Semiconductor Basics Notes FE Sem 2 Semiconductor Physics | MU

Semiconductors such as silicon and germanium sit between metals and insulators: their conductivity can be changed a million-fold by adding a few impurity atoms, by heating them, or by shining light on them. This control is what makes diodes, transistors and integrated circuits possible. This first module of the Semiconductor Physics elective builds the foundation: types of semiconductors, carrier concentration and the Fermi level in intrinsic semiconductors and how it varies with temperature, extrinsic semiconductors and how their Fermi level varies with temperature and impurity concentration, the equation of conductivity, and the Hall effect with calculation of the Hall voltage. It maps to course outcome CO1 (use this knowledge to calculate the Hall voltage). The prerequisites are band theory of solids, the Fermi-Dirac distribution and density of states.

Summary

Types of semiconductors: On the basis of band theory, a semiconductor is a solid with a completely filled valence band and an empty conduction band at 0 K, separated by a small forbidden gap \(E_g\) (about 1.12 eV for Si, 0.67 eV for Ge, 1.42 eV for GaAs at 300 K). At 0 K it is an insulator; at room temperature some electrons gain enough thermal energy to cross the gap, giving a moderate conductivity (roughly \(10^{-6}\) to \(10^{4}\) S/m). By composition, semiconductors are elemental (Si, Ge, from group IV) or compound (GaAs, InP, GaN from groups III-V; CdS, ZnSe from II-VI; SiC). By purity they are intrinsic (pure) or extrinsic (doped), and extrinsic semiconductors are n-type or p-type. By band structure they are direct gap (GaAs) or indirect gap (Si, Ge).

Intrinsic semiconductor: A pure semiconductor in which charge carriers are produced only by thermal excitation is intrinsic. Each Si atom forms four covalent bonds with its neighbours. At room temperature some bonds break; an electron jumps from the valence band to the conduction band and leaves behind a vacancy called a hole, which behaves like a positive charge and moves when neighbouring valence electrons fill it. Electrons and holes are created in pairs, so \(n = p = n_i\), the intrinsic carrier concentration. Both carriers contribute to current.

Carrier concentration in intrinsic semiconductors: The number of electrons in the conduction band is found by multiplying the density of states \(g_c(E)\) by the Fermi-Dirac probability \(f(E)\) and integrating over the band. Since \(E - E_F \gg kT\) for the conduction band, \(f(E) \approx e^{-(E - E_F)/kT}\) (Maxwell-Boltzmann approximation), giving \(n = N_c\,e^{-(E_c - E_F)/kT}\), where \(N_c = 2\left(\dfrac{2\pi m_e^*kT}{h^2}\right)^{3/2}\) is the effective density of states in the conduction band. Similarly the hole concentration in the valence band is \(p = N_v\,e^{-(E_F - E_v)/kT}\), with \(N_v = 2\left(\dfrac{2\pi m_h^*kT}{h^2}\right)^{3/2}\). Multiplying, \(np = N_cN_v\,e^{-E_g/kT}\), which does not depend on \(E_F\); for an intrinsic semiconductor \(n = p = n_i\), so \(n_i = \sqrt{N_cN_v}\,e^{-E_g/2kT}\), and since \(N_c, N_v \propto T^{3/2}\), \(n_i \propto T^{3/2}e^{-E_g/2kT}\). At 300 K, \(n_i\) is about \(1.5\times10^{16}\) m⁻³ (often quoted as \(1.0\times10^{16}\) in newer data) for Si and about \(2.4\times10^{19}\) m⁻³ for Ge. The relation \(np = n_i^2\) holds for any semiconductor in equilibrium, doped or not (law of mass action).

Fermi level of intrinsic semiconductors: Equating n and p: \(N_ce^{-(E_c - E_F)/kT} = N_ve^{-(E_F - E_v)/kT}\). Taking logarithms, \(E_F = \dfrac{E_c + E_v}{2} + \dfrac{kT}{2}\ln\dfrac{N_v}{N_c} = \dfrac{E_c + E_v}{2} + \dfrac{3kT}{4}\ln\dfrac{m_h^*}{m_e^*}\). At T = 0 K, or if \(m_h^* = m_e^*\), the Fermi level lies exactly at the middle of the forbidden gap, \(E_F = (E_c + E_v)/2\).

