. 4. Refrigerators Insulating foam reduces heat entry. 5. Buildings Roof insulation keeps room cool.  6. Spacecraft Special thermal shields used.  Experimental Methods  Searle’s Method Used for good conductors  Lee’s Disc Method Used for poor conductors.  Comparative Method Compare with known material. Numerical Example A rod with:  High k = copper Transfers heat quickly. Another rod:  Low k = wood Transfers heat slowly. Thus k determines speed of heat conduction. Real Life Observations  Why tile floor feels colder than wood floor? Because tile has higher conductivity and removes heat from your feet faster.  Why blanket feels warm? Blanket traps air, and air is poor conductor. Why pan handle made of plastic? Plastic has low conductivity.  Advanced Concept: Phonon Scattering In crystals, phonons carry heat. But phonons are disturbed by:  Defects  Boundaries  Impurities  Temperature rise So conductivity decreases. Modern Materials $

In Nano Materials

$ Modern research studies thermal conductivity in: Graphene Carbon nanotubes Thin films These materials show extraordinary heat conduction. Used in advanced electronics.  Diamond Very high thermal conductivity. Used in electronics cooling.  Graphene Excellent heat conductor. Used in future nano devices.   Material - Thermal Conductivity (W/mK)  Silver  -   429                      Copper  -   401                         Aluminium - 237                          Iron   -   80                           Glass  -   1                           Wood    -   0.15                         Rubber    - 0.13                         Air        0.024                          Summary Thermal conductivity is the ability of a material to conduct heat. It is represented by k and measured in W/mK. Metals conduct heat well because of free electrons. Insulators conduct poorly because heat travels only through lattice vibrations. Fourier’s law explains heat conduction mathematically. Thermal conductivity depends on temperature, structure, purity, and defects. It is useful in engineering, electronics, and solid state physics,and everyday life.      $