Solid State Device Fudametals ENS 345 Lecture Course by Alexader M. Zaitsev alexader.zaitsev@csi.cuy.edu Tel: 718 982 2812 4N101b 1
Thermal motio of electros Average kietic eergy of electro or hole (thermal eergy, E T ): Thermal speed of free electro v th : v th 3kT m 23 31.3810 JK 0.269.110 1 * 31 E 300K kg 3 2 kt 1 2 2 T mv th 2.310 5 m/s 230 km/s 2
Drift of charge carriers i electric field Radom motio of carriers with ad without a applied electric field. Drift is the motio caused by a electric field. Zig-zag motio is due to collisios with imperfectios i the crystal. Net thermal velocity is zero. Mea free time betwee collisios is m ~ 0.1ps Mea free path betwee collisios is l m = m th ~ 1 m 3
Electro ad hole mobility v av Mometum: acceleratio scatterig m* v e E m Average Speed: v av e m* m E v h E p e h m h v e E e e m e h ( h ) is the hole mobility ad e ( ) is the electro mobility 4
Charge carrier mobility i semicoductors v = E ; has dimesios of v/e cm/s V/cm 2 cm V s Electro ad hole mobility at room temperature i selected semicoductors Si Ge GaAs IAs (cm 2 /V s) 1400 3900 8500 30000 p (cm 2 /V s) 470 1900 400 500 Based o the above table aloe, which semicoductor ad which carriers (electros or holes) are attractive for applicatios i high-speed devices? 5
Drift velocity, mea free time, mea free path EXAMPLE: Give p = 470 cm 2 /V s, what is the hole drift velocity at E = 10 3 V/cm? What is p ad what is the distace traveled betwee collisios (mea free path)? Solutio: = p E = 470 cm 2 /V s 10 3 V/cm = 4.7 10 5 cm/s m = p m* p /e =470 cm 2 /V s 0.39 9.110-31 kg/1.610-19 C = 0.047 m 2 /V s 2.210-12 kg/c = 110-13 s = 0.1 ps mea free path = m th ~ 1 10-13 s 2.210 7 cm/s = 2.210-6 cm = 220 Å = 22 m This is smaller tha the typical dimesios of devices, but gettig close. 6
Homework Based o values of mobilities, estimate mea free time, mea free path ad drift velocities of electros ad holes i Si, Ge ad GaAs i a electric field of 1000 V/cm. 7
Charge carrier scatterig Two mai causes of carrier scatterig: 1. Phoo Scatterig Scatterig by lattice vibratios (phoos) ca be preseted as a scatterig o higher desities of lattice atoms. _ - Electro Boro Io _ 2. Ioized-Impurity scatterig Impurities are efficiet scatterig ceters especially whe charged. Ioized doors ad acceptors i semicoductor are a commo example of such impurities. 8
Impurity-io scatterig or Coulomb scatterig Scatterig due to coulomb iteractio betwee charge carriers ad ioized impurities Slow electro μ(t)~t Fast electro μ(n)~1/n Slow electro Fast electro There is less chage i the directio of travel for a electro with high speed. impurity N a v 3 th N d T N a 3/ 2 N d 9
Depedece of mobility o impurity cocetratio What -type silico is more coductive, doped with cocetratio of 10 16, or 10 18 cm -3? Mobility i silico for differet dopats ad dopig desities 10
Temperature effect o mobility Desity of phoos icreases with temperature. Thus the scatterig time ad the mobility decrease with temperature. phoo phoo phoo desity carrier 1 1 3/ 2 T 1/ 2 thermal velocity T T = e/m v th T 1/2 T 11
Combiatio of two scatterig effects τ(n) τ(t) dt/τ(n) dt/τ(t) dt/τ = dt/τ(n) + dt/τ(t) μ = eτ m 1 μ = 1 μ(n) + 1 μ(t) 12
Temperature depedece of mobility of electros i Si 1 1 1 phoo 1 phoo 1 impurity 1 impurity What N d will make dμ /dt = 0 at room temperature? 13
Measurig mobility: Hall Effect R H = E y J x B z Hall coefficiet Hall coefficiet i bipolar semicoductors μ = R H ρ Mobility of charge carriers ca be foud from the resistivity ad the Hall coefficiet. = 1 er H τ = m R H eρ Cocetratio of charge carriers ca be foud from the Hall coefficiet. The relaxatio time ca be foud from the resistivity ad the Hall coefficiet: 14
Homework 1. Estimate the value of Hall coefficiet for itrisic Si, Ge ad GaAs at room temperature. 2. Estimate the value of Hall coefficiet at room temperature for -type Si, Ge ad GaAs doped to a cocetratio of 10 18 cm -3. 15
