CUET PG Physics
Solid State Physics, Devices & Electronics – Complete Notes
- 1. Crystal Structure
- 2. Bravais Lattices and Basis
- 3. Coordination Number and Packing Fraction
- 4. Miller Indices
- 5. X‑ray Diffraction and Bragg’s Law
- 6. Intrinsic and Extrinsic Semiconductors
- 7. Variation of Resistivity with Temperature
- 8. Fermi Level
- 9. P‑N Junction Diode
- 10. Zener Diode and Applications
- 11. Bipolar Junction Transistor (BJT)
- 12. Single Stage Amplifier
- 13. Two Stage R‑C Coupled Amplifiers
- 14. Simple Oscillators
- 15. Operational Amplifier (Op‑Amp) and Applications
- 16. Boolean Algebra
- 17. Binary Number Systems
- 18. Logic Gates
- 19. Combination of Gates
- 20. De Morgan’s Theorem – Applications
- 21. Summary Table: Important Formulas
1. Crystal Structure
A crystal is a solid with a highly ordered, periodic arrangement of atoms. Characteristics:
- Long‑range order
- Discrete symmetry operations
- Sharp melting point
Amorphous vs. Crystalline:
| Property | Crystalline | Amorphous |
|---|---|---|
| Atomic arrangement | Periodic | Random |
| Melting point | Sharp | Broad range |
| Anisotropy | Anisotropic | Isotropic |
| Examples | NaCl, Quartz, Diamond | Glass, Rubber |
2. Bravais Lattices and Basis
A space lattice is an infinite array of points with identical surroundings. There are 14 Bravais lattices distributed in 7 crystal systems.
| Crystal System | Axial relations | Angle relations | Bravais lattices |
|---|---|---|---|
| Cubic | \(a=b=c\) | \(\alpha=\beta=\gamma=90^\circ\) | P, I, F |
| Tetragonal | \(a=b\neq c\) | \(\alpha=\beta=\gamma=90^\circ\) | P, I |
| Orthorhombic | \(a\neq b\neq c\) | \(\alpha=\beta=\gamma=90^\circ\) | P, I, F, C |
| Monoclinic | \(a\neq b\neq c\) | \(\alpha=\gamma=90^\circ,\ \beta\neq90^\circ\) | P, C |
| Triclinic | \(a\neq b\neq c\) | \(\alpha\neq\beta\neq\gamma\neq90^\circ\) | P |
| Trigonal | \(a=b=c\) | \(\alpha=\beta=\gamma\neq90^\circ\) | R |
| Hexagonal | \(a=b\neq c\) | \(\alpha=\beta=90^\circ,\ \gamma=120^\circ\) | P |
Basis: atom(s) attached to each lattice point → Crystal structure = Lattice + Basis.
Unit cell atoms:
- SC: \(8\times\frac18 = 1\)
- BCC: \(8\times\frac18 + 1 = 2\)
- FCC: \(8\times\frac18 + 6\times\frac12 = 4\)
- HCP: 6
3. Coordination Number and Packing Fraction
Coordination number: number of nearest neighbours.
- SC: 6
- BCC: 8
- FCC: 12
- HCP: 12
- Diamond: 4
- NaCl: 6:6
- CsCl: 8:8
Packing fraction \(= \frac{\text{Volume occupied by atoms}}{\text{Volume of unit cell}}\).
- SC: \( \frac{\pi}{6} \approx 52\% \)
- BCC: \( \frac{\sqrt3\pi}{8} \approx 68\% \)
- FCC & HCP: \( \frac{\sqrt2\pi}{6} \approx 74\% \)
- Diamond: 34%
4. Miller Indices \((hkl)\)
Procedure:
- Find intercepts on axes (in units of a,b,c).
- Take reciprocals.
- Reduce to smallest integers.
Interplanar spacing (cubic): \(d_{hkl} = \dfrac{a}{\sqrt{h^2+k^2+l^2}}\).
5. X‑ray Diffraction and Bragg's Law
Bragg’s law: constructive interference when path difference = \(n\lambda\).
Reciprocal lattice: \(\vec{G}_{hkl}=h\vec{a}^*+k\vec{b}^*+l\vec{c}^*\), magnitude \(|\vec{G}| = \frac{2\pi}{d_{hkl}}\).
Brillouin zone: Wigner‑Seitz cell of reciprocal lattice.
6. Intrinsic and Extrinsic Semiconductors
Intrinsic: pure, \(n=p=n_i\).
Extrinsic: doped.
- n‑type: pentavalent donors (P, As) → majority electrons, \(n\approx N_D,\ p=n_i^2/n\).
