. SEMICONDUCTORS Silicon bond model: Electrons and holes; Generation and recombination; Thermal equilibrium; Intrinsic semiconductor; Doping; extrinsic semiconductor. Some questions: How do semiconductors conduct electricity? What is a ”hole”? How many electrons and holes are there in a semiconductor in thermal equilibrium at a certain temperature? How can one engineer the conductivity of semiconductors? Silicon bond model: electrons and holes Si is in Column IV of periodic table. .

Abridged periodic table III A IV A V A VI A B 5 C 6 N 7 O 8 Al 13 Si 14 P 15 S 16 Ga 31 Ge 32 As 33 Se 34 In 49 Sn 50 Sb T e

. Electronic structure of Si atom: I t has, 10 core electrons (tightly bound) and 4 valence electrons (loosely bound, which are responsible for most chemical properties) Other semiconductors which are not limited to these are: Ge, C (diamond form), SiGe, GaAs, InP, InGaAs, InGaAsP, ZnSe, CdTe(on the average, they have 4 valence electrons per atom). Silicon is a crystalline material. It has long range atomic arrangement. Diamond lattice: It has atoms which are tetrahedrally bonded by sharing valence electrons (covalent bonding). Each atom shares 8 electrons. Si has atomic density of 5 x 1022 cm−3

At 0K, all the bonds in the tetrahedral structure are satisfied i. e At 0K, all the bonds in the tetrahedral structure are satisfied i.e. all valence electrons engaged in bonding. There are no ”free” electrons. At finite temperature, there will be finite thermalenergy. Thus some bonds will be broken. Hence ”free” electrons (mobile negative charge, −1.6×10−19C) ”free” holes (mobile positive charge, 1.6×10−19C) ”Free” electrons and holes are called carriers. These are the mobile charged particles. However, electrons and holes in semiconductors are “fuzzier”. They span many atomic sites. .

Lets define some terms: n ≡ (free) electron concentration [cm−3] and p ≡ hole concentration [cm−3] Generation and Recombination Generation = break up of covalent bond to form electron and hole . This requires energy from thermal or optical sources (or other external sources). Generation rate: G = Gth + Gopt + ... [cm−3 · s -1 . In general, atomic density ˃˃ n, p ; hence, G ≠ f(n, p). Thus the supply of breakable bonds is virtually inexhaustible Recombination = formation of bond by bringing together electron and hole. This releases energy in thermal or optical form. .

The recombination rate, is given as, R [cm−3 · s -1 ] .The recombination rate, is given as, R [cm−3 · s -1 ]. A recombination event requires 1 electron + 1 hole, thus, R ∝ n · p . However, generation and recombination are most likely at surfaces where periodic crystalline structure is broken. Thermal equilibrium: Thermal equilibrium = steady state + absence of external energy sources, hence all time derivatives are equal to zero. Hence, δ<θ>/ δt =0 Generation rate in thermal equilibrium: Go = f(T) Recombination rate in thermal equilibrium: Ro ά no·po where o denotes thermal equilibrium. In thermal equilibrium, Go = Ro, hence, nopo=f(T)≡ni 2(T) .

The consequence of this is that in thermal equilibrium, and for a given semiconductor, np product is a constant that depends only on temperature! Electron-hole formation can be seen as chemical rxn : bond ↔ e− + h+ . This is similar to water decomposition reaction: H2O ↔ H+ + OH− The law-of-mass action relates concentration of reactants and reaction products. For water: K = [H+][OH−]/ [H2O]. Since: [H2O] ˃˃ [H+], [OH−] Then: [H2O] ≈ constant Hence: [H+][OH−] ≈ constant. .

Intrinsic semiconductor In a perfectly pure semiconductor in thermal equilibrium at finite temperature, how many electrons and holes are there? Since when a bond breaks, an electron and a hole are produced: we know that, no = po . Also: nopo = ni 2. . Then, no = po = ni . ni ≡ intrinsic carrier concentration [cm−3]. Thus in Si at 300 K (”room temperature”): ni≈1×1010 cm−3 . ni is a very strongly dependent function of temperature: T ↑→ ni ↑ . However, note that an intrinsic semiconductor need not be perfectly pure. .

Doping: This is introduction of foreign atoms to engineer semiconductor electrical properties Donors: introduce electrons to the semiconductor (but not holes). For Si, group-V atoms with 5 valence electrons (As, P, Sb) will serve as donors. 4 electrons of donor atom will participate in bonding while the 5th electron is easy to release. Thus, at room temperature, each donor releases 1 electron that is available for conduction and the donor site becomes positively charged (fixed charge). As+ immobile ionized donor. Lets define: Nd ≡ donor concentration [cm−3] If : Nd ˂˂ ni , doping is irrelevant, because, for intrinsic semiconductor, no = po = ni . .

However if Nd ˃˃ ni , doping controls conc However if Nd ˃˃ ni , doping controls conc. This is extrinsic semiconductor. Here, no = Nd and po = ni 2 / Nd Note that, no ˃˃ po gives rise to n-type semiconductor. Chemical reaction analogy: dissolve a bit of KOH into water ⇒ [OH−] ↑, [H+] ↓ and pH will rise. Acceptors: introduce holes to the semiconductor (but not electrons). For Si, group-III atoms with 3 valence electrons e.g. B. Here 3 electrons are used in bonding to neighbouring Si atoms. Hence 1 bonding site will be ”unsatisfied”. It is now easy to ”accept” neighbouring bonding electron to complete all bonds. At room temp., each acceptor releases 1 hole that is available for conduction. Thus the acceptor site becomes negatively charged (fixed charge) .

. Let us define: Na ≡ acceptor concentration [cm−3] If Na ˂˂ ni , doping is irrelevant bicos for intrinsic semiconductor, no = po = ni . However, if Na ˃˃ ni , doping controls carrier concentrations. This is also an extrinsic semiconductor, but of the p-type. Here, po = Na hence, no = ni 2/ Na . Thus when po ˃˃ no, we have p-type semiconductor. For the chemical reaction analogy: dissolve a bit of H2SO4 in water; [H+] ↑, [OH−] ↓ and therefore, pH will drop.

In summary, for semiconductors, there are two types of ”carriers”: electrons and holes. In thermal equilibrium and for a given semiconductor no po is a constant that only depends on temperature: no po = ni 2 For Si at room temperature: ni = 1010 cm-3 . For intrinsic semiconductor, i.e. “pure” semiconductor, no = po = ni In intrinsic semiconductors, carrier concentrations can be engineered by addition of ”dopants” i.e. Addition of selected foreign atoms. Thus we have n-types and p-types dopants which is a function of valence electrons of dopant. .