Saturatio velocity Whe the kietic eergy of a carrier exceeds a critical value, it geerates a optical phoo ad loses the kietic eergy. Therefore the kietic eergy is capped at large E ad the velocity does ot rise above a saturatio velocity, v sat. v(e) = μe 1 + μe v sat A estimate of the saturatio velocity ca be obtaied by calculatig the carrier velocity of a eergy equal to the optical phoo eergy. v sat = 2E phoo m 16
Drift curret ad coductivity E J p + uit area + Curret desities J p = epv J = ev EXAMPLE: If p = 10 15 cm -3 ad v = 10 4 cm/s, the J p = 1.610-19 C 10 15 cm -3 10 4 cm/s 2 2 = 1.6 C/s cm 1.6 A/cm 17
Drift curret ad coductivity i semicoductor J p,drift = epv = ep p E J,drift = ev = -e E J drift = J,drift + J p,drift = E =e( +p p )E Coductivity [S/cm] of a semicoductor is: = e( + p p ) 18
Coductivity ad Resistivity of Si versus dopig desity 19
Sheet resistace The sheet resistace cocept is used to characterize wafers ad thi doped layers. It is techically easier to measure the sheet resistace rather tha the resistivity. Sheet resistace, R S is measured i uits [Ohm/square, Ω/ ]. R S = ρ l wt = (l = w) = ρ t w l Sheet resistace of a 350 micro thick -type ad p-type silico wafer versus dopig desity. 20
Example: Resistace (a) What is the resistivity () of silico doped with N d = 10 17 cm -3 of arseic? (b) What is the resistace (R) of a piece of this silico material 1m log ad 0.1 m 2 i crosssectioal area? Solutio: (a) Usig the N-type curve i the previous figure, we fid that = 0.084 -cm. (b) R = L/A = 0.084 -cm 1 m / 0.1 m 2 = 0.084 -cm 10-4 cm/ 10-10 cm 2 = 8.4 10-4 21
Homework 1. Estimate resistivity of itrisic Si, Ge ad GaAs at room temperature. 2. Estimate sheet resistace of 300 µm thick wafers of itrisic Si, Ge ad GaAs at room temperature. 3. Estimate resistivity at room temperature of p-type Si doped to a cocetratio of 10 18 cm -3. 4. Estimate sheet resistace at room temperature of 500 µm thick wafer made of p-type Si as described i 3. 22
Temperature depedece of coductivity Schematic illustratio of the temperature depedece of electrical coductivity for a doped semicoductor. I the temperature rage of full activatio of dopats, coductivity decreases with temperature because of reductio i mobility. 23
Example: Temperature depedece of resistace By what factor will R chage with temperature icrease from T=300 K to T=400 K? Solutio: The temperature depedet factor i (ad therefore ) is. From the mobility vs. temperature curve for 10 17 cm -3, we fid that decreases from 770 at 300K to 400 at 400K. As a result, R icreases by 770 400 1.93 24
Diffusio of charge carriers Particles diffuse from a higher-cocetratio locatio to a lowercocetratio locatio. 25
Diffusio curret J, diffusio ed d dx J p, diffusio ed p dp dx D is the diffusio coefficiet p 26
Total curret sum of four curret compoets J total = J + J p J = J,drift + J,diffusio = e E - J p = J p,drift + J p,diffusio = ep p E + ed ed p d dx dp dx 27
Eistei relatioship betwee D ad N e c ( E c E f ) / kt d dx kt ee J e E ed 0 e E e d dx ed kt E 0 at equilibrium. D kt e D p kt e p These are kow as the Eistei relatioship. 28
EXAMPLE: Diffusio coefficiet Calculate hole diffusio coefficiet i a piece of silico with p = 410 cm 2 V -1 s -1 D p kt e p (26 mv) 410 cm 2 V 1 s 1 11cm 2 /s Remember: k B T 26 mev at room temperature. 29
Homework 1. Estimate diffusio coefficiets i itrisic Si, Ge ad GaAs at room temperature. 2. Estimate diffusio coefficiets at room temperature of p-type Si doped to a cocetratio of 10 18 cm -3. 30
EXAMPLE: Thermo-voltage Predict magitude of thermo-voltage across a 1 cm log silico bar doped with phosphorous at a cocetratio of 10 18 cm -3, the eds of which are kept at temperatures 20 ad 25 C. N C N 2 D 1/ 2 e ( E C E D )/ 2 kt D kt e At equilibrium: J e E ed d dx 0 E V l V ~ 10 mv 31
Charge carrier motio summary J, diffusio qd d dx v p v pe - E J p, diffusio qd p dp dx J J p, drift, drift qp pe qe D kt q D p kt q p 32