Variation of the intrinsic Fermi level with temperature: The second term \((3kT/4)\ln(m_h^*/m_e^*)\) grows linearly with T. If \(m_h^* > m_e^*\), the Fermi level rises slightly above mid-gap as temperature increases; if \(m_h^* < m_e^*\), it moves slightly down. Since kT is only 0.026 eV at room temperature and the logarithm is of order 1, the shift is a few hundredths of an eV, small compared with \(E_g\), so for practical purposes the intrinsic Fermi level is taken at the centre of the gap at all ordinary temperatures.

Extrinsic semiconductors: Adding a tiny amount of suitable impurity (doping, typically 1 atom per \(10^6\) to \(10^8\) host atoms) increases conductivity enormously. n-type: a pentavalent impurity (P, As, Sb) uses four of its five valence electrons for bonding; the fifth is loosely bound (ionisation energy about 0.045 eV in Si, 0.01 eV in Ge) and is freed easily into the conduction band. These impurities are donors; their energy level \(E_d\) lies just below \(E_c\). Electrons are majority carriers and holes minority carriers. p-type: a trivalent impurity (B, Al, Ga, In) has only three valence electrons, so one bond is incomplete; it readily accepts an electron from the valence band, creating a hole. These impurities are acceptors; their level \(E_a\) lies just above \(E_v\). Holes are majority carriers. The material stays electrically neutral: \(n + N_A^- = p + N_D^+\).

Fermi level of extrinsic semiconductors: In an n-type semiconductor at room temperature all donors are ionised, so \(n \approx N_D\). From \(n = N_ce^{-(E_c - E_F)/kT}\): \(E_F = E_c - kT\ln(N_c/N_D)\). Equivalently \(E_F - E_i = kT\ln(N_D/n_i)\), where \(E_i\) is the intrinsic level: the Fermi level lies above mid-gap, closer to the conduction band. At very low temperature (near 0 K), when donors are not yet ionised, \(E_F = \dfrac{E_c + E_d}{2} + \dfrac{kT}{2}\ln\dfrac{N_D}{N_c}\), which at 0 K is midway between \(E_d\) and \(E_c\). For p-type, \(p \approx N_A\), \(E_F = E_v + kT\ln(N_v/N_A)\) and \(E_i - E_F = kT\ln(N_A/n_i)\): the Fermi level lies below mid-gap, near the valence band; at 0 K it is midway between \(E_v\) and \(E_a\).

Variation of extrinsic Fermi level with temperature and impurity concentration: With temperature (n-type): at 0 K, \(E_F\) is between \(E_d\) and \(E_c\). As T rises, donors ionise and \(E_F\) moves down below \(E_d\) (extrinsic or saturation range, n ≈ N_D); at still higher temperatures intrinsic generation across the gap becomes comparable to \(N_D\), \(n_i\) grows exponentially, and \(E_F\) approaches the intrinsic level \(E_i\); the material then behaves as intrinsic. For p-type, \(E_F\) starts between \(E_v\) and \(E_a\) and rises towards \(E_i\) as T increases. With impurity concentration: increasing \(N_D\) moves the n-type Fermi level upwards towards (and at very heavy, degenerate doping, into) the conduction band; increasing \(N_A\) moves the p-type Fermi level down towards (or into) the valence band. Since \(E_F - E_i = kT\ln(N_D/n_i)\), every tenfold increase in doping shifts \(E_F\) by \(kT\ln10 \approx 0.06\) eV at 300 K.