- p‑type: trivalent acceptors (B, Al) → majority holes, \(p\approx N_A,\ n=n_i^2/p\).
\(n_i = \sqrt{N_c N_v}\, e^{-E_g/2k_BT}\) with \(N_c = 2\left(\frac{2\pi m_e^* k_B T}{h^2}\right)^{3/2}\), \(N_v\) similar.
7. Variation of Resistivity with Temperature
In semiconductors, resistivity decreases with T (negative temperature coefficient).
- Intrinsic: \(\rho \propto e^{E_g/2k_BT}\).
- Extrinsic: at low T carrier freeze‑out, at mid T saturation, at high T intrinsic dominates.
Metals: resistivity increases with T (positive coefficient).
8. Fermi Level
Fermi‑Dirac distribution: \(f(E)=\frac{1}{e^{(E-E_F)/k_BT}+1}\).
- Intrinsic: \(E_F \approx \frac{E_c+E_v}{2}\) (mid‑gap).
- n‑type: \(E_F = E_c - k_BT\ln(N_c/N_D)\) (moves up).
- p‑type: \(E_F = E_v + k_BT\ln(N_v/N_A)\) (moves down).
9. P‑N Junction Diode
Formation: diffusion → depletion region → built‑in field.
Built‑in potential \(V_0 = \frac{k_BT}{e}\ln\frac{N_A N_D}{n_i^2}\).
I‑V characteristic: \(I = I_0\left(e^{eV/\eta k_BT}-1\right)\).
- Forward bias: barrier reduced, large current.
- Reverse bias: barrier increased, small leakage.
Applications: rectifier, clipper, clamper.
10. Zener Diode and Applications
Operates in reverse breakdown without damage.
- Zener breakdown (\(V_z<5V\)): tunneling, negative temp.coeff.
- Avalanche breakdown (\(V_z>7V\)): impact ionisation, positive temp.coeff.
Voltage regulator: series resistor \(R_s\), Zener in parallel with load. \(V_{out}=V_z\).
11. Bipolar Junction Transistor (BJT)
Three regions: Emitter (heavily doped), Base (thin, light), Collector (moderate). Types: NPN, PNP.
Current relations: \(I_E = I_B + I_C\), \(\alpha = I_C/I_E\), \(\beta = I_C/I_B\).
\(\displaystyle \beta = \frac{\alpha}{1-\alpha}\).
Configurations
| Config | Current gain | Voltage gain | Input impedance | Application |
|---|---|---|---|---|
| CB | \(\alpha<1\) (≈1) | High | Low | High frequency |
| CE | \(\beta\) (high) | High | Medium | Most common amp |
| CC | \(\beta+1\) | ≈1 | Very high | Buffer |
12. Single Stage Amplifier (CE)
Biasing: voltage divider (\(R_1,R_2\)), \(R_C\), \(R_E\), bypass capacitor \(C_E\).
DC analysis: \(V_B = V_{CC}\frac{R_2}{R_1+R_2}\), \(V_E = V_B-0.7\), \(I_C\approx I_E = V_E/R_E\), \(V_{CE}=V_{CC}-I_C(R_C+R_E)\).
AC gain (with bypass): \(A_v = -g_m R_C\), \(g_m = I_C/V_T\) (\(V_T\approx 25\)mV).
13. Two Stage R‑C Coupled Amplifiers
Two CE stages connected via coupling capacitor.
- Advantages: simple, good frequency response, cheap.
- Disadvantages: gain rolls off at low and high frequencies due to capacitors.
- Total gain = \(A_{v1}\times A_{v2}\) (but loading reduces it).
14. Simple Oscillators
Barkhausen condition for sustained oscillations: \(|A\beta| = 1\) and \(\angle A\beta = 0^\circ\) (or 360°).
- RC phase shift oscillator: \(f = \frac{1}{2\pi RC\sqrt{6}}\).
- Wien bridge oscillator: \(f = \frac{1}{2\pi RC}\).
- Colpitts (LC with tapped C): \(f = \frac{1}{2\pi\sqrt{LC_T}}\).
- Hartley (tapped L): \(f = \frac{1}{2\pi\sqrt{(L_1+L_2)C}}\).
15. Operational Amplifier (Op‑Amp) and Applications
Ideal op‑amp: infinite gain, infinite input impedance, zero output impedance, infinite bandwidth.
Inverting amplifier: \(A_v = -\frac{R_f}{R_1}\).
Non‑inverting amplifier: \(A_v = 1+\frac{R_f}{R_1}\).
Summing amplifier: \(V_{out} = -\left(\frac{R_f}{R_1}V_1+\frac{R_f}{R_2}V_2+\frac{R_f}{R_3}V_3\right)\).
