Solid State Physics — Unit I Notes

Solid State Physics

Solid State Physics

Unit I — Band theory, semiconductors & junction devices
8 hrs · master-notes edition

1. Classification of solids (band diagram)
2. Intrinsic semiconductors
3. Extrinsic semiconductors
4. Fermi level (intrinsic)
5. Conductivity derivation
6. P–N junction diode
7. Zener diode
8. Light Emitting Diode
9. Hall effect
10. Chapter summary

UNIT I · SOLID STATE PHYSICS

Band Theory, Semiconductors & Junction Devices

Complete topic-wise notes — starting from the classification of solids using energy bands, through intrinsic/extrinsic semiconductors, the Fermi level, conductivity, and the three key devices built on the p–n junction, ending with the Hall effect. Every derivation is broken into numbered steps so you can follow the logic, not just memorise the result.

Band theoryFermi–Dirac statisticsSemiconductor physics p–n junctionZener & LEDHall effect

Topic 1

Classification of Solids on the Basis of Energy Band Diagram

1.1 Where energy bands come from

In an isolated atom, electrons occupy sharp, discrete energy levels. When atoms are packed together into a crystal, the outer orbitals of neighbouring atoms overlap. By the Pauli exclusion principle no two electrons in the whole crystal can share an identical quantum state, so each discrete atomic level splits into a closely spaced set of $N$ levels (where $N \sim 10^{23}$ is the number of atoms). This set is so dense that it behaves like a continuous energy band rather than a single level.

The two bands that matter most for electrical behaviour are:

• Valence band (VB): the highest energy band that is normally occupied by electrons at 0 K — built from the outer (valence) shell electrons that bind the crystal together.
• Conduction band (CB): the next higher band. Electrons here are not tied to any particular atom and can move freely through the crystal under an applied field, so they conduct current.

Between them lies the forbidden energy gap (or band gap), $E_g$ — a range of energies that no electron in the pure crystal can occupy, because no allowed quantum state exists there.

1.2 The three classes

Solids are classified purely by how the valence band, conduction band and gap are arranged relative to each other and to the Fermi level $E_F$ (the energy level with 50% probability of occupation).

Fig 1.1 — Relative position of conduction band (top, amber), valence band (bottom, green) and forbidden gap (dashed) for the three classes of solids.

(a) Conductors (metals)

The valence and conduction bands overlap, or the valence band is only partially filled. There is effectively no forbidden gap, so electrons at the top of the filled states can move into empty states with an infinitesimally small energy input. This is why metals conduct even at very low temperatures, and resistivity is typically $10^{-8}$–$10^{-6}\ \Omega\text{m}$.

(b) Insulators

The valence band is completely full and the conduction band completely empty, separated by a large gap, typically $E_g > 3\ \text{eV}$ (diamond $\approx 5.5\ \text{eV}$). Thermal energy at room temperature ($k_BT \approx 0.026\ \text{eV}$) is far too small to lift electrons across this gap, so the conduction band stays essentially empty and the material does not conduct. Resistivity is very high, $10^{10}$–$10^{18}\ \Omega\text{m}$.

(c) Semiconductors

Same band structure as an insulator — full valence band, empty conduction band — but the gap is narrow, typically $0.2$–$3\ \text{eV}$ (Ge: $0.72\ \text{eV}$, Si: $1.1\ \text{eV}$, GaAs: $1.43\ \text{eV}$). At 0 K a semiconductor behaves like an insulator. As temperature rises, a small but significant fraction of electrons acquire enough thermal energy ($\sim k_BT$) to jump the gap into the conduction band, leaving behind vacancies (holes) in the valence band. Both electrons and holes then contribute to conduction, and — unlike a metal — resistivity of a semiconductor decreases as temperature increases, because more carriers become available faster than their mobility drops.

KEY DISTINCTION A metal has zero or negative effective gap (overlap); an insulator and a semiconductor have the same qualitative structure — the only difference is the magnitude of $E_g$. This is why raising the temperature enough could, in principle, make any insulator behave like a semiconductor, and why "semiconductor vs insulator" is a matter of degree, not kind.

