Fundamentals of Semiconductor Physics

万 歆Zhejiang Institute of Modern Physics

xinwan@zimp.zju.edu.cnhttp://zimp.zju.edu.cn/~xinwan/

Fall 2007

Transistor technology evokes new physics

The objective of producing useful devices has strongly

influenced the choice of the research projects with which I have

been associated. It is frequently said that having a more-or-less

specific practical goal in mind will degrade the quality of research. I

do not believe that this is necessarily the case and to make my point

in this lecture I have chosen my examples of the new physics of

semiconductors from research projects which were very definitely

motivated by practical considerations.

-- William B Shockley

Nobel Lecture, December 11, 1956

Chapter 3. Junctions & Contacts3.1 Metal-semiconductor contacts

3.2 p-n junctions

3.3 Heterojunctions*

Total 6 hours.

The birth of transistorThe birth of transistor

William Bradford Shockley

John Bardeen Walter Houser Brattain

Nobel Prize in Physics 1956

Why Contacts?

• Bringing materials with vastly different properties together can produce remarkable effects.

• Examples:– Cu/Fe: temperature controlled switch– Superconductor/metal: Andreev reflection– Ferromagnet/semiconductor: spin injection– Ferromagnetic/normal metal: GMR

The birth of nonotechnology/spintronicsThe birth of nonotechnology/spintronics

Nobel Prize 2007: Albert Fert & Peter GruenbergFor the discovery of Giant Magnetoresistance

Chapter 3. Junctions & Contacts3.1 Metal-semiconductor contacts

– A review of the principles

– Idealized metal-semiconductor junctions

– Current-Voltage characteristics

– Ohmic contacts

3.2 p-n junctions

3.3 Heterojunctions

Separated M-S Systems

Bring M-S into Contact

At thermal equilibrium the Fermi levels is constant throughout a system.

f D1 ,2=1

1exp [E−E f1 , 2/kT ]

E f1=E f2

Inhomogeneously Doped SC

Ec

Ef

Ei

Ev

E0

Ec

Ef

Ei

Ev

E0

Inhomogeneously Doped SC

Ideal Density-of-States

Typical Metal Typical Semiconductor

Energy levels

Charge Transfer Equilibrium

Electrons depleted

Assume band structures not changed near the surface.

Real situation: Surface states

No Hope to Solve Analytically• Poisson’s equation

• the electron density, the hole density and the donor and acceptor densities. To solve the equation, we have to express the electron and hole density, n and p, as a function of the potential, φ

d 2dx2

=−s=−qs

p−nN d−N a− , n=ni exp qkT

d 2dx2

= qs 2 n i sinh qkT N a−−N d n=ni exp { E f−E ik BT } , p=ni exp {−E f−E ik BT } , = E i−E f−q

Depletion Approximation

• For the idealized n-type semiconductor, hole density neglected

• Electron density n

Field, Potential and Charge

(Gauss’ law)

Applied Bias• Up to this point, we have been considering thermal equilibrium

conditions at the metal-semiconductor junction.

• Now, we study the case of an applied voltage; i.e., a nonequilibrium condition.

• Electrons transferring from metal to semiconductor see a barrier.

Applied Bias• To the first order, the barrier height is independent of bias because

no voltage can be sustained across the metal.

• Bias changes the curvature of the semiconductor bands, modifying the potential drop from φi.

Junction Charge & Capacitor

Small-Signal Capacitance

Variable Doping

“Semiconductor Profiler”

Schottky Barrier Lowering• We now explore the statement that the barrier to electron flow from

metal to semiconductor is “to first order” unchanged by bias.

• Approximations:

– Free electron theory

– Metal as plane conducting sheet

– Semiconductor: effective mass, relative permittivity

• Root: Metal plane = image charge of opposite sign

#Slide 14

Schottky Barrier Lowering

I-V Characteristics w/out Math• At equilibrium, rate at which electrons cross the barrier into the

semiconductor is balanced by rate at which electrons cross the barrier into the metal. (flow = counter flow).

• When a bias is applied, the potential drop within the semiconductor is changed and we can expect the flux of electrons from the semiconductor toward the metal to be modified.

• The flux of electrons from the metalto the semiconductor is not affected.

• The difference is the net current.

I-V Characteristics• The current of electrons from the semiconductor to the metal is

proportional to the density of electrons at the boundary.

• In equilibrium,

ns=N C exp −qBkT =N D exp −qikT qB=qiEC−E f

∣J MS∣=∣J SM∣=K N Dexp −qiKT

I-V Characteristics• When a bias is applied to the junction,

• Now,

• Therefore, the ideal diode equation reads

ns=N Dexp −q i−V akT J=J MS−J SM=K N D exp −q i−V aKT −K N D exp −qiKT

J=J 0 [exp qV aKT −1 ]

Comments• The ideal diode equation arises when a barrier to electron flow

affects the thermal flux of carriers asymmetrically.