Equation of conductivity with current flow: When an electric field E is applied, electrons drift opposite to E with drift velocity \(v_e = \mu_eE\), and holes drift along E with \(v_h = \mu_hE\), where μ is mobility. Both currents are in the direction of E. Electron current density \(J_e = ne\mu_eE\), hole current density \(J_h = pe\mu_hE\), total \(J = (ne\mu_e + pe\mu_h)E = \sigma E\). Hence the conductivity \(\sigma = e(n\mu_e + p\mu_h)\). Intrinsic: \(\sigma_i = n_ie(\mu_e + \mu_h)\). n-type: \(\sigma_n \approx N_De\mu_e\). p-type: \(\sigma_p \approx N_Ae\mu_h\). Because \(n_i\) increases exponentially with temperature, the conductivity of an intrinsic semiconductor rises rapidly with temperature (negative temperature coefficient of resistance), unlike metals. In addition to drift, carriers also move from high to low concentration by diffusion, giving diffusion current \(J = eD_n\,dn/dx\) for electrons and \(-eD_p\,dp/dx\) for holes; D and μ are linked by the Einstein relation \(D/\mu = kT/e\).

Hall effect: When a current-carrying conductor or semiconductor is placed in a magnetic field perpendicular to the current, a voltage (the Hall voltage) develops across the specimen in the direction perpendicular to both the current and the field. It was discovered by Edwin Hall in 1879. Explanation: take a rectangular slab with current I along x, magnetic field B along z, width w along y and thickness t along z. The moving carriers feel the Lorentz force \(F = q(\mathbf{v}\times\mathbf{B})\) along y and pile up on one face, leaving opposite charge on the other face. This builds a transverse electric field \(E_H\), the Hall field, which pushes carriers the other way. In equilibrium \(eE_H = ev_dB\), so \(E_H = v_dB\). With \(I = nev_d\,wt\), \(v_d = I/(newt)\), and the Hall voltage \(V_H = E_Hw = \dfrac{BI}{net}\). The Hall coefficient is \(R_H = \dfrac{E_H}{JB} = \dfrac{1}{ne}\) for holes (positive) and \(-\dfrac{1}{ne}\) for electrons (negative), so \(V_H = \dfrac{R_HBI}{t}\). Note that t is the thickness along the magnetic field. Importance: the sign of \(V_H\) tells whether the sample is n-type or p-type (for both carriers deflected to the same face, electrons charge it negative and holes positive); the magnitude gives the carrier concentration \(n = 1/(|R_H|e)\); combined with conductivity it gives the mobility \(\mu = \sigma|R_H|\). Hall sensors measure magnetic fields and are used as position, speed and current sensors (for example in brushless motors, wheel speed sensors and phone compasses).

Mind map

Derivation of the Hall voltage and Hall coefficient

Important definitions

Formulas

Key points to remember

• Eg: Si 1.12 eV, Ge 0.67 eV, GaAs 1.42 eV at 300 K.
• ni rises exponentially with T and is larger for smaller Eg (Ge has a much larger ni than Si).
• np = ni² for any non-degenerate semiconductor in equilibrium.
• Intrinsic EF is at mid-gap; it shifts by (3kT/4) ln(mh*/me*) with temperature, which is small.
• n-type EF lies near Ec; p-type EF lies near Ev.
• Rising temperature drives extrinsic EF towards mid-gap; rising doping drives it towards the band edge.
• Electron and hole currents add, even though they drift in opposite directions.
• Electron mobility is greater than hole mobility (Si: about 0.135 and 0.048 m²/V s).
• Hall voltage is inversely proportional to n, so it is large in semiconductors and tiny in metals.
• Sign of RH gives carrier type; magnitude gives concentration; σRH gives mobility.
• In VH = BI/(net), t is the dimension parallel to B.

Solved examples

Example 1. Intrinsic germanium at 300 K has ni = 2.4 × 10^19 m^-3, μe = 0.39 and μh = 0.19 m²/V s. Find its conductivity and resistivity.

\(\sigma = n_ie(\mu_e + \mu_h) = 2.4\times10^{19}\times1.6\times10^{-19}\times0.58 = 3.84\times0.58 = 2.23\) S/m. \(\rho = 1/2.227 = 0.449\) Ω m.

