Lecture 2 OUTLIE Basic Semiconductor Physics (cont d) Carrier drift and diffusion P unction Diodes Electrostatics Caacitance Reading: Chater 2.1 2.2 EE105 Sring 2008 Lecture 1, 2, Slide 1 Prof. Wu, UC Berkeley
Doant Comensation An tye semiconductor can be converted into P tye material by counter doing it with accetors such that A > D. A comensated semiconductor material has both accetors and donors. tye material ( D > A ) P tye material ( A > D ) n D D 2 i n A A n A A 2 i n D D EE105 Sring 2008 Lecture 2, Slide 2 Prof. Wu, UC Berkeley
Tyes of Charge in a Semiconductor egative charges: Conduction electrons (density n) Ionized accetor atoms (density A ) Positive charges: Holes (density ) Ionized donor atoms (density D ) The net charge density (C/cm 3 ) in a semiconductor is ρ q ( n + ) D A EE105 Sring 2008 Lecture 2, Slide 3 Prof. Wu, UC Berkeley
Carrier Drift The rocess in which charged articles move because of an electric field is called drift. Charged articles within a semiconductor move with an average velocity roortional to the electric field. The roortionality constant is the carrier mobility. Hole velocity v h μ E Electron velocity v e μ n E otation: μ hole mobility (cm 2 /V s) μ n electron mobility (cm 2 /V s) EE105 Sring 2008 Lecture 2, Slide 4 Prof. Wu, UC Berkeley
Velocity Saturation In reality, carrier velocities saturate at an uer limit, called the saturation velocity (v sat ). Esat 4 10 V/cm μ0 μ 1+ be μ0 vsat b μ0 v μ0e 1+ v sat E EE105 Sring 2008 Lecture 2, Slide 5 Prof. Wu, UC Berkeley
Drift Current Drift current is roortional to the carrier velocity and carrier concentration: v h ta volume from which all holes cross lane in time t v h t A # of holes crossing lane in time t q v h t A charge crossing lane in time t qv h A charge crossing lane er unit time hole current Hole current er unit area (i.e. current density),drift qv h EE105 Sring 2008 Lecture 2, Slide 6 Prof. Wu, UC Berkeley
Conductivity and Resistivity In a semiconductor, both electrons and holes conduct current: μ μ, drift tot, drift tot, drift q E, drift Conductivity σ qμ + qnμ + σe Resistivity 1 ρ [Unit: Ω-cm] σ Tyical resistivity range for Si: 10 3 ~ 10 3 Ω cm n, drift qn( EE105 Sring 2008 Lecture 2, Slide 7 Prof. Wu, UC Berkeley n n, drift q( μ + nμ ) E n qμ E n E) + qnμ E [unit: mho/cm S/cm] n
Resistivity Examle Estimate the resistivity of a Si samle doed with hoshorus to a concentration of 10 15 cm 3 and boron to a concentration of 10 17 cm 3. The electron mobility and hole mobility are 800 cm 2 /Vs and 300 cm 2 /Vs, resectively. EE105 Sring 2008 Lecture 2, Slide 8 Prof. Wu, UC Berkeley
Electrical Resistance I + V _ W homogeneously doed samle t L Resistance R V I L ρ Wt (Unit: ohms) where ρ is the resistivity EE105 Sring 2008 Lecture 2, Slide 9 Prof. Wu, UC Berkeley
Carrier Diffusion Due to thermally induced random motion, mobile articles tend to move from a region of high concentration to a region of low concentration. Analogy: ink drolet in water Current flow due to mobile charge diffusion is roortional to the carrier concentration gradient. The roortionality constant is the diffusion constant. qd d dx EE105 Sring 2008 Lecture 2, Slide 10 Prof. Wu, UC Berkeley otation: D hole diffusion constant (cm 2 /s) D n electron diffusion constant (cm 2 /s)
Diffusion Examles Linear concentration rofile constant diffusion current x 1 L on-linear concentration rofile varying diffusion current x ex L d, diff d qd dx qd L, diff d qd dx qd ex L d x L d EE105 Sring 2008 Lecture 2, Slide 11 Prof. Wu, UC Berkeley
Diffusion Current Diffusion current within a semiconductor consists of hole and electron comonents:, diff tot, diff qd q( D n d dx dn dx n, diff d dx The total current flowing in a semiconductor is the sum of drift current and diffusion current: tot D ) qd, drift + n, drift +, diff + n dn dx n, diff EE105 Sring 2008 Lecture 2, Slide 12 Prof. Wu, UC Berkeley
The Einstein Relation The characteristic constants for drift and diffusion are related: D μ kt q kt 26mV ote that at room temerature (300K) q This is often referred to as the thermal voltage. EE105 Sring 2008 Lecture 2, Slide 13 Prof. Wu, UC Berkeley
The P unction Diode When a P tye semiconductor region and an tye semiconductor region are in contact, a P junction diode is formed. V D + I D EE105 Sring 2008 Lecture 2, Slide 14 Prof. Wu, UC Berkeley
Diode Oerating Regions In order to understand the oeration of a diode, it is necessary to study its behavior in three oeration regions: equilibrium, reverse bias, and forward bias. V D 0 V D < 0 V D > 0 EE105 Sring 2008 Lecture 2, Slide 15 Prof. Wu, UC Berkeley
