Energy Bands & Semiconductors Energy Bands & Semiconductors In isolated atoms, electrons occupy discrete energy levels. When many atoms join to form a solid, their close spacing causes these levels to split into quasicontinuous allowed ranges called bands. Two bands are central to electronics: the valence band (VB), containing electrons bound in bonds, and the conduction band (CB), where electrons can move freely and conduct current. Between them lies an energy range with no allowed electron states, the forbidden gap (band gap) E g . If E g = 0 or bands overlap, electrons easily move and the solid conducts like a metal. If E g is large (several eV), electrons cannot thermally jump to the CB, so the material is an insulator. If E g is small (about 0.7 to 3 eV), some electrons can be thermally excited across the gap: this is the hallmark of a semiconductor. In semiconductors, not only electrons but also the absence of an electron in the valence band (a hole) behaves like a positive charge carrier. The balance between electrons and holes, set by temperature and doping, controls conductivity. The Fermi level E F is the energy at which the probability of occupancy is 1/2 at a given temperature; it helps predict which band is more likely to be populated. Doping shifts E F and sets a majority carrier: donors (Group V) create extra electrons (n-type) and push E F upward toward E C ; acceptors (Group III) create holes (p-type) and pull E F down toward E V . These band ideas power the operation of p – n junctions, diodes, LEDs, solar cells, and transistors. Analogy: Think of lanes on a highway. The conduction band is an open fast lane; the valence band is a parking lot with almost no motion. The band gap is a toll barrier. In metals the barrier is absent; in semiconductors it is low, so with some energy (temperature/light), cars can cross; in insulators it is a huge barrier. remember Highest range of allowed energies with electrons bound in atomic bonds; conduction is not easy. Valence Band (VB) Lowest range of allowed energies where electrons are mobile and contribute to conduction. Conduction Band (CB) Energy range between VB and CB that has no allowed electron states. Forbidden Energy Gap ( E g ) Fermi Level ( E F ) Energy level with 50% occupancy probability; indicates the tendency of electrons to occupy states. Pure semiconductor (e.g., Si, Ge) where electrons and holes are generated thermally with n e = n h = n i . Intrinsic Semiconductor Extrinsic Semiconductor Doped semiconductor: donors (n-type) add electrons; acceptors (p-type) create holes. Donor (Group V) Impurity atom providing an electron close to E C ; sets majority carriers as electrons (n-type). Impurity atom creating an empty state (hole) close to E V ; sets majority carriers as holes (p-type). Acceptor (Group III) Hole A vacant state in the valence band that behaves like a positively charged mobile particle. Proportionality between drift velocity and electric field: v d = E ; electrons have e , holes have h . Mobility ( ) Measure of how well a material conducts: higher means easier current flow. Conductivity ( ) Carrier type in greater/lesser concentration: electrons are majority in n-type, holes in p-type. Majority/Minority Carriers Band formation arises from quantum mechanics. In a crystal, the overlap of many identical atomic orbitals forms bands of closely spaced energy levels. The Pauli exclusion principle fills states from the lowest energy upward. If the conduction band overlaps the valence band (metals), even at T=0 , K there are mobile electrons. If there is a wide forbidden gap, the valence band is full and the conduction band is empty, so no carriers are available and current is negligible (insulator). In semiconductors, E g is small enough that thermal energy ( k B T ) can excite some electrons from VB to CB, leaving behind holes. Thus, both electrons (in CB) and holes (in VB) carry charge. Typical E g at room temperature: Ge 0.66 , eV , Si 1.12 , eV , GaAs 1.43 , eV , diamond 5.5 , eV (insulator). The Fermi level E F provides a compact way to describe occupancy. In an intrinsic semiconductor, by symmetry of electrons and holes, E F lies close to the midgap: E F E C + E V 2 with a small correction if m e m h . Doping shifts E F . In n-type silicon with donors (e.g., P, As), donor