The story so far: Band structures I

The story so far: • Looking at “bulk” crystalline materials - model as weak periodic potential + confinement. Result: bands of allowed single-particle...
Author: Katrina Berry
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The story so far: • Looking at “bulk” crystalline materials - model as weak periodic potential + confinement. Result: bands of allowed single-particle states. • Single-particle states still labeled by k, but E(k) no longer simple. • Crystal structure leads to lattice in reciprocal space (k-space); unit cell in reciprocal space = Brillouin zone. • Diffraction happens when wavevector change is a reciprocal lattice vector - particle strongly feels lattice potential. • This diffraction is what causes the band gaps to form. • Filling of the bands of single-particle states helps determine whether a material is a metal, a semiconductor, or an insulator.

Band structures I

Conduction band Band gap ~ 10 eV Valence band 0

π/a

Band insulator e.g. diamond, Al2O3 EF said to lie in the gap. Insulating b/c no unoccupied empty states near Fermi level.

Semiconductor (direct gap) e.g. GaAs Band gap ~ 1 eV 0

π/a

Insulating at low T; somewhat conducting at high T because of thermal excitations into conduction band.

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Band structures II Semiconductor (indirect gap) e.g. Si, Ge Minimum in conduction band not Band gap ~ 1 eV at same k value as maximum in valence band. 0

π/a

Metal due to electronic density. Metallic b/c unoccupied empty states near Fermi level available for electrons to scatter into when electric field is applied. 0

π/a

Semimetal, or metal due to band overlap.

0

π/a

Electrons and Holes • Excitations of free Fermi gas are electrons and holes. E [h2/2mL2]

ky

E (k ) =

kx

h 2k 2 2m

|k| [π /L ]

• Excitations of nearly-free Fermi gas are quasiparticles that act like electrons and holes. • The difference: E(k) is no longer simple, and there are interaction effects (that we’ve been neglecting) that can make a small difference.

0

π/a

2

Electrons and Holes II Metal • Since states near EF are empty, easy to create quasiparticles - “gapless”

π/a

0

• Typical excitation energy ε measured from EF is ~ kBT.

Caveats: Excitations are only approximate energy eigenstates possess some finite lifetime, τ, and some width in energy,

Γ= To be well-defined,

h

τ

ε =| ( E − E F ) |~ k BT >> Γ

Electrons and Holes III Semiconductors • Require an energy at least as large as Eg to create an electronhole pair. • Electrons and holes are in different bands.

0

π/a

0

π/a

• Optical processes often used to create e-h pairs, with photon energy ħω > ~ Eg. • Indirect gap case: need an additional excitation to conserve momentum. • Result: GaAs, InP (direct gap) semiconductors used in photodetectors + LEDs; Si & Ge are not. • Nanostructuring can allow some rules to be bent - light emission from nanoporous Si.

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Effective mass Because E(k) no longer simple, nearly-free electrons and holes move as if they have masses not equal to the free electron mass! Near a minimum in the conduction band, define

Ec (k ) ≈ Ec +

h2 (k − k min ) 2 + h.o.t. 2m*

1  ∂ 2 Ec (k )   → m* ≡ 2  h  ∂k 2 

−1

for electrons.

Similarly, near maximum in valence band,

− 1  ∂ 2 Ev (k )  h2  Ev (k ) ≈ Ev − (k − k min ) 2 + h.o.t., → m* ≡ 2  h  ∂k 2  2m*

−1

An electron near top of valence band acts like m* < 0! An absence of such an electron acts like a positively charged hole with m* > 0.

Effective mass II • Effective mass is a tensorial quantity in general, with different values along different crystalline symmetry directions. • Direction-averaged m* = (m*l m*t m*t)1/3 • Crystal structure determines anisotropy in m* • Electrons and holes can have very different effective masses. • Because of band degeneracies & chemical origins of bands, generally two types of holes (“heavy” and “light”). • Remember: the more steeply the band energy curves with k, the lower the effective mass.

