Electronic Properties of Metal

Cheatsheet Content

### Conduction in Metals - Electrical conduction involves motion of charges in a material under an applied electric field. - **Conduction Electrons:** Valence electrons from atoms form a "sea of electrons" free to move. - In presence of electric field, free electrons accelerate opposite to the field, creating electric current. - Flow of electrons governed by Ohm's law ($I \propto V$). #### Mechanism of Electron Motion - **Without electric field:** Random motion, average velocity is zero. - **With electric field:** Electrons gain a drift velocity opposite to the field. ### Drude Model - Classical approach to explain conduction in metals. - Explains Ohm's law and temperature dependence of resistivity. - Introduces concepts of **mean free path** and **mean free time**. #### Basic Assumptions 1. Between collisions, interaction of a given electron with other electrons and ion cores is neglected. 2. In absence of external electric field, each electron moves freely in a straight line. 3. Collisions are instantaneous events abruptly altering electron velocity, attributed to electrons bouncing off impermeable ion cores. 4. Electron experiences a collision with probability $dt/\tau$ per unit time, where $\tau$ is the **relaxation time** (average time since last collision). 5. Electrons achieve thermal equilibrium with surroundings only through collision. #### DC Electrical Conductivity - **Ohm's law:** $V = IR$ - **Electric field:** $\vec{E} = \rho \vec{J}$ - **Current density:** $\vec{J}$ (direction parallel to flow) - **Current density for n electrons with charge -e and average velocity $\vec{v}$:** $\vec{J} = -ne\vec{v}$ - In equilibrium, average velocity is zero (no net electrical current). - In presence of electric field, electron velocity $\vec{v} = \frac{-e\vec{E}t}{m}$ - $v_t$: velocity from last collision (random) - $t$: time since last collision - Average velocity $\vec{v}_{avg} = \frac{-e\vec{E}\tau}{m}$ - **Conductivity:** $\sigma = \frac{ne^2\tau}{m}$ - Thus, $\vec{J} = \sigma\vec{E}$ - **Resistivity:** $\rho = \frac{m}{ne^2\tau}$ #### General Equation of Motion - Electron momentum $\vec{p}(t)$ under force $\vec{f}(t)$. - Current density $\vec{J} = \frac{ne\vec{p}(t)}{m}$. - Momentum per electron at $t + dt$: - Collision with probability $dt/\tau$: loses momentum, emerges with random momentum. - Contribution to total momentum: $\frac{dt}{\tau}\vec{f}(t)dt$ (assuming random component is absorbed into $\vec{f}(t)$). - No collision with probability $(1 - dt/\tau)$: momentum is $\vec{p}(t) + \vec{f}(t)dt$. - Contribution: $(1 - dt/\tau)[\vec{p}(t) + \vec{f}(t)dt]$. - Net momentum: $\vec{p}(t + dt) = (1 - \frac{dt}{\tau})[\vec{p}(t) + \vec{f}(t)dt] + \text{random component} + O(dt)^2$. - Simplified: $\vec{p}(t + dt) \approx \vec{p}(t) - \frac{\vec{p}(t)dt}{\tau} + \vec{f}(t)dt$. - **Equation of motion:** $\frac{d\vec{p}(t)}{dt} = -\frac{\vec{p}(t)}{\tau} + \vec{f}(t)$. - The term $-\frac{\vec{p}(t)}{\tau}$ introduces a frictional damping (energy loss) due to individual electron collisions. #### AC Electrical Conductivity - Time-dependent electrical field $\vec{E}(t) = \vec{E}_0e^{-i\omega t}$. - Equation of motion for momentum per electron: $\frac{d\vec{p}}{dt} = -\frac{\vec{p}}{\tau} -e\vec{E}$. - Solution has form $\vec{p}(t) = \vec{p}_0e^{-i\omega t}$. - In frequency domain, $\vec{p}(\omega)$ satisfies: $-i\omega\vec{p}(\omega) = -\frac{\vec{p}(\omega)}{\tau} -e\vec{E}(\omega)$. - Current density $\vec{J} = -ne\vec{p}/m$. - **AC conductivity $\sigma(\omega)$:** $\vec{J}(\omega) = \frac{ne^2\tau/m}{1-i\omega\tau}\vec{E}(\omega)$. - Thus, $\vec{J}(\omega) = \sigma(\omega)\vec{E}(\omega)$, where $\sigma(\omega)$ is the frequency-dependent AC conductivity. #### Drude Model Limitations 1. **Hall coefficient:** Drude predicts $R_H = -1/(nec)$, which is negative, but often positive values are observed. 