Chapter 2. Electronics I - Semiconductors

Transcription

1 Chapter 2 Electronics I - Semiconductors Fall 2017 talarico@gonzaga.edu 1

2 Charged Particles The operation of all electronic devices is based on controlling the flow of charged particles There are two type of charge in solids Electrons Holes There are two mechanism through which charge can be transported in a material Drift (motion of charge caused by an electric field) Diffusion (motion resulting from a non-uniform charge distribution) talarico@gonzaga.edu 2

3 Electronic Structure of the elements Atom s chemical activity depends on the electrons in the outermost shells (orbits). These electrons are called VALENCE electrons. In extremely pure elements, such as silicon, the atoms arrange themselves in regular patterns called CRYSTALS. The valence electrons determine the exact shape (= LATTICE structure) talarico@gonzaga.edu 3

4 Electronic orbitals in silicon Element Atomic Number Configuration Si 14 (1s) 2 (2s) 2 (2p) 6 (3s) 2 (3p) 2 s, p, d designate orbital shape s holds up to 2 electrons p holds up to 6 electrons d holds up to 10 electrons Source: Howe & Sodini talarico@gonzaga.edu 4

5 Silicon Crystal Lattice Source: Howe & Sodini concentration (atoms/cm 3 ) of atoms in silicon talarico@gonzaga.edu 5

6 2D-Representation of Silicon Crystal Source: Sedra & Smith Figure 3.1 Two-dimensional representation of the silicon crystal. The circles represent the inner core of silicon atoms, with +4 indicating its positive charge of +4q, which is neutralized by the charge of the four valence electrons. Observe how the covalent bonds are formed by sharing of the valence electrons. At 0 K, all bonds are intact and no free electrons are available for current conduction. talarico@gonzaga.edu 6

7 Free electrons and holes Source: Sedra & Smith Figure 3.2 At room temperature, some of the covalent bonds are broken. Each broken bond gives rise to a free electron and a hole, both of which become available for current conduction. talarico@gonzaga.edu 7

8 Periodic Table Source: Pierret 8

9 Energy Band Structure Band gap energy (E g ) is the minimum energy to dislodge an electron from its covalent bond. For Silicon at room temp. (T=300 K) E g = 1.12 ev = Joule Source: Millman & Halkias talarico@gonzaga.edu 9

10 Concentration of free electrons The concentration of electrons (and holes) in pure silicon at room temperature is approximately: As temperature increases, the intrinsic concentration n i approximately doubles every 10 C rise over room temperature (source: Howe & Sodini) Given that the number of bonds is cm 3, at room temperature only an extremely small fraction of the bonds are broken (1 in bonds, that is 1 in atoms) Too few: we need more!!! n " T = 300 K 1 10,- cm 01 n " (T) n " (T 1-- ) ,- cm 01 talarico@gonzaga.edu 10

11 Intrinsic carrier concentration as a function of temperature source: Streetman 300 K = 27 C For an intrinsic semiconductor: n = p = n i / talarico@gonzaga.edu 11

12 Intrinsic carrier concentration E G also depends on T n " = A 9 A : T 1/= e 5 10,C T 1/= e (cm 01 ) The constants A C and A V can be derived from the effective density of the states in conduction band N C (cm -3 ) and valence band N V (cm -3 ). N 9 = A 9 T 1/= N : = A : T 1/= For silicon at T = 300K (source Pierret): N C = 3.22 x cm 3 and N V = 1.83 x cm 3 n " T = 300 K 1 10,- cm 01 Boltzmann constant = K = 8.617e-5 ev/k = J/ K Energy Gap for silicon at room temperature = 1.12 ev talarico@gonzaga.edu 12

13 Energy Band Gap The energy band gap E g is affected by temperature according to the following Varshni equation: (ev) where E g (0) is the band gap energy at absolute zero and a E and b E are material specific constants talarico@gonzaga.edu 13

14 Extrinsic Semiconductors (1) Doping with donor impurities (N-type semiconductor) source: Sedra & Smith Example: N D cm -3 Figure 3.3 A silicon crystal doped by a pentavalent element. Each dopant atom donates a free electron and is thus called a donor. The doped semiconductor becomes n type. talarico@gonzaga.edu 14

15 Extrinsic Semiconductors (2) Doping with acceptor impurities (P-type semiconductor) source: Sedra & Smith Example: N A cm -3 Figure 3.4 A silicon crystal doped with boron, a trivalent impurity. Each dopant atom gives rise to a hole, and the semiconductor becomes p type. talarico@gonzaga.edu 15

16 N-type semiconductor E C E G E D E V source: Howe & Sodini source: Millman & Halkias q = Cb ρ = chargedensity[cb / cm 3 ] = 0 = ( qn) + (qp) + qn D electrons holes donors talarico@gonzaga.edu 16

17 P-type semiconductor (1) E C E G E A E V source: Howe & Sodini source: Millman & Halkias q = Cb ρ = chargedensity[cb / cm 3 ] = 0 = ( qn) + (qp) qn A electrons holes acceptors talarico@gonzaga.edu 17

18 P-type semiconductor (2) Holes can be filled by absorbing free electrons, therefore there is an effective flow of holes Holes are slower than free electrons (due to the probability of a hole to be filled) The effective mass of holes is larger than the effective mass of the free electrons: m* h > m* e talarico@gonzaga.edu 18

19 Mobility of free electrons and holes Electron and Hole mobilities for silicon at 300 K Mobilities vary with doping level source: Howe & Sodini For intrinsic silicon at T=300 K: μ p 480 cm 2 /(V s) μ n 1350 cm 2 /(V s) 2.8 μ p µ T 3/2 N tot = N A + N D = talarico@gonzaga.edu 19