levels E D lie slightly below E C (typically tens of meV). At room temperature most donors ionize, contributing electrons to CB and pushing E F upward, closer to E C . In p-type silicon with acceptors (e.g., B, Al), acceptor levels E A lie above E V , most acceptors ionize to create holes, and E F moves downward, closer to E V . This shift explains majority/minority carrier concentrations and sets diode behavior. Reference levels: We measure energies relative to a fixed zero. In band diagrams here, E C (conduction band edge) is higher than E V (valence band edge), and E g = E C - E V > 0 . Fermi level E F can lie within the gap for semiconductors and insulators. tip Definition of band gap 19.1 Band gap is the energy required to promote an electron from the top of VB to the bottom of CB. This formula defines the minimum energy required to excite an electron from the filled valence band to the empty conduction band in a semico Temperature affects carrier generation and mobility. As temperature rises, more electrons are thermally excited to the conduction band, increasing electron and hole densities. However, increased lattice vibrations reduce mobility due to scattering. In intrinsic semiconductors, the rise in carrier density dominates, so conductivity increases strongly with temperature. In metals, carrier density is almost fixed; mobility drops with temperature, so conductivity decreases. This opposite trend is a signature difference between metals and semiconductors. Conductivity of a semiconductor Total conductivity is the sum of electron and hole contributions. 19.2 Ohmic regime: low electric field so drift velocity is proportional to field. Non-degenerate semiconductor statistics; constant mobilities over the field range. Steady state; thermal equilibrium for carrier generation-recombination balance. = e(n e e + n h h ) = e ,(n e e + n h h ) Total current density is the sum from electrons and holes. Write current densities in terms of carrier densities and drift velocities. Define mobilities as drift per unit field. Substitute drift relations. Ohm's law in microscopic form. Identify conductivity by comparing coefficients of E . Conductivity arises from both electrons and holes; each contributes e n . Same relation highlighting intrinsic vs extrinsic regimes and unit care. Intrinsic carrier concentration n i is the number density of electrons (equal to holes) in a pure semiconductor at a given temperature. In intrinsic material, n e = n h = n i , and conductivity simplifies to = e n i ( e + h ) . For silicon at 300 , K , n i 1.5 10 10 , cm -3 = 1.5 10 16 , m -3 . Mobilities are roughly e 0.135 , m 2 /(V ,s) and h 0.048 , m 2 /(V ,s) , giving a very small , consistent with weak conduction in pure silicon. Mass action law 19.3 In thermal equilibrium, the product of electron and hole densities is constant for a given temperature. When doping changes carrier balance, this law ensures the product of electron and hole concentrations remains constant at equilibrium. Doping changes carrier balance while preserving the mass action law in equilibrium. In n-type material with donor density N D (fully ionized at room temperature), electron concentration n e N D and hole concentration n h = n i 2 /n e . Thus, increasing N D boosts n e and suppresses n h . The reverse holds for p-type with acceptor density N A . Because e > h in most semiconductors (electrons are lighter and scatter less), n-type materials often achieve higher conductivity for the same magnitude of doping. neet-alert Unit trap: Many data are given in cm -3 . Always convert to m -3 by multiplying by 10 6 . Also convert cm 2 /(V ,s) to m 2 /(V ,s) by multiplying by 10 -4 . Metal 0 (overlap) Electrons (fixed density) Decreases (mobility drops) Intrinsic Semiconductor (Si) ≈ 1.12 Electrons = Holes = n i Increases strongly (carrier density rises) n-type Semiconductor ≈ intrinsic E g Electrons (majority), Holes (minority) Generally increases (ionization then mobility-limited) Insulator (Diamond) ≈ 5.5 Almost none Tiny change; remains very low Material Type Band Gap E g (eV) Carrier at 300 K Conductivity vs Temperature Donors (Group V): E D slightly below E C ; provide electrons; shift E F . Acceptors (Group III): E A slightly above E V ; create holes; shift E F . Full ionization at