2. **Overestimation of electronic contribution to specific heat capacity:** Drude predicts $C_V = \frac{3}{2}nk_B$, much larger than observed. 3. **Thermoelectric field (thermopower Q):** Calculated $Q \approx -0.47 \times 10^{-4}$ V/K; Observed $Q$ values are 100 times smaller. 4. **Magnetoresistance:** Predicts zero, but it exists. 5. Problems with DC and AC electrical conductivity in certain aspects. ### Hall Effect and Magnetoresistance #### Hall Effect When an electric current flows in a conductor and a magnetic field is applied perpendicular to the current, a transverse electric field (Hall field) is generated perpendicular to both. - **Scenario:** - Electric field $E_x$ applied along x-direction. - Current density $J_x$ flows in the wire. - Magnetic field $H$ points in +z direction. - **Lorentz force:** $\vec{F}_L = -e(\vec{v} \times \vec{H})$ acts on electrons, deflecting electrons in -y direction. - Electrons accumulate on one side of the wire, building up an electric field $E_y$ (Hall field). - $E_y$ opposes the Lorentz force, leading to a steady state where forces balance. - **Hall coefficient $R_H = E_y/J_xH$:** - Describes magnitude and sign of the Hall voltage. - $R_H = -1/(nec)$ (from Drude model), depends on carrier density $n$. - **Equation of motion for electron momentum in combined $\vec{E}$ and $\vec{H}$ fields:** $\frac{d\vec{p}}{dt} = -e(\vec{E} + \vec{v} \times \vec{H}) - \frac{\vec{p}}{\tau}$. - **Steady state** ($dp/dt = 0$, current is time independent): - $0 = -eE_x - \omega_c P_y - P_x/\tau$. - $0 = -eE_y - \omega_c P_x - P_y/\tau$. - Where $\omega_c = \frac{eH}{mc}$ is the cyclotron frequency. - Assuming transverse current $J_y = 0$, the Hall field is: - $E_y = -\frac{H}{ne} J_x = -\frac{H}{\sigma_0} J_x$. - Where $\sigma_0$ is the DC conductivity without magnetic field. #### Magnetoresistance The change in electrical resistance of a material when subjected to an external magnetic field. - Basic Drude model predicts that resistance does not depend on the magnetic field if $J_y = 0$. - This leads to the Drude model prediction of zero magnetoresistance. - However, real materials show a magnetic field dependence of resistance which can change dramatically in some cases. ### Thermal Conductivity Transfer of thermal energy through a material. - Given by **Fourier's Law:** $\vec{J}_q = -K\nabla T$. - $\vec{J}_q$: heat current density. - $K$: thermal conductivity. - $\nabla T$: temperature gradient. - **Electron energy dependence:** - Energy of an electron depends on the local temperature. - Higher temperature $\rightarrow$ higher electron energy. - **Mechanism of heat transfer:** - Electrons at equilibrium at position $x$ have energy $E(T[x])$. - Electrons arriving from a colder region ($x - v\tau$) have lower energy $E(T[x - v\tau])$. - Electrons arriving from a hotter region ($x + v\tau$) have higher energy $E(T[x + v\tau])$. - Net transfer of energy from hotter to colder regions. - **Thermal conductivity formula** (from kinetic theory considerations): - $K = \frac{1}{3} n v^2 C_V$ (in 3D, for a gas). - $n$: number density of electrons. - $v^2$: mean square electronic speed. - $\tau$: relaxation time. - $C_V$: heat capacity per unit volume, $C_V = dE/dT$ (change in energy w.r.t temperature). - Drude used $C_V = \frac{3}{2}nk_B$ for free electrons. #### Wiedemann-Franz Law Ratio of thermal conductivity ($K$) to electrical conductivity ($\sigma$) times temperature ($T$) is a constant for metals. - $\frac{K}{\sigma