Property	Conductor	Semiconductor	Insulator
Band gap $E_g$	0 (overlap)	0.2 – 3 eV	> 3 eV
Resistivity (Ω·m)	$10^{-8}$–$10^{-6}$	$10^{-5}$–$10^{6}$	$10^{10}$–$10^{18}$
Effect of ↑ temperature	resistivity ↑ (mobility ↓)	resistivity ↓ (carriers ↑ dominates)	almost no change
Charge carriers	free electrons only	electrons & holes	practically none
Example	Cu, Ag, Al	Si, Ge, GaAs	diamond, mica, glass

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Topic 2

Intrinsic Semiconductors

2.1 Definition

An intrinsic semiconductor is a chemically pure semiconductor crystal (e.g. pure Si or Ge) containing no significant impurity atoms. Every charge carrier in it originates purely from the semiconductor's own atoms being thermally excited across the band gap — nothing is "added" from outside.

2.2 Carrier generation: electron–hole pairs

Silicon and germanium are group-IV elements: each atom has 4 valence electrons, all engaged in covalent bonds with its 4 neighbours in the crystal lattice. At $T = 0\ \text{K}$ every bond is intact, the valence band is completely full, the conduction band completely empty — the crystal is a perfect insulator.

As temperature rises, thermal vibrations occasionally supply enough energy ($\geq E_g$) to break a covalent bond. The freed electron jumps into the conduction band, and the vacancy it leaves behind in the valence band is called a hole. Crucially, this process always creates carriers in pairs:

ELECTRON–HOLE PAIR GENERATION $$\text{Si–Si bond} \xrightarrow{\;E \geq E_g\;} e^-\ (\text{in CB}) \;+\; h^+\ (\text{in VB})$$

A hole is not a real particle — it is the absence of an electron in an otherwise full band — but it behaves in every practical sense like a positively charged carrier of magnitude $+e$, because neighbouring valence electrons can hop into the vacancy, making the vacancy itself appear to drift in the direction opposite to electron motion, i.e. along the field.

2.3 Carrier concentration relation

Because electrons and holes are always created in pairs in an intrinsic material,

INTRINSIC CONDITION $$n_e = n_h = n_i$$ where $n_e$ = electron concentration in CB, $n_h$ = hole concentration in VB, and $n_i$ is called the intrinsic carrier concentration — a strong function of temperature, roughly $n_i \propto T^{3/2}\, e^{-E_g/2k_BT}$.

2.4 Conduction mechanism

Under an applied electric field, two independent processes carry current simultaneously:

• Electrons in the conduction band drift opposite to the field (they are negative).
• Holes in the valence band drift along the field direction, as valence electrons successively fill vacancies, making the vacancy migrate the other way.

Total current = electron current + hole current, even though physically only electrons are moving.

2.5 Key properties of intrinsic semiconductors

• Behave as insulators at 0 K; conductivity rises rapidly (roughly exponentially) with temperature.
• Carrier concentration is low compared to a doped semiconductor at the same temperature, so intrinsic conductivity is small and not very useful directly in devices — this is why real devices use doped (extrinsic) material.
• The Fermi level sits, as shown in Topic 4, almost exactly at the middle of the band gap.

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Topic 3

Extrinsic Semiconductors

An extrinsic semiconductor is an intrinsic crystal that has been deliberately "doped" — a controlled, tiny amount ($\sim 1$ atom in $10^6$–$10^8$) of a chosen impurity is introduced to drastically change the number and type of majority carriers. Doping increases conductivity by several orders of magnitude and is what makes semiconductors useful as devices. There are two types, depending on the impurity's valence.

3.1 N-type semiconductor (donor doping)

Formed by doping Si or Ge (group IV, 4 valence electrons) with a pentavalent impurity — group V element with 5 valence electrons, e.g. Phosphorus (P), Arsenic (As), Antimony (Sb).

