B. Diffusion Current Density- Holes ¥ Current Density = (Charge) X (# Carriers Per Second Per Area)

I. Carrier Transport: Drift B.A. DiffusionDrift Velocity Current Density- Holes ¥ Current density = (charge) x (# carriers per second per area): 1 1 (a) Thermal Equilibrium, E =- -0 px()ÐλAλÐ--(b)px Electric()+ FieldλA E λ> 0 diff 2 2 J = q ------Electron # 1 p Electronτ # 1 Ac

E ¥ If we assume the mean free path is much smaller than the dimensions x x x x of our device,i f,then1 we can consider λ = dx xandi xf, 1use Electrona Taylor# 2 expansion on p(x - λ) andElectron p(x #+ 2 λ):

E diff dp x λ2x τ x ¥ J xf,2 =xi ÐqD , where Dp = f, 2/ c is xthei diffusion coefficent p pdx

Electron # 3 Electron # 3

¥ Holes diffuse down the concentration gradient and carryE positive charge. x xf,3 xi x xf,3 xi

* xi = initial position * xf, n = final position of electron n after 7 collisions

¥ Electrons drifting in an Electric Field move (on average) with a drift velocity which is proportional to the Electric Field, E.

EECS 6.012 Spring 1998 Lecture 3

Drift Velocity (Cont.)

Velocity in Direction of E Field

t qE vat==± ------t m

qτ c vave =Ð ------E 2mn

¥ mn is an effective mass to take into account quantum mechanics µ 2 ¥ Lump it into a quantity we call mobility n (units:cm /V-s) µ vd = - nE

EECS 6.012 Spring 1998 Lecture 3

B. Electron and Hole Mobility ¥ mobilities vary with doping concentration-- plot is for 300K

1400

1200

electrons 1000 Vs) / 2 800

600 mobility (cm holes 400

200

0 1013 1014 1015 1016 1017 1018 1019 1020 + −3 Nd Na total dopant concentration (cm )

• “typical values” for bulk silicon -- 300 K

µ 2 n = 1000 cm /(Vs) µ 2 p = 400 cm /(Vs)

EECS 6.012 Spring 1998 Lecture 3

C. Velocity Saturation ¥ At electric ﬁelds greater >~ 104 V/cm ¥ Drift velocities saturate --> max. out at around 107 V/cm. ¥ Velocity saturation is very common in VLSI devices, due to sub- micron dimensions

− vdn, vdp (cm/s) 108

107

electrons 106 holes 105

104

103

10 102 103 104 105 E (V/cm)

EECS 6.012 Spring 1998 Lecture 3

D. Drift Current Density • Electrons drift against electric ﬁeld ¥ Electrons carry negative charge dr -2 -1 -2 ¥ Jn = (-q) n vd units: Ccm s = Acm

dr µ µ Jn = (-q) n (- n E) = q n n E

• Holes drift with electric ﬁeld µ ¥ vd = p E dr ¥ Jp = (+q) p vd

dr µ Jp = q p p E

Caution:: The linear relationship between field and drift velocity breaks down for high electric fields.

EECS 6.012 Spring 1998 Lecture 3 II. Carrier Transport: Diffusion A. Diffusion ¥ Diffusion is a transport process driven by gradients in the concentration of particles in random motion undergoing frequent collisions. 7 ¥ Average carrier velocity = vth = 10 cm/s τ -13 ¥ Average interval between collisions = c = 10 s = 0.1 picoseconds λ τ -6 ¥ mean free path = = vth c = 10 cm = 10 nm

reference plane (area = A) p(x) hole diffusion

diff Jp (positive) − λ p(xr ) volume Aλ: volume Aλ: holes moving p(x + λ) r holes moving in + x direction cross in − x direction cross reference plane within ∆ = τ reference plane within t c. ∆ = τ t c.

− λ + λ x xr xr xr

¥ Since their motion is random, half of the carriers in each volume will pass through the plane before their next collision.

EECS 6.012 Spring 1998 Lecture 3 B. Diffusion Current Density- Holes ¥ Current density = (charge) x (# carriers per second per area):

1 1 -- []px()Ðλ AλÐ--[]px()+λ Aλ diff 2 2 J = q ------p τ Ac : ¥ If we assume the mean free path is much smaller than the dimensions of our device, then we can consider λ = dx and use a Taylor expansion on p(x - λ) and p(x + λ):

diff dp λ2 τ ¥ J = ÐqD where Dp = / c is the diffusion coefficent p pdx

¥ Holes diffuse down the concentration gradient and carry positive charge.

p(x) diff Jp ( > 0)

diff Jp ( < 0)

EECS 6.012 Spring 1998 Lecture 3 C. Diffusion Current Density - Electrons ¥ Electrons diffuse down the concentration gradient, yet carry negative charge --> electron diffusion current density points in the direction of the gradient

n(x) diff Jn ( < 0)

diff Jn ( > 0)

D. Einstein Relation ¥ Both mobility and the diffusion constant are related to the mean time τ between collisions c. There is a relation between these important quantities called the Einstein Relationship.

D D p kT n kT ------=------and ------= ------µ q µ q p n

EECS 6.012 Spring 1998 Lecture 3 III. Total Current Density

dr diff dn J ==J + J qnµ EqD+ n n n n ndx

dr diff dp J==J+ J qpµ EqDÐ ppp p pdx

¥ Fortunately, we will be able to eliminate one or the other component in ﬁnding the internal currents in microelectronic devices.

EECS 6.012 Spring 1998 Lecture 3