Engineering Physics (BT-201) Unit 3 Notes

How unit 3 is examined

Free electrons, Fermi level, density of states, Bloch and Kronig-Penney, the PN junction and Zener diode, the solar cell and the Hall effect; Hall effect (derivation), Fermi-level shift, and the PN and Zener V-I curves carry the marks.

Free electron theory of metals

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Definition. In the free electron theory a metal's valence electrons move freely inside the crystal like gas molecules, in a region of constant potential, and are held only by the surface of the metal.

Key points.

Drude (classical) treated the electrons as a gas that obeys Maxwell-Boltzmann statistics; Sommerfeld (quantum) made them obey Fermi-Dirac statistics and Pauli's exclusion principle.
Electrons collide with ions and with each other only occasionally, with average time between collisions (relaxation time) $\tau$.
Conductivity is $\sigma = \dfrac{n e^2 \tau}{m}$, which explains Ohm's law and the high conductivity of metals.
Fermi energy at 0 K is $E_F = \dfrac{\hbar^2}{2m}(3\pi^2 n)^{2/3}$.
It fails to explain semiconductors and insulators, the small electronic specific heat, and the positive Hall coefficient of some metals.

Fermi level of Intrinsic and extrinsic

Definition. The Fermi level $E_F$ is the energy level at which the probability of occupation by an electron is $\tfrac12$ at any temperature above 0 K; it is the reference level that shows how full the bands are.

Formula. Extrinsic range: n-type $E_F = E_c - kT\ln\dfrac{N_c}{N_d}$, p-type $E_F = E_v + kT\ln\dfrac{N_v}{N_a}$.

Key points.

In an intrinsic semiconductor electrons and holes are equal in number, so $E_F = E_i$ lies almost exactly at the middle of the gap, $\tfrac{E_c+E_v}{2}$, at all temperatures.
In an n-type semiconductor at 0 K the donors are un-ionised and $E_F$ lies midway between the donor level $E_d$ and the conduction band edge $E_c$.
In a p-type semiconductor at 0 K, $E_F$ lies midway between the valence band edge $E_v$ and the acceptor level $E_a$.
With rising donor concentration $N_d$ the Fermi level of n-type moves up towards $E_c$; with rising acceptor concentration $N_a$ the Fermi level of p-type moves down towards $E_v$.
With rising temperature, at low $T$ (freeze-out) the donors ionise and $E_F$ first rises then falls; in the moderate (extrinsic) range all donors are ionised and $E_F$ falls slowly.
At high temperature, electron-hole pairs generated across the gap swamp the doping, the material turns intrinsic and $E_F$ tends to $E_i$ from either side.

Diagram. Fermi level of n-type against temperature.

Region	Temperature	Position of $E_F$
Freeze-out	0 K	Midway between $E_d$ and $E_c$
Extrinsic	Room	Just below $E_c$, falling slowly
Intrinsic	Very high	At $E_i$, mid-gap

<mark>The Fermi level of an n-type semiconductor lies just below the conduction band, rises with donor concentration, and falls back to the mid-gap intrinsic level $E_i$ as temperature increases.</mark>

Answer frame. Open with the definition of $E_F$ and its intrinsic position; draw band diagrams of intrinsic, n-type and p-type with $E_F$ marked, then the $E_F$-versus-$T$ curve falling from $\tfrac{E_c+E_d}{2}$ to $E_i$; develop points 1-6 in order, concentration before temperature; close with the sentence in the highlight.

Asked: [7 marks] (Jun 2022, Dec 2023, Dec 2024) Explain the variation of Fermi level in N-type semiconductor with concentration and temperature. / Explain the Fermi level shifting in semiconductors.

Density of states

Definition. Density of states $g(E)$ is the number of available electron states per unit volume per unit energy range at energy $E$.

Formula. $g(E)\,dE = \dfrac{V}{2\pi^2}\left(\dfrac{2m}{\hbar^2}\right)^{3/2} E^{1/2}\,dE$ for a 3D free electron gas (spin included).

Key points.