• The essence of the ideal diode equation predicts a saturation current J0 for negative Va and a very large steeply rising current when Va is positive.

• J0 in the above ideal case is independent of the applied bias. More careful analysis will modify this slightly.

More Detailed Analysis• Schottky: Integrating the equations for carrier diffusion and drift

across the depletion region near the contact.

Assumes that the dimensions of the space-charge region are sufficiently large (a few electron mean-free path) so that the use of a diffusion constant and a mobility value are meaningful (small field, no drift velocity saturation).

• Bethe: Based on carrier emission from the metal.

Valid even when these abovementioned constraints are not met.

Diffusion and Drift Currents

J x=q [nn E xDn dndx ]=q Dn [−qnkT d dx dndx ]

Trick for IntegrationJ x=q [nn E xDn dndx ]=q Dn [−qnkT ddx dndx ]J x∫0

x d exp −qkT dx=q Dn [nexp −qkT ]0xd

0=0 ; xd =i−V a=B−n−V a

n0=N C exp −qBkT ; n xd =N D=N C exp −qnkT

J x=q Dn N C exp −qBkT ∫0

xdexp −q xkT dx [

exp qV akT −1 ]=J 0 [exp qV akT −1 ]

Schottky Barrier

Mott Barrier

Comments• One important implicit assumption is that the system is in quasi-

equilibrium, i.e., almost at thermal equilibrium even though currents are flowing.

– Electron density at the interface when the bias is applied.

– Using Einstein relation to relate drift and diffusion current.

• The ultimate test: Agreements between measurements and predictions.

• Derivation not valid when Va ~ φi (no barrier). A fraction of the voltage is dropped across the resistance of the semiconductor. The forward voltage across the junction is reduced.

Schottky Diodes

Surface Effects

Pinning of Fermi Energy• To account for the surface effects, the metal-semiconductor contact

is treated as if it contained an intermediate region sandwiched between the two crystals.

• For a large density of surface states, the Fermi energy is said to be pinned by the high density of states.

Nonrectifying (Ohmic) Contacts• Definition: The contact itself offers negligible resistance to current

flow when compared to the bulk.

• The voltage dropped across the ohmic contacts is negligible compared to voltage drops elsewhere in the device.

• No power is dissipated in the contacts.

• Ohmic contacts can be described as being in equilibrium even in when currents are flowing.

• All free-carrier densities at an Ohmic contact are unchanged by current flow. The densities remain at their thermal-equilibrium values.

Tunneling Contacts• By heavily doping the semiconductor, so that the barrier width is

very small and tunneling through the barrier can take place.

Schottky Ohmic Contacts

Chapter 3. Junctions & Contacts3.1 Metal-semiconductor contacts

3.2 p-n junctions

3.3 Heterojunctions

Metal-Semiconductor Contacts• A system of electrons is characterized by a constant Fermi level at

thermal equilibrium.

• Systems that are not in thermal equilibrium approach equilibrium as electrons are transferred from regions with a higher Fermi level to regions with a lower Fermi level.

• The transferred charge causes the buildup of barriers against further flow, and the potential drop across these barriers increases to a value that just equalizes the Fermi levels.

Similar phenomena in a single crystal with non-uniform doping.

p-n Junction

Ec

Ef

Ei

Ev

E0

Ec

Ef

Ei

Ev

E0

electrons

• Systems that are not in thermal equilibrium approach equilibrium as electrons are transferred from regions with a higher Fermi level to regions with a lower Fermi level.

p-n Junction

Ec

Ef

Ei

Ev

E0

Ec

Ef

Ei

Ev

E0

E

• A system of electrons is characterized by a constant Fermi level at thermal equilibrium.

p-n Junction

Ec

Ef

Ei

Ev

E0E

c

Ef

Ei

Ev

E0

φi

• The transferred charge causes the buildup of barriers against further flow, and the potential drop across these barriers increases to a value that just equalizes the Fermi levels

Graded Impurity Distributions• We assume initially the majority carrier density equals the dopant

density everywhere. We ask how equilibrium is approached. • A gradient in the mobile carrier density diffusion of carriers.• Carrier diffusion leaves dopant ions behind.• Separation of charge field opposing the diffusion flow.• Equilibrium is reached when diffusion is balanced by the field.

Potential

The separation of the Fermi level from the conduction-band edge (or intrinsic Fermi level) represents the potential energy of an electron.

Field

Using mass-action law,

J n=qn nxq Dn dndx=0 in equilibrium

Density vs Potential Barrier

Poisson’s Equation

Cannot be solved in the general case!

d 2dx2

= qs 2ni sinh qkT N a−−N d n=ni exp qkT ; p=ni exp −qkT

d 2dx2

=−s=− qs

p−nN d−N a−

(i) Small Gradient Case

Quasi-neutrality approx.:

constant field: kT qλ

(ii) p-n Junction• Depletion approximation ↔ quasi-neutral approximation

Depletion of Mobile Charge

Potential Barrier

2a

i

a

p Nnn N

=

= ( )2

2

00

d as

pnd q N Ndx

φε

==

−= −

2d

i

d

n Nnp N

=

=

Two Idealized Cases• Linearly graded junction: a continuous gradient in dopant between

n-type and p-type regions.