Example 2. Silicon is doped with 10^22 phosphorus atoms per m³. Find the conductivity and the hole concentration (ni = 1.5 × 10^16 m^-3, μe = 0.135 m²/V s).

All donors ionised: \(n \approx N_D = 10^{22}\) m⁻³. \(\sigma \approx N_De\mu_e = 10^{22}\times1.6\times10^{-19}\times0.135 = 216\) S/m. Holes: \(p = n_i^2/n = (1.5\times10^{16})^2/10^{22} = 2.25\times10^{10}\) m⁻³, negligible compared with n. (With the newer value \(n_i = 1.0\times10^{16}\) m⁻³, \(p = 1.0\times10^{10}\) m⁻³.)

Example 3. For the sample of Example 2, find the position of the Fermi level relative to the intrinsic level and to the conduction band at 300 K (Nc = 2.8 × 10^25 m^-3).

\(kT = 8.617\times10^{-5}\times300 = 0.02585\) eV. \(E_F - E_i = kT\ln(N_D/n_i) = 0.02585\ln(6.67\times10^{5}) = 0.02585\times13.41 = 0.347\) eV above mid-gap. \(E_c - E_F = kT\ln(N_c/N_D) = 0.02585\ln(2800) = 0.02585\times7.94 = 0.205\) eV below the conduction band edge. (Check: 0.347 + 0.205 ≈ 0.55 eV, about half the band gap. With \(n_i = 1.0\times10^{16}\) m⁻³, \(E_F - E_i = 0.02585\ln(10^{6}) = 0.357\) eV.)

Example 4. In a semiconductor the effective mass of holes is twice that of electrons. Find the shift of the intrinsic Fermi level from mid-gap at 300 K.

\(\Delta E = \dfrac{3kT}{4}\ln\dfrac{m_h^*}{m_e^*} = 0.75\times0.02585\times\ln2 = 0.75\times0.02585\times0.693 = 0.0134\) eV above mid-gap. This is only about 1 percent of a 1.1 eV gap, so the Fermi level is practically at the centre.

Example 5. An n-type sample 5 mm wide and 1 mm thick carries 10 mA along its length in a magnetic field of 0.5 T applied along its thickness. The electron concentration is 10^21 m^-3. Find the Hall coefficient, Hall voltage and drift velocity.

\(R_H = -1/(ne) = -1/(10^{21}\times1.6\times10^{-19}) = -6.25\times10^{-3}\) m³/C. \(V_H = \dfrac{BI}{net} = \dfrac{0.5\times0.01}{10^{21}\times1.6\times10^{-19}\times10^{-3}} = \dfrac{0.005}{0.16} = 0.03125\) V \(= 31.25\) mV. \(v_d = \dfrac{I}{newt} = \dfrac{0.01}{160\times5\times10^{-3}\times10^{-3}} = 12.5\) m/s. Check: \(V_H = v_dBw = 12.5\times0.5\times5\times10^{-3} = 31.25\) mV.

Example 6. A semiconductor has Hall coefficient 3.66 × 10^-4 m³/C and conductivity 112 S/m. Find the carrier concentration and mobility.

\(n = \dfrac{1}{R_He} = \dfrac{1}{3.66\times10^{-4}\times1.6\times10^{-19}} = 1.71\times10^{22}\) m⁻³. \(\mu = \sigma R_H = 112\times3.66\times10^{-4} = 0.041\) m²/V s. Since R_H is positive, the carriers are holes (p-type).

Important questions and answers

Q1. Derive an expression for the carrier concentration in an intrinsic semiconductor. (5 marks)