room temperature if E D or E A is within a few k B T of band edges. Very low temperature can freeze-out dopants, reducing free carriers sharply. Doping essentials NCERT Semiconductors, Qualitative PYQs Classify each as conductor, semiconductor, or insulator. easy Case A: VB and CB overlap. Case B: E g = 1.2 , eV . Case C: E g = 6.0 , eV . Use the size of E g and presence of overlap as the criterion. Identify the material type from band diagrams (qualitative). Mobility differences matter. In silicon, e is typically about 2 – 3 times h . Since depends on n , an n-type sample with the same magnitude of doping as a p-type one often shows higher conductivity. Device engineers exploit this by choosing doping types and levels to reach target resistivities and switching speeds. However, very heavy doping (degenerate doping) can reduce mobility due to impurity scattering, so there is an optimal range for device performance. Calculate the intrinsic conductivity of silicon at 300 K. Convert units to SI, use = e n i ( e + h ) , then = 1/ . Intrinsic silicon: n i = 1.5 10 10 , cm -3 . e = 1350 , cm 2 /(V ,s) , h = 480 , cm 2 /(V ,s) . e = 1.6 10 -19 , C . medium Intrinsic Si properties at room temperature Conductivity in S/m and resistivity in , m . S/m, Ω·m Majority vs minority: In n-type, electrons are majority and holes are minority. In p-type, holes are majority and electrons are minority. Do not flip this when asked about diffusion or drift currents in p – n junctions. neet-alert Energy band diagrams visualize how doping shifts the Fermi level. For n-type, E F sits closer to E C than in intrinsic. For p-type, E F sits closer to E V . At absolute zero, an ideal intrinsic semiconductor would have a completely full VB and empty CB. With donors present and at sufficiently low temperature, the donors may not ionize (freeze-out), so n e can be much less than N D . At moderate temperatures (room temperature for Si), donors are almost fully ionized and n e N D . At very high temperatures, intrinsic excitation dominates again, and n e n h n i grows rapidly. Diode current depends on voltage in a non-linear way (Shockley equation). Linearize about operating point Q. Define dynamic (incremental) resistance as the local slope inverse. r d = V I r d = V I Small-signal operation around a Q-point on the diode I–V curve. Temperature constant; junction parameters do not change with the perturbation. Local linearization of a non-linear characteristic. Dynamic resistance is the reciprocal slope of the I–V curve at the operating point. Equivalent small-signal definition r d = dV/dI ; for finite small changes use V/ I . In a transistor (beyond this concept), = I C /I E in common-base; introduced here to connect carriers to device gain. In common-emitter, = I C /I B ; larger than because I B I E . DC version dc = I C /I B ; relevant when linking band-based carrier flow to transistor operation. NAND: Y = A B ; a universal logic gate implemented by semiconductor devices. NOR: Y = A + B ; also universal; realizable using doped semiconductor junctions. While logic gates and transistor gains are device-level topics, they rely on the same carrier physics: electron and hole populations set by band structure and doping, mobilities controlled by scattering, and junction phenomena determined by Fermi level alignment. Mastering energy bands helps predict how p – n junctions rectify, how r d varies along a characteristic, and why depends on minority carrier transport in the base. N-type vs P-type Semiconductors Type Majority Carrier Minority Carrier Dopant Valency Energy Level Created N for Negative Electrons (5-Pentavalent), P for Positive Holes (3-Trivalent). N-type Electrons ( e - ) Holes ( h + ) 5 (Pentavalent) Donor level ( E d ) just below Conduction Band P-type Holes ( h + ) Electrons ( e - ) 3 (Trivalent) Acceptor level ( E a ) just above Valence Band n type vs p type semiconductors 300 Room temperature Si sigma(Si) small but non-zero Temperature arb. units Conductivity Metals instead show a gentle decrease; insulators stay near zero. custom sigma sigma = A exp(-B/T) 2D PLOT Intrinsic semiconductor conductivity vs temperature Prefactor Eg/2k Metal dependent dependent Intrinsic Semiconductor dependent Insulator Metal: gentle decrease; Intrinsic semiconductor: sharp increase; Insulator: