course230B1

ECE 230B: Winter 2013

Solid-State Electronic Devices

Professor Yuan Taur

Electrical & Computer Engineering

Electrical & Computer Engineering

University of California, San Diego

ECE 230B: Winter 2013

Solid-State Electronic Devices

Prerequist: ECE 103 135A B or equivalent and ECE 230A Prerequist: ECE 103, 135A, B, or equivalent and ECE 230A Instructor: Yuan Taur

Office: EBU1-3801 Office: EBU1 3801 Phone: 534-3816

[email protected]

Office hours: Wednesday, Friday, 11-12, or by appointmenty, y, , y pp Course website: http://ted.ucsd.edu

TA: no

This course covers the physics of solid-state electronic devices, including p-n junctions, MOS devices, field-effect transistors, bipolar transistors, etc.

Principles of CMOS and bipolar scaling to nanometer dimensions and their high frequency performance in digital and analog circuits will be taught. q y p g g g

Topics to be Covered

1) Band diagram, Fermi level, Poisson’s

eq., Carrier transport (1.5 wks)

2.1

2) P-n junction (1 wk)

3) MOS device (1 wk)

2.2 2.3

4) Schottky diodes, High field effects (1 wk)

5) MOSFETs (1.5 wks)

2.4-2.5 3

6) CMOS scaling and design (1.5 wks)

7) CMOS performance factors (1 wk)

4 5

8) Bipolar transistors and SiGe (1.5 wks)

Textbook: “Fundamentals of Modern VLSI Devices, 2nd _ed.”

6-8 ,

Homework and Grading Policy

g

y

•

Biweekly homework without handing in.

Answers will be posted.

•

20%--Two quizes (open book, calculator

ll

d)

allowed).

•

30%--Midterm (Closed book, calculator

d

f

id d

t

and one page of one-sided notes

allowed).

•

50% Final (Closed book calculator and

•

50%--Final (Closed book, calculator and

Electron Energy Levels and Bands

gy

Discrete electron energy

levels in an atom:

Broadening into electron

energy bands in a solid:

levels in an atom:

energy bands in a solid:

There are forbidden energy gaps between

There are forbidden energy gaps between

The Origin of Energy Gap in a

Crystalline Solid

The Origin of Energy Gap in a

Crystalline Solid

Metals, Insulators, and Semiconductors

Insulators: E

> 4-5 eV

Metals, Insulators, and Semiconductors

Energy Band Diagram of Silicon

E = 1 12 eV

E

= 1.12 eV

kT/q = 0.026 V

@ 300 K

@ 300 K

Density of States

N(E)dE : Number of electronic states per unit volume with an

energy between E and (E + dE) in the conduction band. From quantum mechanics, there is one allowed state in a phase space of volume ( x p_x)( y p_y)( z p_z) = h3_.

N E dE

g

dp dp dp

h

x y z

( )

2

where dp_xdp_ydp_z is the volume in the momentum space

within which the electron energy lies between E and E + dE,

g is the number of equivalent minima in the conduction band g is the number of equivalent minima in the conduction band,

and the factor of two arises from the two possible directions of electron spin. The conduction band of silicon has a six-fold degeneracy, so g = 6. g y, g

Density of States

If the electron kinetic energy is not too high, one can apply the parabolic approximation,

p

p

p

where (E E_c) is the electron kinetic energy, and m_x, m_y, m_z are the effective masses inversely proportional to the

E E

p

m

p

m

p

m

2

2

2

are the effective masses inversely proportional to the

curvatures of the band. For silicon conduction band in the 100 direction, the longitudinal mass is m_x = 0.92m₀ and the transverse masses are m = m = 0 19m

transverse masses are m_y = m_z = 0.19m₀.

