Lecture 2

OUTLINE • Basic Semiconductor Physics (cont’d) – Carrier drift and diffusion • PN Junction Diodes – Electrostatics – Capacitance

Reading: Chapter 2.1‐2.2

EE105 Spring 2008Lecture 1,2, Slide 11Prof. Wu, UC Berkeley Dopant Compensation

• An N‐type semiconductor can be converted into P‐ type material by counter‐doping it with acceptors

such that NA > ND. • A compensated semiconductor material has both acceptors and donors. N‐type material P‐type material

(ND > NA) (NA > ND)

−≈ NNn AD ≈ − NNp DA n 2 n 2 p ≈ i n ≈ i − NN AD − NN DA

EE105 Spring 2008 Lecture 2, Slide 2Prof. Wu, UC Berkeley Types of Charge in a Semiconductor

• Negative charges: – Conduction electrons (density = n)

– Ionized acceptor atoms (density = NA) • Positive charges: – Holes (density = p)

– Ionized donor atoms (density = ND)

• The net charge density (C/cm3) in a semiconductor is

ρ = ( − + − NNnpq AD )

EE105 Spring 2008 Lecture 2, Slide 3Prof. Wu, UC Berkeley Carrier Drift

• The process in which charged particles move because of an electric field is called drift. • Charged particles within a semiconductor move with an average velocity proportional to the electric field. – The proportionality constant is the carrier mobility. → → Hole velocity = μ ph Ev → →

Electron velocity e −= μn Ev

Notation: 2 μp ≡ hole mobility (cm /V∙s) 2 μn ≡ electron mobility (cm /V∙s)

EE105 Spring 2008 Lecture 2, Slide 4Prof. Wu, UC Berkeley Velocity Saturation

• In reality, carrier velocities saturate at an upper limit, called the saturation velocity (vsat).

μ μ = 0 1+ bE μ v = 0 sat b μ v = 0 E μ E 4 1+ 0 Esat ≈10 V/cm vsat

EE105 Spring 2008 Lecture 2, Slide 5Prof. Wu, UC Berkeley Drift Current • Drift current is proportional to the carrier velocity and carrier concentration:

vh tA= volume from which all holes cross plane in time t pvh t A = # of holes crossing plane in time t q p vh t A = charge crossing plane in time t qpvh A = charge crossing plane per unit time = hole current

Î Hole current per unit area (i.e. current density) Jp,drift = qpvh EE105 Spring 2008 Lecture 2, Slide 6Prof. Wu, UC Berkeley Conductivity and Resistivity

• In a semiconductor, both electrons and holes conduct current:

,driftp = μ p ,driftn = − −μn EqnJEqpJ )(

,drifttot ,driftp ,driftn p +=+= μμn EqnEqpJJJ

,drifttot np )( ≡+= σμμEEnpqJ • Conductivity

σ ≡+qpμμpn qn [unit: mho/cm = S/cm] • Resistivity 1 ρ ≡Ω [Unit: -cm] σ • Typical resistivity range for Si: 10‐3 ~ 103 Ω‐cm

EE105 Spring 2008 Lecture 2, Slide 7Prof. Wu, UC Berkeley Resistivity Example

• Estimate the resistivity of a Si sample doped with phosphorus to a concentration of 1015 cm‐3 and boron to a concentration of 1017 cm‐3. The electron mobility and hole mobility are 800 cm2/Vs and 300 cm2/Vs, respectively.