T} = (\frac{k_B}{e})^2 \frac{\pi^2}{3} = 2.44 \times 10^{-8}$ W$\cdot\Omega$/K$^2$. - This constant is known as the **Lorenz number** ($L_0$). - This law suggests that $K$ and $\sigma$ are both determined by free electrons and their collisions. ### Thermoelectric Effect Generation of an electric field in response to a temperature gradient. - Electronic velocity does not vanish even in presence of a temperature gradient. - Electrons from high-temperature side have higher energy and speed $\rightarrow$ net motion of electrons from high energy to low energy side. - Under **open-circuit condition** (no net current flow): - Electrons accumulate at the low-temperature side. - This accumulation produces an electric field. - This phenomenon is known as the **Seebeck effect**. - The generated electric field is given by $\vec{E} = Q\nabla T$. - $Q$ is the **thermopower** (Seebeck coefficient). - Mean electronic velocity due to temperature gradient: $\vec{v}_T = -\frac{1}{3}\frac{d\vec{v}}{dT}\nabla T$. - Mean velocity due to induced electric field: $\vec{v}_E = -\frac{e\vec{E}\tau}{m}$. - At equilibrium (no net current), $\vec{v}_T + \vec{v}_E = 0$. - Solving for $\vec{E}$: $\vec{E} = \frac{m}{3e\tau}\frac{d\vec{v}}{dT}\nabla T$. - From this, Drude model predicts thermopower $Q = -\frac{\pi^2 k_B^2 T}{2eE_F}$ (simplified form). - Calculated value $Q \approx -0.47 \times 10^{-4}$ V/K. - Observed values are typically 100 times smaller. - This discrepancy is a drawback of the Drude model due to overestimation of $C_V$. ### Sommerfeld Theory (Quantum Mechanical Consideration) Applies quantum mechanics (Fermi-Dirac statistics) to explain electronic properties of metals, improving upon the classical Drude model. - **Drude vs. Sommerfeld:** - Drude: electronic velocity distribution like ordinary classical density, uses Maxwell-Boltzmann statistics. - Sommerfeld: applies quantum mechanics, uses Fermi-Dirac statistics. - Valence electrons are free in the metal but confined within the crystal boundaries. - They have lower potential energy inside the crystal than outside. - Allowed energy levels for electrons are quantized. - Specific heat of metal is very low (a key observation that Drude failed to explain). #### Fermi Sphere and Velocity Distribution - In equilibrium, valence electrons perform random motion with no preferential velocity. - **Fermi velocity ($V_F$):** maximum velocity of electrons at 0K. - All points inside the **Fermi sphere** (in k-space) correspond to occupied electron states. - At equilibrium, velocity vectors cancel pairwise $\rightarrow$ no net velocity. #### Effect of Electric Field - Application of an electric field causes a **displacement of the Fermi sphere** in k-space. - Direction of displacement is opposite to the field direction. - Most electron velocities still cancel pairwise. - The uncompensated electrons (those in the shifted region of the Fermi sphere) are the cause of the observed current. #### Conductivity Derivation - Only specific electrons (near the Fermi surface) participate in conduction. - These electrons drift with Fermi velocity $V_F$. - Largest energy electrons occupy the Fermi energy $E_F$ at $T=0$. - Only a small energy $\Delta E$ is needed to accelerate a substantial number of electrons. - **Current density $\vec{J}$:** - $\vec{J} = V_F e N'$ where $N'$ = number of electrons displaced per unit volume. - $N' = N(E_F)\Delta E$, where $N(E_F)$ is the density of states at the Fermi energy. - $\vec{J} = V_F e N(E_F)\Delta E = V_F e N(E_F)\frac{dE}{dk}\Delta k$. - **Relation between energy and wave vector:** $E = \frac{\hbar^2 k^2}{2m} \Rightarrow \frac{dE}{dk} = \frac{\hbar^2 k}{m} = \hbar V_F$. - **Displacement of Fermi sphere in k-space ($\Delta k$) due to electric field:** - Force $\vec{F} = \hbar\frac{d\vec{k}}{dt} = -e\vec{E} \Rightarrow \Delta k = \frac{-e\vec{E}}{h} \Delta t$. - $\Delta t = \tau$ (relaxation time). - Thus, $\Delta k = \frac{-e\vec{E}\tau}{\hbar}$. - **Current density for spherical Fermi surface:** $\vec{J} = \sigma \vec{E} = e^2 N(E_F) \tau V_F^2 \vec{E}$. - **Electrical conductivity:** $\sigma = e^2 N(E_F) \tau V_F^2$. - Depends on: 1. Fermi velocity ($V_F$). 