Department of Electronics

Introductory Nanotechnology

~ Basic Condensed Matter Physics

~

Atsufumi Hirohata

Quick Review over the Last Lecture

Classic model : Dulong-Petit empirical law

T

c V, mol

3R

0 E

DEinstein model :

• E : Einstein temperature

• c V, mol ~ 3R for E << T

• c V, mol exp (- E / T) for E << T

Debye model :

• D : Debye temperature

• c V, mol ~ 3R for D << T

• c V, mol T 3 for D << T

Contents of Introductory Nanotechnology

First half of the course : Basic condensed matter physics

1. Why solids are solid ?

2. What is the most common atom on the earth ?

3. How does an electron travel in a material ?

4. How does lattices vibrate thermally ?

5. What is a semi-conductor ?

6. How does an electron tunnel through a barrier ?

Second half of the course : Introduction to nanotechnology (nano-fabrication / application)

7. Why does a magnet attract / retract ?

8. What happens at interfaces ?

What Is a Semi-Conductor ?

• Elemental / compound semiconductor

• Intrinsic / extrinsic semiconductors

• n / p-dope

• Temperature dependence

• Schottky junctions

• pn junctions

What is semi-conductor ?

Band diagrams :

metal conductors insulators

Forbidden

Allowed

Allowed

semiconductors

Forbidden

Allowed

Allowed

Forbidden

AllowedForbidden

Allowed

Allowed

With very small energy,

electrons can overcome

the forbidden band.

EF

Energy Band of a semiconductor

Schematic energy band diagram :

E

Conduction band

Valence band

Band gapconduction electron

hole

Elemental Semiconductors

In the periodic table,

Semimetal : conduction and valence bands are slightly over ｌ aped.

As (semimetal) ~ 10 20 cm -3

Sb (semimetal) ~ 10 19 cm -3

C (semimetal) ~ 10 18 cm -3

Bi (semimetal) ~ 10 17 cm -3

Carrier density : Cu (metal) ~ 10 23 cm -3

Ge (semiconductor) ~ 10 13 cm -3

Fabrication of a Si-Based Integrated Circuit

Czochralski method :Si purity (99.999999999 %)

* http://www.wikipedia.org/

Compound Semiconductors

In the periodic table,

III-V compounds : GaAs, InAs, InSb, AlP, BP, ...

II-VI compounds : ZnO, CdS, CdTe, ...

IV-IV compounds : SiC, GeSi

IV-VI compounds : PbSe, PbTe, SnTe, ...

Shockley Model

Contributions for electrical transport :

E

Conduction band

Valence band

Band gapconduction electron (number density : n)

positive hole(number density : p)

Ambipolar conduction

Intrinsic semiconductor

€

σ =σ e +σ h = nqμ e + pqμ h = niq μ e + μ h( ) n i ≡ n = p( )

* http://www.wikipedia.org/

Carrier Number Density of an Intrinsic Semiconductor

Carrier number density is defined as

€

n = f E( )g E( )dE∫Here, the Fermi distribution function is

€

f E( ) =1

exp E − EF( ) kBT[ ] +1For the carriers like free electrons with m*, the density of states is

€

g E( ) = 21

2π( )2

2m*

h2

⎛

⎝ ⎜

⎞

⎠ ⎟

3 2

E

For electrons with effective mass me*, g(E) in the conduction band is written

with respect to the energy level EC,

* H. Ibach and H. Lüth, Solid-State Physics (Springer, Berlin, 2003).

€

gC E( ) = 21

2π( )2

2me*

h2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

3 2

E − EC

For holes with effective mass mp*,

g(E) in the valence band is written

with respect to the energy level EV = 0,

€

gV E( ) = 21

2π( )2

2mp*

h2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

3 2

−E

Carrier Number Density of an Intrinsic Semiconductor (Cont'd)

fp(E) for holes equals to the numbers of unoccupied states by electrons :

€

fp E( ) = 1− fe E( )

n is an integral in the conduction band from the bottom EC to top Ect :

€

n = fe E( )ge E( )dEEC

ECt∫ =1

2π 2

2me*

h2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

3 2

E − EC1

exp E − EF( ) kBT[ ] +1dE

EC

ECt∫p is an integral in the valence band from the bottom -EVb to top 0 :

€

p = fp E( )gp E( )dE−EVb

0

∫ =1

2π 2

2mp*

h2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

3 2

−E 1−1

exp E − EF( ) kBT[ ] +1

⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪dE

−Evb

0

∫

=1

2π 2

2mp*

h2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

3 2

−E1

exp − E − EF( ) kBT[ ] +1dE

−Evb

0

∫

Here, EC (= Eg = EC - EV) >> kBT E - EF EC /2 for EC E Ect (EF ~ EC /2)

€

fe E( ) ≈ exp − E − EF( ) kBT[ ]

Similarly, EC >> kBT -(E - EF) EC /2 for EVb E 0

€

fp E( ) ≈ exp E − EF( ) kBT[ ]

Carrier Number Density of an Intrinsic Semiconductor (Cont'd)