20 Mass Action Law The mass-action law is valid for both intrinsic (pure) and extrinsic (doped) semiconductors n i 2 = n p If n then p A larger number of free electrons causes the recombination rate of free electrons with holes to increase talarico@gonzaga.edu 20

21 Doping with donors (n-type) Charge neutrality: ρ = 0 = q(p n + N D ) Using mass-action law: n i 2 flip sides multiply both sides by n 2 n n + N = 0 n i D n + n N = 0 D n2 N D n n 2 i = 0 n = N D ± N D 2 4n i 2 2 N D Free electrons are Majority Carriers p! n 2 i N D Holes are Minority Carriers doping with N D >> n i talarico@gonzaga.edu 21

22 Doping with acceptors (p-type) Charge neutrality: Doping with N A >> n i ρ = 0 = q(p n + N D ) p N A n! n 2 i N A Holes are Majority Carriers Free electrons are Minority Carriers talarico@gonzaga.edu 22

23 Doping with both donors and acceptors Charge neutrality: ρ = 0 = q(p n + N D N A ) Assuming that N D -N A >> n i (nearly always true) For N D > N A n! N D N A and p! n i 2 N D N A For N A > N D p! N A N D and n! n i 2 N A N D talarico@gonzaga.edu 23

24 First Carriers Transport Mechanism: Drift!!!" drift v p = µ p E!!!" drift v n = µn E The process in which charged particles move because of an electric field is called drift. Charged particles will move at a velocity that is proportional to the electric field (this is true as long as the field doesn t become too large) talarico@gonzaga.edu 24

25 Drift velocity in silicon v n,drift, v p,drift velocity start to saturate talarico@gonzaga.edu 25

26 Saturation of the drift velocity v drift source: Razavi source Gray & Meyer: v FG"HI E μe 1 + E = μe 1 + μe E 9 v NOI Eventually the drift velocity saturates: there are too many collisions among ` carriers and between carriers and lattice v sat for silicon is 10 7 cm/s = 10 5 m/s talarico@gonzaga.edu 26

27 Drift Current source: Streetman H drift E P = F: R S: R = : 7 = : 7 FP SP 0T T talarico@gonzaga.edu 27

28 Drift current and current density electric current: amount of charge that flows through a reference plane per unit time charge per unit volume [Cb/cm 3 ] I FG"HI = ΔQ Δt source: Streetman = Δx W H n q Δt volume per unit time [cm 3 /s] = v FG"HI W H n q v drift! Δx Δt crosssection Area [cm 2 ] charge per unit volume (aka charge density) [Cb/cm 3 ] Cb s = A + + Figure 4 16 Current entering and leaving a volume ΔxA J drift = I drift W H [A / m2 ] Current Density source: Howe & Sodini talarico@gonzaga.edu 28

29 Drift Current Density Source: Howe & Sodini J n drift v drift n J p drift v p drift J drift = J drift n + J drift p = v drift n q e n + v drift p q p p = µ n Eq e n + µ p Eq h n = = µ n Eqn + µ p Eqp = ( µ n qn + µ p qp)e = σ = 1/ρ conductivity [Ωm] 1 q h = q e q = Cb J drift = σ E Ohm Law talarico@gonzaga.edu 29

30 Conversion between resistivity and dopant density of silicon at room temperature source: Hu 30

31 Second Carriers Transport Mechanism: Diffusion The thermal motion of an electron or a hole changes direction frequently by scattering off imperfections in the semiconductor crystal source: Streetman In a material where the concentration of particles is uniform the random motion balances out and no net movement result (drunk sail-man walk Brownian walks) Random thermal motion of an electron or hole in a solid. talarico@gonzaga.edu 31

32 Diffusion Current If there is a difference (gradient) in concentration between two parts of a material, statistically there will be more particles crossing from the side with higher concentration to the side with lower concentration than vice versa Therefore we expect a net flux of particles q p = q e! q Source: Razavi I n diff I p diff dn Aq e dx dp Aq p dx = Aq dn dx = Aq dp dx The more non uniform is the concentration the more is the current talarico@gonzaga.edu 32

33 Electron and hole diffusion current Assuming the charge concentration decreases with increasing x it means that dn/dx and dp/dx are negative quantities so to conform with conventions we have to put a sign in front of the proportionality constant D I n diff I p diff dn = D n Aq e dx = D naq dn dx dp = D p Aq p dx = D dp paq dx Source: Howe & Sodini talarico@gonzaga.edu 33

34 Diffusion current densities J diff = J p diff + J n diff J n diff J p diff dn = D n q e dx = D nq dn dx dp = D p q p dx = D q dp p dx Source: Howe & Sodini talarico@gonzaga.edu 34

35 Einstein s Relation Since both μ and D are manifestation of thermal random motion (i.e. are due to statistical thermodynamics phenomena) they are not independent D p = D n = KT µ p µ n q Einstein s Relation K=Boltzmann Constant = J/ K = ev/ K T = temperature in K q = charge of proton = Cb V T! KT q Thermal Voltage At room temperature V T 25.9 mv talarico@gonzaga.edu 35

36 Total current density The electron and hole total current density is: J = J p + J n = J p drift + J p diff + J n drift + J n diff J p = J p drift + J p diff J n = J n drift + J n diff = qpµ p E qd p dp dx = qnµ n E + qd n dn dx dp J = qpµ p E qd p dx + qnµ dn ne + qd n dx talarico@gonzaga.edu 36