The formula is found by counting $k$-space states in a sphere of radius $k$ and dividing by the energy interval, with two spin states per $k$ value.
$g(E) \propto E^{1/2}$, so the curve is a parabola starting at zero at $E=0$ and rising with energy.
The carrier concentration is $n=\int g(E)f(E)\,dE$, so it fixes conduction properties.

Asked: [5 marks] (Dec 2023) Write short notes on "Density of State".

Bloch's theorem for particles in a periodic potential

Definition. Bloch's theorem: for an electron in a periodic potential $V(x+a)=V(x)$, the wavefunction is $\psi_k(\mathbf r)=e^{i\mathbf k\cdot\mathbf r}\,u_k(\mathbf r)$, where $u_k(\mathbf r)$ has the periodicity of the lattice.

Key points.

Proof: since $V(x+a)=V(x)$, $\psi(x+a)$ is also a solution, so the translation operator $T$ commutes with $H$ and they share eigenfunctions: $\psi(x+a)=\lambda\psi(x)$.
Periodic boundary condition over $N$ cells, $\psi(x+Na)=\psi(x)$, gives $\lambda^N=1$, hence $\lambda=e^{ika}$ with $k=\dfrac{2\pi m}{Na}$.
Define $u(x)=e^{-ikx}\psi(x)$; then $u(x+a)=u(x)$, giving $\psi=e^{ikx}u(x)$.

Asked: [4 marks] (Dec 2023) State and prove Bloch Theorem.

Kronig-Penney model and origin of energy bands

Definition. The Kronig-Penney model represents the crystal lattice as a one-dimensional periodic array of rectangular potential wells of width $a$ separated by barriers of width $b$ and height $V_0$.

Key points.

Potential: $V=0$ for $0<x<a$ and $V=V_0$ for $-b<x<0$, with period $a+b$.
Schrodinger equations: $\dfrac{d^2\psi}{dx^2}+\alpha^2\psi=0$ in the well, $\alpha^2=\dfrac{2mE}{\hbar^2}$; $\dfrac{d^2\psi}{dx^2}-\beta^2\psi=0$ in the barrier, $\beta^2=\dfrac{2m(V_0-E)}{\hbar^2}$.
Bloch's theorem and continuity of $\psi$ and $d\psi/dx$ give $P\dfrac{\sin\alpha a}{\alpha a}+\cos\alpha a=\cos ka$, with $P=\dfrac{mV_0ba}{\hbar^2}$.
Conclusion: solutions exist only where the left side lies between $-1$ and $+1$, so energy splits into allowed bands separated by forbidden gaps; bands widen at higher energy, and $P\to0$ gives free electrons, $P\to\infty$ isolated levels.

Asked: [7 marks] (Jun 2023) Explain Kronig Penney model for periodic potential. Write down the Schrodinger equation and discuss conclusion of this model.

V-I characteristics of PN junction

Definition. A PN junction diode is a single crystal doped p-type on one side and n-type on the other; at the junction electrons and holes diffuse across and leave a depletion region with a barrier potential ($\approx0.7$ V Si, $0.3$ V Ge).

Formula. $I=I_0\left(e^{V/\eta V_T}-1\right)$, with $V_T=kT/e\approx26$ mV at room temperature.

Key points.

In forward bias (p to positive) the applied voltage opposes the barrier, so the depletion layer narrows.
Below the knee (cut-in) voltage the current is negligible; above it the current rises exponentially and the diode conducts like a small resistance.
In reverse bias the depletion layer widens and only minority carriers move, giving a tiny reverse saturation current $I_0$ (microamperes) almost independent of voltage.
At the breakdown voltage the reverse current rises sharply, by avalanche or Zener breakdown.

Diagram. V-I curve, drawn with $I$ in mA for forward and $\mu$A for reverse.

Region	Voltage	Current
Forward, below knee	0 to 0.7 V	Almost zero
Forward, above knee	Above 0.7 V	Rises exponentially (mA)
Reverse	0 to $-V_{BR}$	Constant $I_0$ ($\mu$A)
Breakdown	At $-V_{BR}$	Sharp rise

Comparison. Avalanche versus Zener breakdown.