EE105 Spring 2008 Lecture 2, Slide 8Prof. Wu, UC Berkeley Electrical Resistance V I _ +

W t homogeneously doped sample

L

V L Resistance R =≡ ρ (Unit: ohms) I Wt

where ρ is the resistivity

EE105 Spring 2008 Lecture 2, Slide 9Prof. Wu, UC Berkeley Carrier Diffusion

• Due to thermally induced random motion, mobile particles tend to move from a region of high concentration to a region of low concentration. – Analogy: ink droplet in water • Current flow due to mobile charge diffusion is proportional to the carrier concentration gradient. – The proportionality constant is the diffusion constant. dp −= qDJ p p dx Notation: 2 Dp ≡ hole diffusion constant (cm /s) 2 Dn ≡ electron diffusion constant (cm /s) EE105 Spring 2008 Lecture 2, Slide 10 Prof. Wu, UC Berkeley Diffusion Examples

• Linear concentration profile • Non-linear concentration profile Æ constant diffusion current Æ varying diffusion current ⎛ x ⎞ − x Np ⎜1−= ⎟ = Np exp ⎝ L ⎠ Ld

dp dp −= qDJ −= qDJ ,diffp p dx ,diffp p dx

N p NqD − x = qDp = exp L Ld Ld

EE105 Spring 2008 Lecture 2, Slide 11 Prof. Wu, UC Berkeley Diffusion Current

• Diffusion current within a semiconductor consists of hole and electron components: dp dn −= qDJ = qDJ ,diffp p dx ,diffn n dx dn dp (DqJ −= D ) ,difftot n dx p dx • The total current flowing in a semiconductor is the sum of drift current and diffusion current:

= ,driftptot + ,driftn + , + JJJJJ ,diffndiffp

EE105 Spring 2008 Lecture 2, Slide 12 Prof. Wu, UC Berkeley The Einstein Relation

• The characteristic constants for drift and diffusion are related: kTD = μ q

kT • Note that ≅ mV26 at room temperature (300K) q – This is often referred to as the “thermal voltage”.

EE105 Spring 2008 Lecture 2, Slide 13 Prof. Wu, UC Berkeley The PN Junction Diode

• When a P‐type semiconductor region and an N‐type semiconductor region are in contact, a PN junction diode is formed. VD – +

ID

EE105 Spring 2008 Lecture 2, Slide 14 Prof. Wu, UC Berkeley Diode Operating Regions

• In order to understand the operation of a diode, it is necessary to study its behavior in three operation regions: equilibrium, reverse bias, and forward bias.

VD = 0 VD < 0 VD > 0

EE105 Spring 2008 Lecture 2, Slide 15 Prof. Wu, UC Berkeley Carrier Diffusion across the Junction

• Because of the difference in hole and electron concentrations on each side of the junction, carriers diffuse across the junction:

Notation: ‐3 nn ≡ electron concentration on N‐type side (cm ) ‐3 pn ≡ hole concentration on N‐type side (cm ) ‐3 pp ≡ hole concentration on P‐type side (cm ) ‐3 np ≡ electron concentration on P‐type side (cm )

EE105 Spring 2008 Lecture 2, Slide 16 Prof. Wu, UC Berkeley Depletion Region

• As conduction electrons and holes diffuse across the junction, they leave behind ionized dopants. Thus, a region that is depleted of mobile carriers is formed. – The charge density in the depletion region is not zero. – The carriers which diffuse across the junction recombine with majority carriers, i.e. they are annihilated.

quasi- quasi- neutral neutral region width=Wdep region EE105 Spring 2008 Lecture 2, Slide 17 Prof. Wu, UC Berkeley The Depletion Approximation

In the depletion region on the N side: dE ρ qN == D Gauss’s Law dx si εεsi

qN D 12 E ()+= bx ε si =10 F/cm ε si

ρ(x) In the depletion region on the P side: qN D dE ρ − qN a == A dx εε -b x si si -qNA qN E A ()−= xa ε si

A = bNaN D

EE105 Spring 2008 Lecture 2, Slide 18 Prof. Wu, UC Berkeley Potential Distribution

• In the depletion region, the electric potential is quadratic since the electric field is linear • The potential difference between the N and the P

side is called built‐in potential, V0

dV V(x) E =− V0 dx VEdx=− ⋅ ∫ -b a x 0

EE105 Spring 2008 Lecture 2, Slide 19 Prof. Wu, UC Berkeley PN Junction in Equilibrium

• In equilibrium, the drift and diffusion components of current are balanced; therefore the net current flowing across the junction is zero.