For E - EF > 3kBT,

€

fe EF + 3kBT( ) < 0.05 and hence ECt ∞

Similarly,

€

fp EF − 3kBT( ) < 0.05 and hence EVb -∞

€

n =1

2π 2

2me*

h2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

3 2

E − EC exp − E − EF( ) kBT[ ]dEEC

∞

∫

€

p =1

2π 2

2mp*

h2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

3 2

−E exp E − EF( ) kBT[ ]dE−∞

0

∫As a result,

€

n = NC exp − EC − EF( ) kBT[ ] ≈ NC fe EC( )

€

NC ≡ NCeT3 2

NCe ≡ 22πme

*kB

h2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

3 2

⎧

⎨ ⎪ ⎪

⎩ ⎪ ⎪

€

n = NV exp −EF kBT[ ] ≈ NV fp 0( )

€

NV ≡ NVpT3 2

NVp ≡ 22πmp

*kB

h2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

3 2

⎧

⎨ ⎪ ⎪

⎩ ⎪ ⎪

Fermi Level of an Intrinsic Semiconductor

For an intrinsic semiconductor,

€

n = p ≡ ni

€

NC exp − EC − EF( ) kBT[ ] = NV exp −EF kBT[ ]

€

∴EF =1

2EC +

3

4kBT ln

mp*

me*

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

Assuming, me* = mp

* = m*

€

EF =1

2EC =

1

2Eg

np product is calculated to be

€

np = ni2 = NCNV exp −EC kBT[ ] = 4

2πkBT

h2

⎛

⎝ ⎜

⎞

⎠ ⎟3

me*mp

*( )

3 2exp −EC kBT( )

constant for small ni

can be applied for an extrinsic (impurity) semiconductor

Extrinsic Semiconductors

Doping of an impurity into an intrinsic semiconductor :

n-type extrinsic semiconductor :

e.g., Si (+ P, As, Sb : donor)

* M. Sakata, Solid State Physics (Baifukan, Tokyo, 1989).

p-type extrinsic semiconductor :

e.g., Si (+ Ga, Al, B : acceptor)

As neutral donorAs positive donor

conduction electron

conduction band

B neutral acceptorholes

B negative acceptor

Carrier Number Density of an Extrinsic Semiconductor

Numbers of holes in the valence band EV should equal to

the sum of those of electrons in the conduction band EC and in the acceptor level EA :

€

p = n + nA

Similar to the intrinsic case,

€

p = NV fp 0( ) ≈ NV exp −EF

kBT

⎛

⎝ ⎜

⎞

⎠ ⎟

€

n = NC fe Eg( ) ≈ NC exp −Eg − EF

kBT

⎛

⎝ ⎜

⎞

⎠ ⎟

Assuming numbers of neutral acceptors are NA,

€

nA =NA

1 + 2expEA − EF

kBT

⎛

⎝ ⎜

⎞

⎠ ⎟

valence band

EC (Eg)

EF

EA

EV (0)

n

nA

p

For EA - EF > kBT,

€

nA ≈ NA

€

∴NV exp −EF

kBT

⎛

⎝ ⎜

⎞

⎠ ⎟≈ NC exp −

Eg − EF

kBT

⎛

⎝ ⎜

⎞

⎠ ⎟+NA

Carrier Number Density of an Extrinsic Semiconductor (Cont'd)

At low temperature, one can assume p >> n,

€

p ≈ nA

As nA is very small, EA > EF

€

nA =NA

2exp −

EA − EF

kBT

⎛

⎝ ⎜

⎞

⎠ ⎟

€

∴NV exp −EF

kBT

⎛

⎝ ⎜

⎞

⎠ ⎟≈NA

2exp −

EA − EF

kBT

⎛

⎝ ⎜

⎞

⎠ ⎟

€

∴exp −EF

kBT

⎛

⎝ ⎜

⎞

⎠ ⎟≈

NA

2NV

exp −EA

2kBT

⎛

⎝ ⎜

⎞

⎠ ⎟

€

∴EF ≈kBT

2ln

2NV

NA

+EA

2

€

NV ≡ NVpT3 2

By substituting

€

∴EF ≈kBT

2ln

2NVp

NA

T 3 2 ⎛

⎝ ⎜

⎞

⎠ ⎟+EA

2

For T ~ 0,

€

EF ≈EA

2

valence band

EC (Eg)

EA

EF

EV (0)

n

p

nA

At high temperature, one can assume n >> nA,

€

p ≈ n

Similar to the intrinsic case, for me* = mp

*,

€

EF ≈Eg

2

Temperature Dependence of an Extrinsic Semiconductor

* H. Ibach and H. Lüth, Solid-State Physics (Springer, Berlin, 2003).