Electrons in the conduction band: \(n = \int_{E_c}^{\infty}g_c(E)f(E)\,dE\), with \(g_c(E) = \dfrac{4\pi}{h^3}(2m_e^*)^{3/2}(E - E_c)^{1/2}\) and \(f(E) \approx e^{-(E - E_F)/kT}\) since \(E - E_F \gg kT\). Substituting \(x = (E - E_c)/kT\) and using \(\int_0^\infty x^{1/2}e^{-x}dx = \sqrt{\pi}/2\) gives \(n = 2\left(\dfrac{2\pi m_e^*kT}{h^2}\right)^{3/2}e^{-(E_c - E_F)/kT} = N_ce^{-(E_c - E_F)/kT}\). Similarly, holes in the valence band: \(p = N_ve^{-(E_F - E_v)/kT}\). For an intrinsic semiconductor \(n = p = n_i\), so \(n_i^2 = np = N_cN_ve^{-(E_c - E_v)/kT}\), hence \(n_i = \sqrt{N_cN_v}\,e^{-E_g/2kT} = 2\left(\dfrac{2\pi kT}{h^2}\right)^{3/2}(m_e^*m_h^*)^{3/4}e^{-E_g/2kT}\). Thus \(n_i\) depends strongly on temperature and band gap.

Q2. Show that the Fermi level of an intrinsic semiconductor lies at the middle of the band gap. Discuss its variation with temperature. (5 marks)

Set \(n = p\): \(N_ce^{-(E_c - E_F)/kT} = N_ve^{-(E_F - E_v)/kT}\). Take logs: \(-(E_c - E_F)/kT + \ln N_c = -(E_F - E_v)/kT + \ln N_v\), so \(2E_F = E_c + E_v + kT\ln(N_v/N_c)\) and \(E_F = \dfrac{E_c + E_v}{2} + \dfrac{3kT}{4}\ln\dfrac{m_h^*}{m_e^*}\), using \(N_v/N_c = (m_h^*/m_e^*)^{3/2}\). If \(m_h^* = m_e^*\), \(E_F = (E_c + E_v)/2\), exactly at mid-gap at all temperatures. If \(m_h^* > m_e^*\), \(E_F\) rises linearly with T above mid-gap; if \(m_h^* < m_e^*\), it falls. At 0 K the second term vanishes. The shift is only of order kT, so \(E_F\) is effectively at mid-gap. Sketch: E_F versus T as a straight line starting at mid-gap with small positive or negative slope.

Q3. Explain with diagrams how the Fermi level of n-type and p-type semiconductors varies with temperature and impurity concentration. (6 marks)

n-type: band diagram with Ec, Ed (dashed, just below Ec), Ei (dotted, mid-gap), Ev and EF. At 0 K, \(E_F = (E_c + E_d)/2\). As T increases, donors ionise; \(E_F = E_c - kT\ln(N_c/N_D)\) moves down; at high T, intrinsic carriers dominate and \(E_F \rightarrow E_i\). Higher \(N_D\) keeps \(E_F\) closer to \(E_c\) at any temperature and delays the approach to \(E_i\). p-type: Ea just above Ev. At 0 K, \(E_F = (E_v + E_a)/2\). As T increases \(E_F\) rises towards \(E_i\). Higher \(N_A\) keeps \(E_F\) closer to \(E_v\). Plot: E_F versus T for three doping levels; all curves start near the band edge and converge to E_i at high temperature, the higher-doped curves converging later. Physical reason: at high temperature intrinsic carriers swamp the impurity carriers, so every doped semiconductor becomes intrinsic; this sets the maximum operating temperature of devices.

Q4. Derive the expression for the conductivity of a semiconductor. (3 marks)

In a field E, electrons drift with \(v_e = \mu_eE\) (opposite to E) and holes with \(v_h = \mu_hE\) (along E). Current densities: \(J_e = nev_e = ne\mu_eE\), \(J_h = pev_h = pe\mu_hE\); both are along E because electrons carry negative charge in the opposite direction. Total \(J = e(n\mu_e + p\mu_h)E\). Comparing with \(J = \sigma E\), \(\sigma = e(n\mu_e + p\mu_h)\). Special cases: intrinsic \(\sigma_i = n_ie(\mu_e + \mu_h)\); n-type \(\sigma \approx N_De\mu_e\); p-type \(\sigma \approx N_Ae\mu_h\).