near-flat near zero. MSI for band gap sizes: Metal = 0, Semiconductor = Small, Insulator = Immense. Recall order: 0, small, large. A hole is the absence of an electron in the valence band. It behaves like a positive charge carrier due to how electrons redistribute, but no proton moves. A hole is a real positive particle like a proton moving in the crystal. At high doping, impurity scattering reduces mobility, and at very low temperature dopants may not ionize (freeze-out), so conductivity does not keep rising linearly. Doping always increases conductivity linearly without limit. Carrier transport in semiconductors combines drift and diffusion. Drift current results from an applied electric field: electrons move opposite to the field, holes move along it. Diffusion current arises from carrier concentration gradients: electrons and holes diffuse from high to low concentration regions. In p – n junctions, these two processes balance in equilibrium, setting up a built-in potential. Although device-level details are beyond this concept, recognizing that drift scales with mobility and field while diffusion scales with the gradient helps explain junction behavior and the shape of I–V curves. Checklist for semiconductor classification Estimate or recall E g . If E g 3 , eV , likely a semiconductor. Check temperature trend of . Upward with T suggests a semiconductor. Consider doping: If majority carriers are electrons (donors), it is n-type; if holes (acceptors), p-type. Locate E F : Near E C indicates n-type, near E V indicates p-type, midgap intrinsic. Use n e N D , mass action n e n h = n i 2 , and = e(n e e + n h h ) . An n-type silicon sample has donor density N D = 5.0 10 21 , m -3 at 300 K with full ionization. Given e = 0.12 , m 2 /(V ,s) , h = 0.045 , m 2 /(V ,s) , n i = 1.5 10 16 , m -3 , compute and minority hole density n h . hard N D = 5.0 10 21 , m -3 , full ionization. e = 1.6 10 -19 , C . e = 0.12 , h = 0.045 , m 2 /(V ,s) . n i = 1.5 10 16 , m -3 . S/m, m -3 Doping and mass action application and n h . Effective mass m is a band-structure concept that captures how electrons respond to forces in a periodic crystal. Near band extrema, the energy– k relation is approximately parabolic, and carriers behave like free particles with modified inertia m . Different bands (and different directions) can have different effective masses. This influences density of states, mobility, and even the exact position of the intrinsic Fermi level within the gap (a small correction to midgap). Qualitative band edges: Donor levels E D lie within a few tens of meV below E C in Si; acceptor levels E A lie a few tens of meV above E V . At room temperature, this is enough for near-complete ionization. tip Photons can bridge the band gap. In photodiodes and solar cells, incident photons with energy h E g generate electron–hole pairs, increasing conductivity (photoconductivity) and enabling current under bias or even power generation under open-circuit load. In LEDs, carriers recombine across the gap and emit photons with energy approximately equal to E g , explaining why the color of an LED is tied to the semiconductor’s band gap. Indirect vs direct band gaps matter for optical processes. Silicon has an indirect gap; electron transitions across the gap require a momentum change assisted by phonons, making light emission inefficient. GaAs has a direct gap; electrons can recombine radiatively with high probability, which is why GaAs and related III–V compounds are preferred for LEDs and laser diodes. For conduction and basic diode action, both direct and indirect semiconductors work, but their optical properties diverge. Intrinsic Si has very low conductivity at 300 K (e.g., 10 -4 , S/m ). Doping is required to obtain useful conduction for devices. Intrinsic semiconductors conduct well at room temperature without doping. Resistivity tailoring in wafers is done by choosing dopant species and dose, followed by thermal treatments to activate and diffuse dopants. For device layers, designers might target resistivities from a few , cm (lightly doped) down to milliohm-cm (heavily doped contacts). However, as dopant concentration rises above about 10 25 , m -3 (roughly 10 19 , cm -3 ), mobility degradation becomes severe, and the semiconductor can become degenerate, where