€

−Eg

2kB

€

−EA

2kB

Semiconductor Junctions

Work function :

Na 11+

1s

2s

2p3s

vacuum level

EF

Current density of thermoelectrons :

€

J = AT 2 exp −φ

kBT

⎛

⎝ ⎜

⎞

⎠ ⎟

Richardson-Dushman equation

A : Richardson constant (~120 Acm 2/K)

Metal - metal junction :

vacuum level

A B

EFA

EFB A - B

= EFA - EFB

A - B : contact potential ---

+ + +vacuum level

A B

EFA

EFB

EFA = EFB

A B

EFA EFB

A B

barrier

Metal - Semiconductor Junction - n-Type

Metal - n-type semiconductor junction :

vacuum level

M

S : electron affinity

EFM

EFS

Mn-S

EV : valence band

ED : donor levelEC : conduction band

S

qVd = M - S : Schottky barrier height

EFM EFS

Mn-S

EV

ED

EC M - S ----+ + + +

depletion layer

* http://www.dpg-physik.de/

Metal - Semiconductor Junction - p-Type

Metal - p-type semiconductor junction :

vacuum level

M S : electron affinity

EFM

Mp-S

EV

EC S

qVd = S - M

EFM EFS

Mp-S

EV

EC

S - M ++++

depletion layer

EA : acceptor levelEFS

EA

Einstein Relationship

At the equillibrium state,

Numbers of electrons diffuses towards -x direction are

€

−Dedn

dx

(n : electron number density, De : diffusion coefficient)

EF

ED

EC

EV

x

Drift velocity of electrons with mobility e under E is

€

vd = −μ eE

Numbers of electrons travel towards +x direction under E are

€

nvd = −μ enE

As E is generated by the gradient of EC, E is along -x and vd is +x.

(-x direction)

(+x direction)

€

−enE −Dedn

dx= 0 (equillibrium state)

Assuming EV = 0, electron number density is defined as

€

n = N e exp −EC − EF

kBT

⎛

⎝ ⎜

⎞

⎠ ⎟

Einstein Relationship (Cont'd)

Now, an electric field E produces voltage VCF = VC - VF

€

EC − EF = −qVCF = −q VC −VF( )

€

∴E = −dVCF

dx=

1

q

d EC − EF( )

dx

Accordingly,

€

dn

dx=

dn

d EC − EF( )⋅d EC − EF( )

dx= −

1

kBTn ⋅qE

€

∴enE = DenqE

kBT

€

∴De = μ ekBT

q Einstein relationship

Therefore, a current density Jn can be calculated as

€

Jn = − −qμ enE − qDedn

dx

⎛

⎝ ⎜

⎞

⎠ ⎟= qDe −

qn

kBT⋅dVxdx

+dn

dx

⎛

⎝ ⎜

⎞

⎠ ⎟

€

∴Jn = B exp −q Vd −V( )

kBT

⎛

⎝ ⎜

⎞

⎠ ⎟− exp −

qVd

kBT

⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

Rectification in a Schottky Junction

By applying a bias voltage V onto a metal - n-type semiconductor junction :

* H. Ibach and H. Lüth, Solid-State Physics (Springer, Berlin, 2003).

forward bias

reverse bias

M - S q(Vd - V)

€

JFoward = JM→S − JS→M = B exp −q Vd −V( )

kBT

⎛

⎝ ⎜

⎞

⎠ ⎟− exp −

qVd

kBT

⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

= J0 expqV

kBT

⎛

⎝ ⎜

⎞

⎠ ⎟−1

⎡

⎣ ⎢

⎤

⎦ ⎥≈ J0 exp

qV

kBT

⎛

⎝ ⎜

⎞

⎠ ⎟ V : large( )

M - S q(Vd - V)

€

JReverse = JM→S − JS→M = B exp −q Vd +V( )

kBT

⎛

⎝ ⎜

⎞

⎠ ⎟− exp −

qVd

kBT

⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

= J0 exp −qV

kBT

⎛

⎝ ⎜

⎞

⎠ ⎟−1

⎡

⎣ ⎢

⎤

⎦ ⎥≈ J0 V : large( )

J

V

€

JFoward = J0 expqV

kBT

⎛

⎝ ⎜

⎞

⎠ ⎟

€

JReverse = J0

pn Junction

Fabrication method :

• Annealing method :

n-type : Spread P2O5 onto a Si substrate and anneal in forming gas.

p-type : Spread B2O3 onto a Si substrate and anneal in forming gas.

• Epitaxy method (“epi” = on + “taxy” = arrangement) :

Oriented overgrowth

n-type : thermal deformation of SiH4 (+ PCl3) on a Si substrate

p-type : thermal deformation of SiH4 (+ BBr3) on a Si substrate

pn Junction Interface

By connecting p- and n-type semiconductors,

p : Most of accepters become - ions

Holes are excited in EV.

* H. Ibach and H. Lüth, Solid-State Physics (Springer, Berlin, 2003).

Fermi level EF needs to be connected.

Built-in potential : qVd = Efn - EFp

n : Most of donors become + ions

Electrons are excited in EC.

qVd

Electron currents balances

p n = n p

Majority carriers :

p : holes, n : electrons

Minority carriers :

p : electrons, p : holes

Rectification in a pn Junction

Under an electrical field E,

Current rectification :

** H. Ibach and H. Lüth, Solid-State Physics (Springer, Berlin, 2003).* M. Sakata, Solid State Physics (Baifukan, Tokyo, 1989).

hole electron

forward bias

drift current

hole

reverse bias

very small drift current