The volume of the ellipsoid in momentum space is (4 /3)(8m_xm_ym_z)1/2_{(E E}

c)3/2. The volume dpxdpydpz within

which the electron energy lies between E and E + dE is which the electron energy lies between E and E + dE is 4 (2m_xm_ym_z)1/2_{(E E} c)1/2dE. N E dE( ) 8 g 2m m m₃x y z E E dE N E dE h E E dEc ( ) ₃

Fermi-Dirac Statistical Distribution

f E

e

( )

1

1 e

1

Fermi Level and Thermal Equilibrium

Consider two electronic systems in contact with Fermi levels

E_f1 and E_f2. The corresponding distribution functions are f_D1(E) and f_D2(E). If E_f1 > E_f2, then f_D1(E) > f_D2(E) for all E.

Electrons flow from 1 to 2.

E kT E E D kT E E D _{f f}

e

E

f

e

E

f

1

1

)

(

1

1

)

(

e

Electron and Hole Densities

Free electron ( ) N(E) f (E) n and p

1/2 1 ( ) E 2 8 ) ( ₃x y z E E_c h m m m g E N

Electron and Hole Densities

f E

e

( )

1

1

f E

( )

e

f E

( )

1

e

Carrying out the integrations,

n N e

p N e

v

E E_{f v} kT

( )/

2

N_c, N_v: Effective density of states, 3/2

3 (2 ) 2 kT h m m m g N_c x y z

V lid f

d i

fi ld

diti

Intrinsic Silicon

n N e

p N e

v

E E_{f v} kT

( )/

Intrinsic silicon: Charge neutrality requires n = p = n_i,

E E E E kT N N i f c v c v 2 2 ln

Intrinsic Fermi level:

Intrinsic carrier n_i N N e_{c v} E E kT N N e c v E kT c v g ( )/2 /2 Intrinsic carrier density: kT E E i i f

e

n

n

p

n

e

n

np

p

Atomic/molecular weight 28.09 60.08

At l l / 3 _{5 0 10}22 _{2 3 10}22

Physical Properties Si SiO₂

Atoms or molecules/cm3 _{5.0 10}22 _{2.3 10}22

Density (g/cm3_{) 2.33 2.27}

Crystal structure Diamond Amorphous

L tti t t (Å) 5 43

Lattice constant (Å) 5.43

---Energy gap (eV) 1.12 8-9

Dielectric constant 11.7 3.9

Intrinsic carrier concentration (cm 3_{) 1 4 10}10

Intrinsic carrier concentration (cm 3_{) 1.4 10}10

---Carrier mobility (cm2_{/V-s) Electron: 1430}

---Hole: 470

---Effective density of states (cm 3_{) Conduction band N : 3 2 10}19

Effective density of states (cm 3_{) Conduction band N}

c: 3.2 1019 ---Valence band N_v: 1.8 1019 ---Breakdown field (V/cm) 3 105 _>107 Melting point ( C) 1415 1600 1700 Melting point ( C) 1415 1600-1700 Thermal conductivity (W/cm- C) 1.5 0.014 Specific heat (J/g- C) 0.7 1.0

Extrinsic (n-type and p-type) Silicon

Band Diagrams of n-type and

p-type Silicon

Conduction band Conduction band

Ec _E Ec Eg Eg c E Ei Ei E_d Ea E_f E_f

Valence band _{Valence band}

Ev Ev a

Free electron ( ) Free hole (+)

( ) t (b) t

Fermi Level in Extrinsic Silicon

n N e

p N e

v

E E_{f v} kT

( )/

For n-type silicon with donor concentration N_d, charge neutrality requires n = p + N_d+ _N

d (assuming complete ionization).