,driftp −= JJ ,diffp

,driftn −= JJ ,diffn

= ,driftptot + ,driftn + , + JJJJJ ,diffndiffp = 0

EE105 Spring 2008 Lecture 2, Slide 20 Prof. Wu, UC Berkeley Built‐in Potential, V0 • Because of the electric field in the depletion region, there exists a potential drop across the junction: dp ⎛⎞dV dp −b a qpμμpp E=⇒−= qD p p⎜⎟D p dx ⎝⎠dx dx p a p dp ⇒− μ dV = D pp∫∫p −bpn DpkT N ⇒−−= VbVa ( ) ( )pp ln = ln A 2 μ pnpq()nNiD/

kT NN DA ln V0 = ln 2 (Unit: Volts) q ni

EE105 Spring 2008 Lecture 2, Slide 21 Prof. Wu, UC Berkeley Built‐In Potential Example

• Estimate the built‐in potential for PN junction below.

N P

18 -3 15 -3 ND = 10 cm NA = 10 cm

18 15 kT ⎛⎞NNDA ⎛⎞10 10 13 V0 ==ln ⎜⎟22()26mV ln ⎜⎟0 =()26mV ln() 10 qn⎝⎠i ⎝⎠10 kT Note: ln(10)≅×≅ 26mV 2.3 60mV q

V0 =×=60mV 13 780mV

EE105 Spring 2008 Lecture 2, Slide 22 Prof. Wu, UC Berkeley PN Junction under Reverse Bias

• A reverse bias increases the potential drop across the junction. As a result, the magnitude of the electric field increases and the width of the depletion region widens.

2ε ⎛ 11 ⎞ si ⎜ ⎟ Wdep = ⎜ + ⎟()0 +VV R ⎝ NNq DA ⎠

EE105 Spring 2008 Lecture 2, Slide 23 Prof. Wu, UC Berkeley Diode Current under Reverse Bias

• In equilibrium, the built‐in potential effectively prevents carriers from diffusing across the junction. • Under reverse bias, the potential drop 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 depletion region.

EE105 Spring 2008 Lecture 2, Slide 24 Prof. Wu, UC Berkeley PN Junction Capacitance • A reverse‐biased PN junction can be viewed as a capacitor. The depletion width (Wdep) and hence the junction capacitance (Cj) varies with VR.

ε si C j = Wdep [F/cm2 ]

EE105 Spring 2008 Lecture 2, Slide 25 Prof. Wu, UC Berkeley Voltage‐Dependent Capacitance

C C = j0 j V 1+ R V0

ε siq NN DA 1 VD C j0 = 2 + DA VNN 0

‐12 εsi ≅ 10 F/cm is the permittivity of silicon

EE105 Spring 2008 Lecture 2, Slide 26 Prof. Wu, UC Berkeley Reverse‐Biased Diode Application

• A very important application of a reverse‐biased PN junction is in a voltage controlled oscillator (VCO),

which uses an LC tank. By changing VR, we can change C, which changes the oscillation frequency.

1 1 f = res 2π LC

EE105 Spring 2008 Lecture 2, Slide 27 Prof. Wu, UC Berkeley Summary • Current flowing in a semiconductor is comprised of drift dn dp and diffusion components: tot p μμn qDEqnEqpJ n −++= qDp dx dx • A region depleted of mobile charge exists at the junction between P‐type and N‐type materials.

– A built‐in potential drop (V0) across this region is established by the charge density profile; it opposes diffusion of carriers across the junction. A reverse bias voltage serves to enhance the potential drop across the depletion region, resulting in very little (drift) current flowing across the junction.

– The width of the depletion region (Wdep) is a function of the bias voltage (VD). 2ε ⎛ 11 ⎞ kT NN W = si ⎜ + ⎟()−VV DA dep ⎜ ⎟ 0 D V0 = ln 2 ⎝ NNq DA ⎠ q ni EE105 Spring 2008 Lecture 2, Slide 28 Prof. Wu, UC Berkeley