Basis	Zener	Avalanche
Mechanism	Strong field pulls electrons from bonds (field ionisation)	Collision of carriers creates a chain of new pairs (impact ionisation)
Doping	Heavy	Light
Depletion layer	Thin	Wide
Voltage	Below about 5 V	Above about 6 V
Temperature coefficient	Negative	Positive
Sharpness	Sharp knee	Less sharp

<mark>A PN junction conducts in forward bias above the knee voltage and blocks in reverse bias except for a small saturation current until breakdown.</mark>

Answer frame. Open with the definition and depletion region; draw the V-I curve with knee and breakdown labelled; then develop forward bias, reverse bias, breakdown; for the comparison question add the table; close with the highlight.

Asked: [7 marks] (Nov 2022, Dec 2024) What is P-N junction diode? Discuss its I-V characteristics. / What is P-N junction diode, explain its working and discuss its V-I Characteristics. Asked: [7 marks] (Jun 2023) Draw and explain the V-I characteristic curve of P-N junction diode. Differentiate between Avalanche and Zener breakdown.

Zener diode

Definition. A Zener diode is a heavily doped PN junction designed to operate in the reverse breakdown region, where it holds a nearly constant voltage $V_Z$.

Key points.

Heavy doping makes the depletion layer very thin, so a modest reverse voltage gives a field of about $10^7$ V/m.
In forward bias it acts as an ordinary diode, conducting above the knee voltage of about 0.7 V.
In reverse bias only a small leakage current flows until $V_Z$ is reached.
At $V_Z$ the field breaks covalent bonds and the current rises sharply while the voltage stays almost fixed, which is the constant-voltage property.
Use: as a voltage regulator, connected in reverse across the load with a series resistor.

Diagram. Same V-I curve as the PN diode: forward part in quadrant I, and in quadrant III a flat leakage line ending at a sharp vertical drop at $-V_Z$, labelled with the knee current.

<mark>A Zener diode operates in reverse breakdown, where the voltage stays almost constant at $V_Z$ while the current changes widely.</mark>

Answer frame. Open with the definition; draw the V-I curve with $V_Z$ marked; develop forward bias then reverse bias and breakdown; close with the voltage-regulator use.

Asked: [7 marks] (Jun 2022, Jun 2025) What is Zener diode? Discuss its I-V characteristics.

Solar Cell

Definition. A solar cell is a PN junction that converts light energy directly into electrical energy by the photovoltaic effect.

Key points.

Construction: a thin n-type top layer over a p-type base, with a front metal grid, a back contact and an anti-reflective coating.
Generation: photons with $h\nu>E_g$ create electron-hole pairs near the junction.
Separation: the junction field sweeps electrons to the n side and holes to the p side.
Collection: the carriers gather at the electrodes, producing photovoltage and current through the load.

Diagram. Sunlight then anti-reflective coating, thin n-layer, depletion region, p-base, back contact.

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Asked: [7 marks] (Nov 2022) Explain the construction and working of solar cell with neat diagrams.

Hall Effect

Definition. <mark>When a current-carrying conductor or semiconductor is placed in a magnetic field perpendicular to the current, a transverse voltage, the Hall voltage, appears across the sides perpendicular to both current and field.</mark>

Diagram. Slab of width $w$ (y) and thickness $t$ (z): current $I$ along x, field $B$ along z, Hall voltage $V_H$ across y.

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Derivation.

Take an n-type slab; the carriers (electrons) drift with velocity $v_d$ against the current, so the current density is $J=\dfrac{I}{wt}=nev_d$.
The magnetic force $F=ev_dB$ deflects the electrons to one side (y), charging that face negatively and the opposite face positively.
This builds up a Hall field $E_H$ that exerts a force $eE_H$ opposing the magnetic force. At equilibrium $eE_H=ev_dB$, so $E_H=v_dB$.
Substituting $v_d=\dfrac{J}{ne}$ gives $E_H=\dfrac{BJ}{ne}$.
Hall voltage: $V_H=E_Hw=\dfrac{BJw}{ne}=\dfrac{BI}{net}$.
Hall coefficient: $R_H=\dfrac{E_H}{JB}=\dfrac{1}{ne}$, negative for electrons and $+\dfrac{1}{pe}$ for holes in p-type.
Hall angle: the angle between the total field and current direction satisfies $\tan\theta_H=\dfrac{E_H}{E_x}=\dfrac{v_dB}{E_x}=\mu B$, since mobility $\mu=\dfrac{v_d}{E_x}$.