Four of the impurity atom's five valence electrons form covalent bonds with the four neighbouring Si atoms. The 5th electron is not part of any bond — it is very weakly bound (binding energy only $\sim 0.01$–$0.05\ \text{eV}$) and is easily donated to the conduction band even at room temperature. Hence the impurity is called a donor, and it creates a discrete donor energy level $E_D$ just below the conduction band edge $E_C$.

RESULT Free electrons become the majority carriers; holes (from the small intrinsic generation that still occurs) are the minority carriers. Since the donated electrons come from neutral donor atoms, the crystal as a whole remains electrically neutral — the fixed donor atom becomes a stationary positive ion ($N_D^+$) once it donates its electron, but there is no net charge.

3.2 P-type semiconductor (acceptor doping)

Formed by doping with a trivalent impurity — group III element with 3 valence electrons, e.g. Boron (B), Aluminium (Al), Gallium (Ga), Indium (In).

The impurity atom's 3 valence electrons complete only 3 of the 4 covalent bonds needed with its Si neighbours; the 4th bond is left with a vacancy — a hole — right from the start, with no need for thermal bond-breaking. This impurity readily accepts an electron from a neighbouring bond to complete its own 4th bond, so it is called an acceptor, and it creates a discrete acceptor level $E_A$ just above the valence band edge $E_V$.

RESULT Holes become the majority carriers; electrons are the minority carriers. The acceptor atom becomes a fixed negative ion ($N_A^-$) after accepting an electron; overall charge neutrality is still preserved.

Fig 3.1 — Donor level $E_D$ sits just below the conduction band in n-type material; acceptor level $E_A$ sits just above the valence band in p-type material.

	N-type	P-type
Dopant valence	Pentavalent (Group V)	Trivalent (Group III)
Examples	P, As, Sb	B, Al, Ga, In
Impurity role	Donor — gives up e⁻	Acceptor — takes e⁻
Extra level created	$E_D$, just below $E_C$	$E_A$, just above $E_V$
Majority carrier	Electrons	Holes
Minority carrier	Holes	Electrons
Fixed ion left behind	Positive ($N_D^+$)	Negative ($N_A^-$)

MASS-ACTION LAW Even after doping, the product of electron and hole concentrations stays fixed at a given temperature: $$n_e \, n_h = n_i^{\,2}$$ So increasing one carrier type (via doping) automatically suppresses the other — e.g. heavy donor doping raises $n_e$ far above $n_i$, which forces $n_h$ to fall below $n_i$.

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Topic 4

Fermi Level in an Intrinsic Semiconductor

4.1 What the Fermi level means

The Fermi level $E_F$ is the energy at which the Fermi–Dirac occupation probability is exactly $\tfrac12$. Occupation probability of any state of energy $E$ is given by the Fermi–Dirac distribution function:

FERMI–DIRAC FUNCTION $$f(E) = \dfrac{1}{1 + e^{(E - E_F)/k_BT}}$$

$E_F$ itself need not correspond to any real, allowed energy state (it can fall inside the forbidden gap, where no states exist) — it is simply a reference energy that fixes the whole distribution.

4.2 Derivation: position of $E_F$ for an intrinsic semiconductor

Let $E_C$ = bottom of conduction band, $E_V$ = top of valence band, $E_g = E_C - E_V$.

1

The electron concentration in the conduction band, obtained by integrating (density of states) × (occupation probability) over the band, works out to $$n_e = N_C\, e^{-(E_C - E_F)/k_BT}$$ where $N_C = 2\left(\dfrac{2\pi m_e^{*} k_BT}{h^2}\right)^{3/2}$ is the effective density of states at the conduction band edge, and $m_e^{*}$ is the electron effective mass.

2

Similarly, the hole concentration in the valence band is $$n_h = N_V\, e^{-(E_F - E_V)/k_BT}$$ where $N_V = 2\left(\dfrac{2\pi m_h^{*} k_BT}{h^2}\right)^{3/2}$, and $m_h^{*}$ is the hole effective mass. (Both expressions use the approximation $E_C - E_F \gg k_BT$ and $E_F - E_V \gg k_BT$, valid away from band edges — the "non-degenerate" limit.)