• Step junction (abrupt junction): a constant n-type dopant density changes abruptly to a constant p-type dopant density – for example, formed by epitaxial deposition

Step Junctions

Step Junction Analysis

In n-type region,

Integration leads to

Similarly,

Continuation at x = 0 requires

This is just the charge neutrality in the depletion region.

Step Junction Analysis

In n-type region,

In p-type region,

Totally,

Comments• At high dopant concentrations, use F-D distribution in stead.

However, the practical result is that the Fermi level is very near the band edge. For example, for heavily doped p-type silicon and lightly doped silicon:

• Total depletion-region width is:

• There is only partial depletion near the edge of the depletion region.

Partial Depletion

Debye Length

Typically , N d~1016 cm−3 , LD~40 nm？？？

Linearly Graded Junction

Do It Yourself!

Applied Bias• Everything works as before. Just replace

• For positive Va, the barrier to the majority carrier is reduced; depletion region is narrowed; the junction is forward biased; appreciable currents can flow under small forward bias.

• For negative Va, the barrier to the majority carrier increases; depletion region widens; the junction is reverse biased; there is very little current flow under reverse bias.

by i i aVφ φ −

Depletion Width & Maximum Field

Abrupt p-n junction Linearly graded junction

Capacitance

VaractorUnder reverse bias V

R

Question: Which one is more sensitive? Linearly

graded junction? Or abrupt junction?

• LGJ: n = 1/3

• AJ: n = 1/2 (more sensitive)

• Can you design an even more sensitive varactor?

C∝iV R−n

Junction Breakdown

Avalanche Breakdown

Avalanche breakdown is confines to the central portion of the space-charge region, where the field is sizeable.

Q: Conservation of energy and momentum requires the original electron possess kinetic energy of at least ?

32 gE

Zener Breakdown

The WKB approximation

Currents in p-n junctions

• Generation and recombination• Continuity equation• Current-voltage characteristics• Charge storage and diode transients

Recombination through Traps

Additional reading: Shockley-Hall-Read recombination

Generation and Recombination

Carrier lifetime

0'

n n

n nU nτ τ

= = −Excess carrier concentrationRecombination rate:

Minority Carriers Matter

==> current flow (I-V characteristics) & charge storage (transient behavior)

Quasi-neutral zoneQuasi-neutral zone

Ideal-Diode AnalysisConsider excess holes injected into the n-regions, where bulk

recombination through generation-recombination centers is dominant.

In the quasi-neutral region, we have roughly E = 0.

Stationary:

∂ pn∂ t

=− pn p∂E∂ x

− p E∂ pn∂ x

D p∂2 pn∂ x2

G p−R p

∂ pn∂ t

=D p∂2 pn∂ x2

pn− pn0 p

Minority-Carrier Boundary Values

• Change of majority-carrier populations negligible.• Detailed balance “nearly” applied for small enough applied bias.

φi

Minority-Carrier Boundary Values

• Change of majority-carrier populations negligible.• Detailed balance “nearly” applied for small enough applied bias.

Case 1: Long-Base Diode• Length scale: diffusion length Lp

Hole Current

Total Current• Minority-carrier electron current injected into p-region:

• Total current:

Case 2: Short-Base Diode• Length scale: lengths of n- and p-type regions, WB and WE

Ohmic contact

Current• Assumption: No recombination occurs in n-type region.

• Hole current through n-type region:

• Total current:

Higher Order Corrections• The ideal-diode equation is based on events in the quasi-neutral

regions. The space-charge region is purely a barrier to diffusion of majority carriers and it plays a role only in the establishment of minority-carrier density at its boundaries.

• This is a reasonable first-order description of all the events.

• It is inaccurate, especially for silicon p-n junctions.

• Corrections due to events in the space-charge region are required.

Qualitative Picture

Diode Transients

On:

buildup time = stored charge / source current

Off:

turn-off time limited by how fast charge can be removed from the quasi-neutral region.

Charge Storage• Mechanism 1: Majority carriers near the edges of the depletion

region move as the depletion region expands or contracts in response to a changing bias. The charge storage in the depletion region is modeled by a small-signal capacitance, e.g., the abrupt-junction depletion capacitance.

• Mechanism 2: Minority-carrier charge changes in the quasi-neutral region when applied bias is switched on or off. This can be modeled by another small-signal capacitance - diffusion capacitance.

Equivalent circuit

Chapter 3. Junctions & Contacts3.1 Metal-semiconductor contacts

3.2 p-n junctions

3.3 Heterojunctions

Will be discussed in the preparation of two-dimensional electron gas.