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Common materials - silicon

From Blakemore

Eg = 1.12 eV at 300 K Electron m* = 0.98 m0 long., 0.19 m0 trans. Hole m* = 0.16 m0 light, 0.49 m0 heavy. “Valley degeneracy” From Sze

Common materials: GaAs

Eg = 1.42 eV at 300 K Electron m* = 0.067 m0 Hole m* = 0.082 m0 Electrons in GaAs conduction band form spherical Fermi surface - easy to think about. From Sze

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Equilibrium populations How many electrons and holes are there in a 3d intrinsic semiconductor in thermal equilibrium? Density of states in the electron and conduction bands matter. Treat electronic states at bottom of conduction band like a 3d Fermi gas w/ energy measured from Ec. Conduction band result:

Free Fermi gas result:

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ν 3d ( E ) =



2

 2m   2  h 

3/2

E 1/ 2

ν e (E ) =

M c  2m*e    2π 2  h 2 

3/ 2

( E − E c )1 / 2

Mc = number of equivalent minima in conduction band.

Equilibrium populations Number of electrons in conduction band in equilibrium:

n=∫

Etop

Ec

ν ( E ) f ( E , E F , T )dE

M  2m  ν e ( E ) = c2  2*e  2π  h 

1

f ( E , EF , T ) = e

3/ 2

   

E − EF k BT

   

+1

( E − E c )1 / 2

In nondegenerate limit (usually good at 300 K), expand f to recover Boltzmann statistics, and get

n=

  2πm k T 3 / 2   *e B M  2  ⋅ exp − C 2 h     

EC − E F k BT

   

Nc(T), effective density of states in conduction band

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Equlibrium populations Do same for holes (no valley degeneracy this time):   2πm k T 3 / 2   *h B   ⋅ exp  − 2 h     

p = 2

EF − EV k BT

   

Nv(T), effective density of states in valence band Equating n and p determines EF Usually very close to middle of gap. Intrinsic carrier density:



Eg



k BT 

ni2 = np = N C NV ⋅ exp  − → ni = N C NV

 ⋅ exp  − 

 

Eg 2 k BT

   

  m*e m*h  1/ 2 3 / 2 M C T exp  − 2    m0 

→ ni = 4.9 × 1015 

Eg 2k BT

per cm3

   

Intrinsic semiconductors   m*e m*h  1 / 2 3 / 2 M C T exp − 2    m0 

ni = 4.9 × 1015 

Eg 2 k BT

   

per cm3

• Strongly temperature dependent! • At low T, all carriers freeze out exponentially - system is a band insulator. Si, 300 K

1 x 1010 cm-3

Ge, 300 K

2 x 1013 cm-3

GaAs, 300 K

2 x 106 cm-3

Can we manipulate the number of carriers available somehow, rather than relying purely on thermodynamics and crystal parameters? Yes - by doping.

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Doping & common dopants Doping means adding or removing charge from the equilibrium value for a homogeneous crystal, usually by the controlled addition of chemical impurities. Example: Si has 4 valence electrons (Group IV). As has 5 valence electrons (Group V). A single As impurity in Si uses 4 valence electrons to satisfy crystal bonding, and has one extra electron. Si

Si

Si Si Si

Si

Si

Si

Si Si

As+ Si

Si Si

Si

Si

Si

Si Si

Si Si

Si Si

Si

Si Si

Si Si

As

Si

Si

Si

Si Si

Si

Si

Si

Si Si

Si

Doping & common dopants Another example: Si has 4 valence electrons (Group IV). B has 3 valence electrons (Group V). A single B impurity in Si borrows an electron from the rest of the crystal to satisfy bonding, leaving behind a hole. Si

Si

Si

Si

Si

Si

Si

Si

Si Si

BSi

Si Si

Si Si

Si Si

Si Si

Si

Si Si

Si Si

P Si

Si

Si Si

Si Si

Si Si

Si

Si Si

Si

Si Si

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Common dopants Dopants that add “extra” electrons are called n-type, or donors. Some n-type dopants in Si include P, As, and Sb. Some n-type dopants in GaAs are Si and Ge. Dopants that remove an electron are called p-type, or acceptors. Some p-type dopants in Si include Al, Ga, and In. Some p-type dopants in GaAs are Zn, Be, Mg, and C.