Therefore,

E

E

kT

N

N

ln

Or, in terms of n_i and E_i,

E

E

kT

N

n

ln

Note that np = n2 _{independent of E (n- or p-type)}

For N_d > N_c, E_f > E_c degenerate n+ _{silicon. Need full F-D eq.}

Fermi Level in Extrinsic Silicon

If incomplete ionization, N_d+ _{< N}

d where

E_f is then solved from

N

N

f E

N

e

1

1

1

1

1 2

(

)

( / )

N e

N

e

N e

1 2

Condition for complete ionization:

N

e

d (E_c E_d )/kT

₁

Requires the donor energy level to be within a few kT of the bottom of the conduction band

N

e

c d

( )

₁

Carrier Transport: Drift

kT

v

m

2

3

2

1

Mean free path :

Average distance between collisions, ~10-5 _cm

Mean free time _cn:

Average time between collisions, ~ 1 ps

*

/ m

q

Mobilityy

E

v

1E+140 1E+15 1E+16 1E+17 1E+18 1E+19 1E+200

Doping concentration (cm )-3

1

1

1

1

1

1

Resistivity of Silicon

y

E

qnv

qn

J

Velocity Saturation

y

1E+2 1E+3 1E+4 1E+5

1E+5 C a Holes T=300 K

cm/s

10

v

1E+2 1E+3 1E+4 1E+5

Electric field (V/cm)

cm/s

10

Carrier Transport: Diffusion

J

qD

dn

dx

D

kT

q

q

D

kT

q

Poisson’s Equation

i

i

E

q

Define intrinsic potential:

q

dx

d

-=

E

dx

x

dx

d

dx

d

(

)

E

)

(

)

(

)

(

)

(

x

N

x

N

x

n

x

p

q

dx

d

dx

d

E

Q

dx

x)

(

1

E

2-D Poisson’s Equation

) , 0 ( ) , 0 ( _{2 2} 1 1Ex y Ex y

Also (x)E_x must be continuous at x = 0

FIGURE 2.11. Diagram for

discussing the boundary conditions (x y) (x y)

1 2

of electric field at the interface between two dielectric media. (x,y)

1 2(x,y)

x y

Carrier Concentration in terms

of Electrostatic Potential

n

n e

n e

Oxide p-type silicon

x

Debye Length

Current Density Equations

Combining drift and diffusion components,

dx dn qD + qn J_{n nE n} Or, dx dx dp qD qp J_{p pE p} dn kT d Or, dx dn qn kT dx d qn J i n n dx dp qp kT dx d qp J i p p dx d qn J_{n n} f d d qp J_{p p} f dx d J J J _{n p} 1 f

If both n and p take on their equilibrium values, then , and

dx qp dx

dx p p dx dx

Current proportional to the gradient of Fermi level assuming local equilibrium and local Fermi level.q

Current Density Equations

When electrons and holes are not in local equilibrium, define quasi-Fermi levels E_fn, E_fp such that

dx dp qD qp

J_{p pE p}

define quasi Fermi levels E_fn, E_fp such that

Then _{d kT dn d} kT E E i i fn e n n ( )/ p n_ie(Ei Efp)/kT J qn d dx kT qn dn dx qn d dx n n i n n J qp d dx kT qp dp dx qp d dx p p i p p where dx qp dx dx p p p i i fn n _n n q kT q E ln i i fp p _n p q kT q E ln

are the quasi-Fermi potentials.

Current Continuity Equations

From conservation of mobile charge,

n J R G n 1 t q x R G n n n p J R G p 1

where G

, G

, R

, R

are electron-hole generation

and recombination rates.

t q x Rp Gp

In the steady state with negligible electron-hole

generation and recombination rates,

g

,

dJ

/dx = 0

dJ

/dx = 0

Si

l

ti it

f l t

d h l

t

Dielectric Relaxation Time

Neglect R

, G

in the current continuity equation,

n

1

J

From Ohm’s law, J

_{= E/}

n

t

q

J

x

1

And from Poisson’s equation with majority carrier

density n, E/ x = qn/

S bstit ting into c rrent contin it eq ation

Substituting into current continuity equation:

n

t

n

The solution is of the form n(t) exp( t/

)

where

10

_{s is the majority carrier}