Integrator: \(V_{out}(t) = -\frac{1}{RC}\int V_{in}(t)dt\).
Differentiator: \(V_{out}(t) = -RC\frac{dV_{in}}{dt}\).
16. Boolean Algebra
| Law | AND form | OR form |
|---|---|---|
| Identity | \(A\cdot1=A\) | \(A+0=A\) |
| Null | \(A\cdot0=0\) | \(A+1=1\) |
| Idempotent | \(A\cdot A=A\) | \(A+A=A\) |
| Complement | \(A\cdot\overline{A}=0\) | \(A+\overline{A}=1\) |
| Involution | \(\overline{\overline{A}}=A\) | |
| Commutative | \(A\cdot B = B\cdot A\) | \(A+B = B+A\) |
| Associative | \(A\cdot(B\cdot C)=(A\cdot B)\cdot C\) | \(A+(B+C)=(A+B)+C\) |
| Distributive | \(A\cdot(B+C)=A\cdot B+A\cdot C\) | \(A+B\cdot C=(A+B)\cdot(A+C)\) |
| Absorption | \(A\cdot(A+B)=A\) | \(A+A\cdot B=A\) |
17. Binary Number Systems
Binary (base 2), Octal (8), Decimal (10), Hexadecimal (16).
Conversion examples:
- \((1101.101)_2 = 1\cdot2^3+1\cdot2^2+0\cdot2^1+1\cdot2^0+1\cdot2^{-1}+0\cdot2^{-2}+1\cdot2^{-3}=13.625_{10}\).
- \((11001011)_2 = CB_{16}\).
Binary addition: \(0+0=0,\ 0+1=1,\ 1+0=1,\ 1+1=0\) carry 1.
Binary subtraction via 2's complement.
18. Logic Gates
Truth tables:
| AND \(A\cdot B\) | OR \(A+B\) | NOT \(\overline{A}\) |
|---|---|---|
| 00→0, 01→0, 10→0, 11→1 | 00→0, 01→1, 10→1, 11→1 | 0→1, 1→0 |
NAND: \(\overline{A\cdot B}\) (output 0 only when all inputs 1).
NOR: \(\overline{A+B}\) (output 1 only when all inputs 0).
XOR: \(A\oplus B = A\overline{B}+\overline{A}B\) (1 when inputs differ).
XNOR: \(\overline{A\oplus B} = AB+\overline{A}\,\overline{B}\) (1 when inputs same).
19. Combination of Gates
Using NAND as universal gate:
- NOT: short inputs → \(\overline{A\cdot A}=\overline{A}\).
- AND: NAND + NOT → \(\overline{\overline{A\cdot B}}\).
- OR: using De Morgan → \(\overline{\overline{A}\cdot\overline{B}} = A+B\).
20. De Morgan’s Theorem – Applications
\(\overline{A\cdot B} = \overline{A}+\overline{B}\)
Used to simplify Boolean expressions and convert between gate types.
21. Summary Table: Important Formulas
| Topic | Formula |
|---|---|
| Interplanar spacing (cubic) | \(d_{hkl} = \dfrac{a}{\sqrt{h^2+k^2+l^2}}\) |
| Bragg's law | \(2d\sin\theta = n\lambda\) |
| Packing fraction SC | \(\pi/6\) |
| Packing fraction BCC | \(\sqrt3\pi/8\) |
| Packing fraction FCC | \(\sqrt2\pi/6\) |
| Intrinsic carrier concentration | \(n_i = \sqrt{N_c N_v}e^{-E_g/2k_BT}\) |
| n‑type carrier conc. | \(n\approx N_D,\ p=n_i^2/n\) |
| p‑type carrier conc. | \(p\approx N_A,\ n=n_i^2/p\) |
| Fermi level n‑type | \(E_F = E_c - k_BT\ln(N_c/N_D)\) |
| Fermi level p‑type | \(E_F = E_v + k_BT\ln(N_v/N_A)\) |
| Diode current | \(I = I_0(e^{eV/\eta k_BT}-1)\) |
| BJT current gain | \(I_C=\beta I_B,\ \beta=\alpha/(1-\alpha)\) |
| CE voltage gain (bypass) | \(A_v = -g_m R_C\) |
| Barkhausen condition | \(|A\beta|=1,\ \angle A\beta=0^\circ\) |
| Inverting op‑amp | \(A_v = -R_f/R_1\) |
| Non‑inverting op‑amp | \(A_v = 1+R_f/R_1\) |
| De Morgan | \(\overline{A+B}=\overline{A}\cdot\overline{B},\ \overline{A\cdot B}=\overline{A}+\overline{B}\) |