Single dopants The “extra” electron from a donor is bound, at low temperatures, to the dopant atom. The bound states look like hydrogenic states, but the semiconductor acts like a dielectric. Result: ionization energies are much lower than for the isolated atoms in vacuum. Effective Bohr radius of a donor’s electron (or an acceptor’s hole):

aeff =

4πεε 0 h 2 m*e 2

Ionization energy of a donor’s electron (or an acceptor’s hole):

EI ,eff =

e 4 m* 2( 4πεε 0 h ) 2

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Plug in some numbers

aeff = EI ,eff =

4πεε 0 h 2 m*e 2

e 4 m* 2(4πεε 0h ) 2

For electrons in Si, m* ~ 0.33m0, ε = 11.7. aeff = 1.9 nm EI,eff = 0.033 eV

For electrons in GaAs, m* ~ 0.067m0,

ε = 13.13.

aeff = 10.4 nm EI,eff = 0.005 eV

• Most donors are ionized at room temperature! • Effective Bohr radius encompasses many atoms - means using ε was reasonable. • Donor wavefunctions can overlap - metal-insulator transition.

Energetics E EF

+ +As +As + AsAs

• Donor states lie close to conduction band. • Donated electrons boost EF into conduction band. • Now + donor ions are left behind.

ν(E) E • Acceptor states lie close to valence band. • Accepted electrons lower EF into valence band.

B- B- B- B-

EF

• Now - acceptor ions are left behind.

ν(E)

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Doping - how is it done? Two main techniques: 1. Diffusion

diffusion constant depends on materials, T

Just what it sounds like. Fick’s law:

J = − D∇n concentration of diffusing atoms

“Current density” of diffusing atoms

Conservation of particles gives the diffusion equation:

∂n = D∇ 2 n ∂t D usually depends very steeply on T, e.g. D ~ exp(-∆/kBT).

Diffusion Consider the following situation at t = 0: nAs n0 As

Si x

x=0 Solution to the diffusion equation for these boundary conditions:

x  n( x, t ) − n(t = 0)  = 1 − erf   n0 − n(t = 0)  2 Dt  n0 x

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Diffusion - reasonable numbers & problems D = D0 exp(− ∆ / k BT ) Dopants are substitutional diffusers. For silicon, Element

D0(cm2/s)

∆(eV)

B Al Ga In P As

10.5 8 3.6 16.5 10.5 0.32

3.69 3.47 3.51 3.90 3.69 3.56

• Width of dopant profile could be tricky for nm-scale devices! • Substantial diffused densities for reasonable concentration gradients in reasonable times require high temperatures > ~1500 K.

2. Ion implantation

Image from Jaeger, Vol. 5

Fire beam of energetic ions directly into wafer. Can implant nearly anything ionizable. Depth controlled by ion energy. Long high-T annealing not necessary.

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Ion implantation range

peak concentration 

n( x) = nP exp − 

( x − RP ) 2   2(∆RP ) 2 

“straggle” ∞

N = ∫ n( x)dx = 2π (nP )(∆RP ) 0

Image from Jaeger, Vol. 5

Masking:

Ion implantation - problems • Hard to reach high doping levels - slow. • Causes damage to lattice that must be annealed away. • Electronic activity requires annealing, too. • Channeling - preferred crystallographic directions.

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Finite size problems Consider reasonable dopant density of 1017 cm-3. Suppose an advanced transistor has an active volume ~ 50 nm on a side. Number of dopant atoms in that volume ~ 12.5 (!) Fluctuations could be very significant from device to device. Clustering of dopants can reduce electrical activity…. Can we even check for issues like this?

Individual dopants

Voyles, P. M. et al. Nature 416, 826-829 (2002)

State-of-the-art TEM image of ultrathinned (5 nm) sample. Only now, and in restricted circumstances, able to study single dopants.

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Summary • Band structure and electronic density determine electrical properties of real materials. • Excitations still act like electrons and holes, but have directiondependent effective masses determined by band curvature. • Minimum in conduction band not necessarily near k=0. • Can add or remove carriers from semiconductors (shift Fermi level up or down) by chemical doping. • Dopants introduce additional energy and size scales that are of concern in nanoscale structures. • At really small scales, properties sensitive to positions of individual atoms - variability and fluctuations are a real concern.

Next time: • Interfaces between different materials - basics • Surface states

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