2. Relaxation time ($\tau$). 3. Population density of electrons near the Fermi surface ($N(E_F)$). - **Key distinction from Drude:** Sommerfeld's result uses $N(E_F)$ (density of states at Fermi level) instead of total electron density $n$, emphasizing that only electrons near the Fermi surface contribute to conduction. #### Sommerfeld Prediction Problems 1. **Linear term in specific heat:** Works for alkali metals, but not noble metals or transition metals. 2. **Specific heat:** Still has some discrepancies. 3. **Compressibility of metal:** Not fully explained. - **Main problem for both Drude and Sommerfeld:** Free electron approximation (electrons are treated as non-interacting particles). ### Resistivity in Metals #### Pure Metals - Resistivity $\rho$ decreases linearly with decreasing temperature until it reaches a finite value. - **Empirical equation:** $\rho_T = \rho_0[1 + \alpha(T - T_0)]$. - $\alpha$ = linear temperature coefficient of resistivity. - **Electron scattering mechanisms:** - Thermal energy causes lattice atoms to oscillate about equilibrium positions. - This increases incoherent scattering of electron waves. - Leads to electron-atom collisions. #### Matthiessen's Rule Total resistivity of a metal is the sum of resistivities from different scattering mechanisms. - $\rho = \rho_{th} + \rho_{imp} + \rho_{def} = \rho_{th} + \rho_{res}$. - $\rho_{th}$: **Thermally induced resistivity** (ideal resistivity). - Due to scattering by thermal vibrations of the lattice. - Temperature dependent. - $\rho_{imp}$: **Resistivity due to impurities**. - $\rho_{def}$: **Resistivity due to defects** in a crystal. - $\rho_{res}$: **Residual resistivity** ($= \rho_{imp} + \rho_{def}$). - Caused by imperfections, impurities, vacancies, grain boundaries, and dislocations. - Not temperature dependent. - Number of impurity atoms is fixed, but vacancies and grain boundaries can change with temperature. #### Alloys - Resistivity increases with increasing amount of **solute content**. - **Causes of resistivity increase:** 1. Different sizes of atoms. 2. Atoms with different valences (introducing local charge difference). 3. Solutes with different concentration compared to host element. - Resistivity of alloy depends on the fractional atomic composition of constituents. #### Nörheim's Rule Describes the resistivity of an alloy as a function of its composition. - $\rho = X_A \rho_A + X_B \rho_B + C X_A X_B$. - $X_A, X_B$ = fractional atomic composition of constituents A and B. - $\rho_A, \rho_B$ = resistivities of pure constituents. - $C$ = materials constant. - Often simplified for impurity (minority component) scattering: - $\rho_r = C x(1-x)$. - $x$ = concentration of impurity. - $C$ = Nörheim's coefficient. - Total resistivity of alloy: $\rho = \rho_{matrix} + C x(1-x) = \rho_T + \rho_R$. - $\rho_T$ is temperature-dependent matrix resistivity. - $\rho_R$ is residual resistivity due to impurities. - **Note that:** - Number of free electrons $N_e$ does not change significantly with temperature. - $N(E_F)$ changes very little with temperature. - Mean free path $\tau$ decreases with increasing temperature (due to increased collision rate). - Thus, conductivity $\sigma$ decreases with increasing temperature.