Fermi level enters a band and classical n e n h = n i 2 relations need quantum corrections. From a band perspective, rectification in a diode comes from asymmetric carrier barriers across a p – n junction. Forward bias lowers the barrier, allowing majority carriers to inject and recombine, while reverse bias raises the barrier, leaving only a tiny saturation current due to minority carriers and generation-recombination processes. The small-signal slope at any operating point defines r d , linking band-governed currents to circuit-level parameters. Thermal generation and recombination obey detailed balance in equilibrium. Recombination can be radiative (common in direct-gap materials) or non-radiative via defect (trap) states or Auger processes. Doping and crystal quality set the density of such recombination centers. This microphysics determines minority carrier lifetimes, diffusion lengths, and ultimately device switching speeds and efficiencies. Practical values to remember at 300 K: For Si, E g 1.12 , eV , n i 1.5 10 10 , cm -3 , e 1350 , cm 2 /(V ,s) , h 480 , cm 2 /(V ,s) . For Ge, E g 0.66 , eV and n i is much larger (so Ge is more conductive intrinsically). For GaAs, E g 1.43 , eV and it is direct-gap, favorable for optoelectronics. Carrier statistics set the occupancy of states. At non-zero temperature, the Fermi–Dirac distribution f(E) = 1/[1 + ((E - E F )/(k B T))] governs the probability of occupancy. In non-degenerate semiconductors where E is several k B T away from E F , the Maxwell–Boltzmann approximation holds, simplifying analysis. That is why relations like n e n h = n i 2 and exponential temperature dependence of n i are good approximations for standard doping ranges used in basic devices. Edge cases: At cryogenic temperatures, dopant ionization is incomplete (freeze-out), reducing majority carrier density below dopant density. At very high temperatures, intrinsic carriers dominate even in doped materials (intrinsic takeover). In degenerate doping (very heavy), the Fermi level can enter the conduction or valence band, and classical semiconductor equations need Fermi–Dirac integrals; mobility also drops due to ionized impurity scattering. Big picture: Band structure sets whether carriers exist; doping sets which carrier dominates; mobility sets how fast they move. Conductivity is the product of all three: = e ,(n e e + n h h ) . remember 1 , cm -3 = 10 6 , m -3 1 , cm 2 /(V ,s) = 10 -4 , m 2 /(V ,s) 1 , eV = 1.6 10 -19 , J Common data conversions Device context link: In BJTs, and connect to how efficiently injected minority carriers traverse the base without recombining. Narrow bases, proper doping profiles, and clean crystal structures increase carrier survival, boosting . Although this concept focuses on bands and carriers, recognizing how these microscopic parameters feed into device gains helps form a coherent understanding from physics to circuits. Experimental signatures of semiconductors include: exponential temperature dependence of carrier concentration, Hall effect revealing carrier sign and density, photoconductivity when illuminated with photons of energy E g , and rectification at p – n junctions. These measurements confirm the band model and quantify parameters like e , h , and n i . Summary checkpoint: Metals have overlapping bands; semiconductors have a small gap; insulators have a large gap. The Fermi level’s position signals doping type. Conductivity increases with temperature in semiconductors due to rising carrier density, while it decreases in metals due to mobility loss. Doping selects the majority carrier and shifts E F toward the relevant band edge. The simple yet powerful formula = e(n e e + n h h ) connects microscopic physics to measurable macroscopic behavior. Key Terms Recap Filled or nearly filled band; electrons are bound. Valence band (VB) Empty or partly filled band enabling conduction. Conduction band (CB) E g Energy separation between CB and VB. Band gap E g Energy with 50% occupancy probability. E F Fermi level E F Intrinsic carrier density n i n i Electron (and hole) density in pure material at temperature T . Proportionality constant in v d = E . Mobility Donor/Acceptor Dopants creating extra electrons/holes. Majority/Minority carriers Dominant/sparse carrier species set by doping.