3

For an intrinsic semiconductor, every electron in the CB left behind exactly one hole in the VB, so we impose $$n_e = n_h$$

4

Substituting steps 1 and 2 into step 3: $$N_C\, e^{-(E_C-E_F)/k_BT} = N_V\, e^{-(E_F-E_V)/k_BT}$$

5

Take natural log of both sides: $$\ln N_C - \dfrac{E_C - E_F}{k_BT} = \ln N_V - \dfrac{E_F - E_V}{k_BT}$$

6

Rearranging and solving for $E_F$: $$E_F = \dfrac{E_C + E_V}{2} + \dfrac{k_BT}{2}\ln\!\left(\dfrac{N_V}{N_C}\right)$$

7

Since $m_e^{*}$ and $m_h^{*}$ are usually of comparable order, $N_C \approx N_V$, so the correction term $\dfrac{k_BT}{2}\ln(N_V/N_C) \approx 0$, giving the standard result: $$\boxed{E_F \approx \dfrac{E_C + E_V}{2}}$$

RESULT In an intrinsic semiconductor the Fermi level lies almost exactly at the middle of the forbidden gap — equidistant from $E_C$ and $E_V$ — reflecting that electrons and holes are equally likely (probability exactly $\tfrac12$ midway between two symmetric exponential tails). A slight shift towards the band with the smaller effective density of states appears only when $m_e^{*} \neq m_h^{*}$.

Fig 4.1 — Fermi level of an intrinsic semiconductor sits at the centre of the band gap.

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Topic 5

Semiconductor Conductivity — Derivation

5.1 Physical picture

Unlike a metal (only free electrons), a semiconductor carries current via two independent species — electrons in the CB and holes in the VB — both driven by the same applied field $E$, but in effect moving in opposite real directions while contributing current in the same direction (conventional current is along $E$ for both).

5.2 Drift velocity and mobility

Under a field $E$, carriers accelerate but repeatedly scatter off lattice vibrations/impurities, reaching a steady average drift velocity proportional to the field:

DRIFT VELOCITY $$v_{de} = \mu_e E \qquad v_{dh} = \mu_h E$$ $\mu_e$, $\mu_h$ = electron and hole mobility (drift velocity per unit field), units $\text{m}^2\text{V}^{-1}\text{s}^{-1}$.

5.3 Step-by-step derivation of conductivity

1

Consider a semiconductor bar of length $L$, cross-section $A$, carrying electron density $n_e$ and hole density $n_h$. Apply field $E$ along its length.

2

Current due to electrons: each electron carries charge $e$ and drifts with $v_{de}$, so the electron current density is $$J_e = n_e\, e\, v_{de} = n_e\, e\, \mu_e E$$

3

Similarly, current due to holes: $$J_h = n_h\, e\, v_{dh} = n_h\, e\, \mu_h E$$

4

Both currents flow in the same direction (along $E$), so the total current density adds: $$J = J_e + J_h = e\,(n_e \mu_e + n_h \mu_h)\,E$$

5

By definition, Ohm's law in local form is $J = \sigma E$. Comparing with step 4: $$\boxed{\sigma = e\,(n_e \mu_e + n_h \mu_h)}$$ This is the general conductivity of any semiconductor.

5.4 Special cases

INTRINSIC CASE Since $n_e = n_h = n_i$: $$\sigma_i = n_i\, e\, (\mu_e + \mu_h)$$

HEAVILY-DOPED N-TYPE $n_e \gg n_h$, so the hole term is negligible: $$\sigma_n \approx n_e\, e\, \mu_e$$

An analogous simplification $\sigma_p \approx n_h\, e\, \mu_h$ holds for heavily-doped p-type material.

5.5 Temperature dependence

$n_i$ grows exponentially with $T$ (since carriers must be thermally excited across $E_g$), while mobility falls only slowly (as a power law) due to increased lattice scattering. The exponential carrier growth always wins, so:

$$\sigma \propto e^{-E_g/2k_BT} \quad\Rightarrow\quad \text{conductivity of a semiconductor increases with temperature}$$

This is the opposite of a metal, where conductivity falls with temperature because carrier density is already fixed and only mobility decreases.