Q5. What is the Hall effect? Derive expressions for the Hall voltage and Hall coefficient. (6 marks)

Definition as above. Derivation: slab of width w (along y) and thickness t (along z), current I along x, field B along z. Carriers of charge q and drift velocity v_d feel \(F = qv_dB\) along y and accumulate on one face, creating Hall field \(E_H\). Equilibrium: \(qE_H = qv_dB\), \(E_H = v_dB\). Since \(J = nqv_d\), \(E_H = JB/(nq)\). Define \(R_H = E_H/(JB) = 1/(nq)\): \(+1/(ne)\) for holes and \(-1/(ne)\) for electrons. Hall voltage \(V_H = E_Hw = \dfrac{BI}{nqt}\), so \(|V_H| = \dfrac{BI}{net}\). Diagram: slab with axes, I, B, accumulated charges on the faces and V_H measured across the width.

Q6. State the applications of the Hall effect. (3 marks)

(i) Determining whether a semiconductor is n-type or p-type from the sign of V_H or R_H. (ii) Finding carrier concentration \(n = 1/(|R_H|e)\). (iii) Finding mobility \(\mu = \sigma|R_H|\). (iv) Measuring magnetic field with a calibrated Hall probe (gaussmeter). (v) Hall-effect sensors for position, proximity and speed (brushless DC motors, ABS wheel speed, keyboard switches), and contactless current sensors. (vi) Magnetic compasses in phones.

Q7. Why is the Hall voltage larger in semiconductors than in metals? (2 marks)

\(V_H = BI/(net)\) is inversely proportional to carrier concentration. Metals have about \(10^{28}\) to \(10^{29}\) electrons per m³, while semiconductors have perhaps \(10^{19}\) to \(10^{23}\), so for the same current, field and thickness the Hall voltage is many orders of magnitude larger in a semiconductor; this is why Hall sensors use semiconductors such as InSb, GaAs or Si.

Q8. Distinguish between n-type and p-type semiconductors. (3 marks)

n-type: doped with pentavalent donors (P, As, Sb); majority carriers electrons; donor level just below Ec; EF near Ec; R_H negative; \(\sigma \approx N_De\mu_e\). p-type: doped with trivalent acceptors (B, Al, Ga, In); majority carriers holes; acceptor level just above Ev; EF near Ev; R_H positive; \(\sigma \approx N_Ae\mu_h\). Both are electrically neutral overall.

Exam tips

• Draw band diagrams for intrinsic, n-type and p-type side by side with Ec, Ev, Ed or Ea, Ei and EF clearly labelled.
• In Hall numericals identify t as the side parallel to B; using the width instead is the commonest mistake.
• Keep all concentrations in m⁻³ and mobilities in m²/V s; convert from cm units (×10⁶ for concentration, ×10⁻⁴ for mobility).
• Use kT = 0.0259 eV (0.02585 eV) at 300 K; kT ln10 ≈ 0.06 eV is a handy shortcut.
• State the sign convention: RH negative for n-type, positive for p-type.
• Write the law of mass action whenever minority carrier concentration is asked.

FAQs

Is a hole a real particle?

No. A hole is the absence of an electron in the valence band. The collective motion of valence electrons filling vacancies is equivalent to a positive charge moving the other way, so it is treated as a positive carrier with its own effective mass and mobility.

Why does doping not make a semiconductor charged?

Each donor atom that releases an electron becomes a fixed positive ion, so the total charge remains zero. Only the mobile carriers are unequal in number.

Why do devices fail at high temperature?

At high temperature ni becomes comparable to the doping concentration, the Fermi level moves to mid-gap and the n or p character is lost, so junctions stop working properly. Silicon tolerates higher temperatures than germanium because its band gap is larger.

Why is the Hall voltage zero in an intrinsic sample with equal mobilities?

Electrons and holes are both deflected to the same face; with equal concentrations and equal mobilities their charges cancel, so no net Hall voltage builds up. Real intrinsic samples give a small negative V_H because electrons are more mobile.

These notes follow the University of Mumbai first-year engineering syllabus (NEP 2020, Semiconductor Physics, Semester II) and were prepared with AI help and checked.