Key points.

$R_H$ gives carrier concentration $n=\dfrac{1}{eR_H}$ and its sign shows whether the material is n-type or p-type.
With conductivity $\sigma=ne\mu$, mobility is $\mu=|R_H|\sigma$.
Applications: finding carrier type, concentration and mobility, and Hall probes that measure magnetic fields.

Pitfall: Draw the slab with $I$, $B$ and $V_H$ mutually perpendicular and state the axes, and do not forget that the thickness $t$ (along $B$) appears in $V_H$ while the width $w$ cancels.

Answer frame. Open with the highlighted definition; draw the slab with axes labelled; then develop steps 1-7 in order, writing the force balance $eE_H=ev_dB$ in a box; close with $V_H=\dfrac{BI}{net}$ and $R_H=\dfrac{1}{ne}$. For "Hall angle" end with $\tan\theta_H=\mu B$; for the short note add applications.

Asked: [7 marks] (Jun 2022, Nov 2022, Dec 2024) Deduce an expression for Hall coefficient and Hall voltage. Asked: [7 marks] (Jun 2023) Explain Hall effect. Derive expression for Hall coefficient and Hall angle. Asked: [14 marks] (Jun 2025) Write short notes on: a) Solar cell b) Hall effect c) Bloch's theorem.

Last-minute revision

Fermi level: probability $\tfrac12$; intrinsic at mid-gap, n-type near $E_c$, p-type near $E_v$; all tend to $E_i$ at high $T$.
$g(E)=\dfrac{V}{2\pi^2}\left(\dfrac{2m}{\hbar^2}\right)^{3/2}E^{1/2}$, a parabola.
Bloch: $\psi=e^{ikr}u_k(r)$ with $u_k$ periodic.
Kronig-Penney: $P\dfrac{\sin\alpha a}{\alpha a}+\cos\alpha a=\cos ka$, giving bands and gaps.
Knee voltage: Si 0.7 V, Ge 0.3 V.
Zener: heavy doping, below 5 V, negative temperature coefficient; avalanche above 6 V, positive.
Solar cell: $h\nu>E_g$, generation, separation, collection.
$V_H=\dfrac{BI}{net}$, $R_H=\dfrac{1}{ne}$, $\tan\theta_H=\mu B$, $\mu=|R_H|\sigma$.

Memory hooks

Fermi level moves like a seesaw: donors push it up, acceptors push it down, heat brings it back to the middle.
Bloch: "wave times lattice": plane wave times periodic $u$.
Hall: F for magnetic force equals F for electric force, so $E_H=v_dB$.
Zener = Zero-ish voltage change; Avalanche = Above 6 V.

Coverage checklist

Free electron theory of metals: no past questions (unasked).
Fermi level of Intrinsic and extrinsic: Q2 (Jun 2022, Dec 2023, Dec 2024).
density of states: Q10 (Dec 2023).
Bloch's theorem for particles in a periodic potential: Q11 (Dec 2023), Q1c (Jun 2025).
Kronig-Penney model(no derivation) and origin of energy bands: Q5 (Jun 2023).
V-I characteristics of PN junction: Q7 (Nov 2022, Dec 2024), Q8 (Jun 2023).
Zener diode: Q9 (Jun 2022, Jun 2025).
Solar Cell: Q6 (Nov 2022), Q1a (Jun 2025).
Hall Effect: Q3 (Jun 2022, Nov 2022, Dec 2024), Q4 (Jun 2023), Q1b (Jun 2025).

Account

Engineering Physics (BT-201) - Unit 3 Short Notes

How unit 3 is examined

Free electron theory of metals

Fermi level of Intrinsic and extrinsic

Density of states

Bloch's theorem for particles in a periodic potential

Kronig-Penney model and origin of energy bands

V-I characteristics of PN junction

Zener diode

Solar Cell

Hall Effect

Last-minute revision

Memory hooks

Coverage checklist