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Topic 6

P–N Junction Diode

6.1 Formation of the junction

A p–n junction is formed when a p-type and an n-type block of the same semiconductor crystal are joined (in practice, grown as a single crystal with doping changed partway through, not glued together). At the instant of contact:

• The n-side has a high concentration of free electrons; the p-side has a high concentration of free holes.
• Because of this concentration gradient, electrons diffuse from n → p, and holes diffuse from p → n, across the junction.

6.2 Depletion region and barrier potential

1

Electrons crossing into the p-region recombine with holes there; holes crossing into the n-region recombine with electrons. This leaves the n-side atoms near the junction as fixed positive ions (having lost their donated electron) and the p-side atoms near the junction as fixed negative ions.

2

A narrow region around the junction becomes stripped of mobile carriers — the depletion region (width $\sim 0.1$–$1\ \mu\text{m}$) — containing only these immobile ions.

3

The positive ions (n-side) and negative ions (p-side) set up an internal electric field pointing from n → p across the depletion region, which opposes further diffusion of majority carriers. Equilibrium is reached when diffusion tendency is exactly balanced by this field.

4

This field corresponds to a built-in potential difference called the barrier potential $V_0$ (typically $0.7\ \text{V}$ for Si, $0.3\ \text{V}$ for Ge at room temperature) which any external carrier must overcome to cross the junction.

Fig 6.1 — Depletion region at a p–n junction, flanked by fixed acceptor (−) and donor (+) ions.

6.3 Forward biasing

External battery's positive terminal → p-side, negative terminal → n-side. The applied field opposes the internal field, so the depletion width and barrier shrink. Once the applied voltage exceeds $V_0$, majority carriers flow freely across the junction and current rises rapidly (exponentially) with voltage — low resistance path.

6.4 Reverse biasing

Battery reversed: positive → n-side, negative → p-side. The applied field adds to the internal field, so the depletion region widens and the barrier increases — majority carrier flow is blocked almost entirely. Only a very small reverse saturation current $I_0$ flows, carried by thermally-generated minority carriers, until the voltage becomes large enough to trigger breakdown (Topic 7).

6.5 V–I characteristic and diode equation

DIODE EQUATION $$I = I_0\left(e^{\,V/\eta V_T} - 1\right)$$ $I_0$ = reverse saturation current, $V_T = k_BT/e \approx 26\ \text{mV}$ at room temperature (thermal voltage), $\eta$ = 1 for Ge, ≈2 for Si (empirical ideality factor).

• Forward region ($V>0$): current rises exponentially once $V$ exceeds the cut-in / knee voltage ($\approx 0.3\ \text{V}$ Ge, $\approx 0.7\ \text{V}$ Si).
• Reverse region ($V<0$): current stays at a nearly constant, tiny $-I_0$ (μA or nA) until breakdown voltage is reached.

Because current flows easily in one direction and is blocked in the other, the p–n junction acts as a rectifier — its principal application (half-wave and full-wave rectifiers, clippers, clampers).

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Topic 7

Zener Diode

7.1 What makes it different

A Zener diode is a p–n junction diode specially designed (heavily doped, giving a very thin depletion region) to operate safely and repeatably in the reverse breakdown region — a regime an ordinary diode is never intended to survive. Its circuit symbol is a diode with a bent line at the cathode.

7.2 Breakdown mechanisms

ZENER BREAKDOWN Dominant for heavily-doped diodes with $V_Z < 5\text{–}6\ \text{V}$. The depletion region is so thin that even a moderate reverse voltage creates an extremely high field ($\sim 10^7\ \text{V/m}$), strong enough to directly rip electrons out of covalent bonds (a form of quantum tunnelling) — no collision needed.