Carrier Diffusion across the unction Because of the difference in hole and electron concentrations on each side of the junction, carriers diffuse across the junction: otation: n n electron concentration on tye side (cm 3 ) n hole concentration on tye side (cm 3 ) hole concentration on P tye side (cm 3 ) n electron concentration on P tye side (cm 3 ) EE105 Sring 2008 Lecture 2, Slide 16 Prof. Wu, UC Berkeley
Deletion Region As conduction electrons and holes diffuse across the junction, they leave behind ionized doants. Thus, a region that is deleted of mobile carriers is formed. The charge density in the deletion region is not zero. The carriers which diffuse across the junction recombine with majority carriers, i.e. they are annihilated. quasineutral region widthw de quasineutral region EE105 Sring 2008 Lecture 2, Slide 17 Prof. Wu, UC Berkeley
The Deletion Aroximation In the deletion region on the side: de dx E ρ ε q ε si si D q ε si D ( x + b) Gauss s Law 12 ε si 10 F/cm ρ(x) q D a -b -q A x In the deletion region on the P side: de dx E ρ ε q ε si si A q ε si ( a x) A a A b D EE105 Sring 2008 Lecture 2, Slide 18 Prof. Wu, UC Berkeley
Potential Distribution In the deletion region, the electric otential is quadratic since the electric field is linear The otential difference between the and the P side is called built in otential, V 0 E dv dx V 0 V(x) V E dx -b 0 a x EE105 Sring 2008 Lecture 2, Slide 19 Prof. Wu, UC Berkeley
P unction in Equilibrium In equilibrium, the drift and diffusion comonents of current are balanced; therefore the net current flowing across the junction is zero., drift n, drift, diff n, diff tot, drift + n, drift +, diff + n, diff 0 EE105 Sring 2008 Lecture 2, Slide 20 Prof. Wu, UC Berkeley
Built in Potential, V 0 Because of the electric field in the deletion region, there exists a otential dro across the junction: d dv d qμe qd μ D dx dx dx μ dv D a b n d D kt A V( b) V( a) ln ln μ q n 2 ( / ) n i D V 0 kt q A ln 2 n EE105 Sring 2008 Lecture 2, Slide 21 Prof. Wu, UC Berkeley i D b a (Unit: Volts)
Built In Potential Examle Estimate the built in otential for P junction below. P D 10 18 cm -3 A 10 15 cm -3 18 15 kt D A 10 10 0 2 20 q ni 10 0 ( ) ( ) ( 13 ) V ln 26mV ln 26mV ln 10 kt ote: ln(10) 26mV 2.3 60mV q V 60mV 13 780mV EE105 Sring 2008 Lecture 2, Slide 22 Prof. Wu, UC Berkeley
P unction under Reverse Bias A reverse bias increases the otential dro across the junction. As a result, the magnitude of the electric field increases and the width of the deletion region widens. W 2ε si 1 1 de q + A + D ( V V ) 0 R EE105 Sring 2008 Lecture 2, Slide 23 Prof. Wu, UC Berkeley
Diode Current under Reverse Bias In equilibrium, the built in otential effectively revents carriers from diffusing across the junction. Under reverse bias, the otential dro across the junction increases; therefore, negligible diffusion current flows. A very small drift current flows, limited by the rate at which minority carriers diffuse from the quasi neutral regions into the deletion region. EE105 Sring 2008 Lecture 2, Slide 24 Prof. Wu, UC Berkeley
P unction Caacitance A reverse biased P junction can be viewed as a caacitor. The deletion width (W de ) and hence the junction caacitance (C j ) varies with V R. C j ε W si de 2 [F/cm ] EE105 Sring 2008 Lecture 2, Slide 25 Prof. Wu, UC Berkeley
EE105 Sring 2008 Lecture 2, Slide 26 Prof. Wu, UC Berkeley Voltage Deendent Caacitance ε si 10 12 F/cm is the ermittivity of silicon 0 0 0 0 1 2 1 V q C V V C C D A D A si j R j j + + ε V D
Reverse Biased Diode Alication A very imortant alication of a reverse biased P junction is in a voltage controlled oscillator (VCO), which uses an LC tank. By changing V R, we can change C, which changes the oscillation frequency. 1 1 f res 2π LC EE105 Sring 2008 Lecture 2, Slide 27 Prof. Wu, UC Berkeley
Summary Current flowing in a semiconductor is comrised of drift dn and diffusion comonents: qμ E + qnμ E + qd qd A region deleted of mobile charge exists at the junction between P tye and tye materials. A built in otential dro (V 0 ) across this region is established by the charge density rofile; it ooses diffusion of carriers across the junction. A reverse bias voltage serves to enhance the otential dro across the deletion region, resulting in very little (drift) current flowing across the junction. The width of the deletion region (W de ) is a function of the bias voltage (V D ). 2ε 1 1 si A D W + ( V V ) V de q A EE105 Sring 2008 Lecture 2, Slide 28 Prof. Wu, UC Berkeley tot D 0 D n 0 n kt q dx ln 2 n i d dx