AVALANCHE BREAKDOWN Dominant for lightly-doped diodes with $V_Z > 6\ \text{V}$. Thermally generated minority carriers are accelerated by the field until they gain enough energy to knock other electrons out of bonds on collision — each such collision multiplies carriers, cascading into a large current.

Commercially, both mechanisms are lumped under the name "Zener diode," and the manufacturer simply specifies the Zener breakdown voltage $V_Z$ at which conduction sets in.

7.3 Key characteristic: voltage regulation

Once reverse voltage reaches $V_Z$, the current can increase over a wide range while the voltage across the diode stays essentially constant at $V_Z$ (as long as a series resistor limits current below the diode's power rating, and current stays below its maximum $I_{Z(max)}$ and above a minimum $I_{Z(min)}$ needed to sustain breakdown). This near-vertical V–I curve in reverse breakdown is what makes it useful.

PRIMARY APPLICATION — VOLTAGE REGULATOR Connected in reverse bias with a series resistor $R_S$ across an unregulated supply, a Zener diode holds the output voltage fixed at $V_Z$ even if the input voltage fluctuates or the load current changes — because any excess current simply diverts through the Zener while $V_Z$ stays essentially unchanged. Also used in over-voltage protection clamps, and as a fixed reference voltage in comparators.

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Topic 8

Light Emitting Diode (LED)

8.1 Principle: electroluminescence

An LED is a p–n junction diode, operated in forward bias, made from a direct-band-gap compound semiconductor (e.g. GaAs, GaP, GaAsP, GaN — not Si or Ge, which are indirect-gap and emit almost no light). In forward bias, electrons from the n-side and holes from the p-side are injected across the junction in large numbers and recombine there.

RADIATIVE RECOMBINATION $$e^-\ (\text{CB}) + h^+\ (\text{VB}) \;\longrightarrow\; \text{photon of energy } h\nu \approx E_g$$ In a direct-gap material, the electron can drop straight from CB to VB releasing its energy as a photon (momentum is automatically conserved, since CB minimum and VB maximum occur at the same crystal momentum). In an indirect-gap material like Si, the same transition needs a phonon as well to conserve momentum, making photon emission far less probable — energy is lost as heat instead. This is why LEDs are never made of ordinary silicon.

8.2 Colour of emitted light

The emitted photon energy is approximately equal to the band gap, so wavelength is fixed by material:

$$\lambda \approx \dfrac{hc}{E_g}$$

Material	$E_g$ (eV)	Emission colour
GaAs	≈1.4	Infrared
GaAsP	≈1.9	Red
GaP	≈2.2	Green / yellow
GaN	≈3.4	Blue / UV

8.3 Why LEDs differ from ordinary diodes

• Higher forward voltage drop (1.5–3.5 V, depending on colour/gap) than a Si/Ge signal diode.
• Light output intensity is roughly proportional to forward current (up to a rated maximum) — hence LED brightness is current-controlled, and a series resistor is essential to limit current.
• Cannot usefully operate in reverse bias — very low reverse breakdown rating; reverse bias just blocks current with no light.

8.4 Advantages / applications

Low operating voltage and power, very fast switching (µs–ns), long lifetime, no filament to burn out, robust to vibration. Used in indicator lamps, seven-segment displays, optical fibre communication sources, remote controls (IR LEDs), backlighting, and general illumination.

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Topic 9

Hall Effect

9.1 Statement

When a current-carrying conductor (or semiconductor) is placed in a magnetic field perpendicular to the current, a small transverse electric field — and hence a transverse voltage — is developed across the sample, perpendicular to both the current and the magnetic field. This is the Hall effect, discovered by Edwin Hall in 1879, and the transverse voltage is the Hall voltage $V_H$.

9.2 Physical origin

Consider a rectangular slab carrying current $I$ along $x$, with magnetic field $B$ applied along $z$ (perpendicular to the slab's flat face). Charge carriers moving with drift velocity $v_d$ along $x$ experience a Lorentz force:

$$\vec{F} = q\,\vec{v_d} \times \vec{B}$$

This force pushes carriers sideways, towards one face of the slab (along $y$), where they pile up. This accumulation of charge builds a transverse electric field $E_H$ (the Hall field) which grows until the electric force on further carriers exactly balances the magnetic force — a steady state.

9.3 Derivation of the Hall voltage and Hall coefficient

1

At equilibrium, the electric force balances the magnetic force on a carrier of charge $q$: $$qE_H = q\,v_d B \quad\Rightarrow\quad E_H = v_d B$$

2

Current density in terms of carrier concentration $n$ and drift velocity: $$J = nq\,v_d \quad\Rightarrow\quad v_d = \dfrac{J}{nq}$$

3

Substitute $v_d$ from step 2 into step 1: $$E_H = \dfrac{JB}{nq}$$

4

Define the Hall coefficient $R_H$ as the constant of proportionality between $E_H$ and $JB$: $$E_H = R_H\, J B \quad\Rightarrow\quad \boxed{R_H = \dfrac{1}{nq}}$$ (For electrons $q=-e$, giving $R_H$ negative; for holes $q=+e$, giving $R_H$ positive — this sign is exactly how the Hall effect experimentally distinguishes n-type from p-type material.)

5

If the slab has thickness $t$ (along field direction) and the current-carrying face has width $w$ perpendicular to both $I$ and $B$, then $J = I/(wt)$ and the Hall voltage is $V_H = E_H \cdot w$: $$V_H = E_H w = R_H\, J B\, w = R_H \dfrac{I}{wt} B w$$

6

Simplifying: $$\boxed{V_H = \dfrac{R_H\, I B}{t} = \dfrac{IB}{n q t}}$$ This is the standard working formula: $V_H$ is directly proportional to both the current $I$ and the field $B$, and inversely proportional to the sample's thickness and carrier concentration.

Fig 9.1 — Current $I$ flows along the slab, field $B$ is perpendicular to it; the Hall voltage develops across the width $w$.

9.4 Hall mobility

Combining the Hall coefficient with the ordinary conductivity $\sigma = nq\mu$ gives a purely electrical way to measure carrier mobility, without knowing $n$ separately:

$$\mu = \sigma\, |R_H| = \sigma \cdot \dfrac{1}{n|q|}$$

9.5 Importance / applications

• Determining carrier type — sign of $V_H$ (or $R_H$) tells whether a semiconductor is n-type or p-type.
• Measuring carrier concentration $n = 1/(|R_H|\,q)$ and, combined with conductivity, carrier mobility $\mu$.
• Hall sensors — measuring magnetic field strength, contactless current sensing, position/proximity sensors, brushless motor commutation.

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Topic 10

Chapter Summary — Quick Recall

Formula sheet

• Intrinsic condition: $n_e = n_h = n_i$, and mass-action law $n_e n_h = n_i^2$
• Fermi level (intrinsic): $E_F \approx (E_C+E_V)/2$ — mid-gap
• Conductivity: $\sigma = e(n_e\mu_e + n_h\mu_h)$; intrinsic case $\sigma_i = n_i e(\mu_e+\mu_h)$
• Diode equation: $I = I_0\left(e^{V/\eta V_T}-1\right)$, $V_T \approx 26\ \text{mV}$
• Zener: operates in reverse breakdown at fixed $V_Z$ — tunnelling (<6 V) or avalanche (>6 V)
• LED: forward-biased direct-gap junction; $\lambda \approx hc/E_g$
• Hall coefficient: $R_H = 1/(nq)$; Hall voltage: $V_H = IB/(nqt)$

One-line takeaways

• Conductors, semiconductors and insulators differ only in the size of the band gap $E_g$, not in kind.
• Doping type is decided by the valence of the impurity relative to the host: pentavalent → n-type (donor), trivalent → p-type (acceptor).
• The p–n junction's depletion region and barrier potential are the mechanism behind rectification, Zener regulation, and LED emission alike — all three devices are variations on the same junction.
• The Hall effect is the experimental tool that tells you the sign, density, and mobility of a semiconductor's carriers — turning an abstract band-theory prediction into a measured number.