Download Tunneling - a primer

Tunneling - a primer
“Nano” often appears in real technology in the form of thin layers
or barriers.
We’re going to look at several ways electrons can transport over or
through these barriers under various conditions.
• Thermionic emission (classical over-barrier)
• Poole-Frenkel behavior (hopping out of tilted potential
wells)
• Tunneling, rectangular barrier (simple quantum problem)
• Tunneling, generic barrier
• Fowler-Nordheim tunneling (triangular barrier - field
emission)
• Resonant tunneling
(Semi)Classical thermionic emission
E
φbe
semiconductor,
voltage = 0.
eVbi
metal,
voltage= V
• Distribution of incident particle energies.
• Those with high enough energy can make it
over classical barrier.
• Net current density = Js-m – Jm-s
Thermionic emission: applications
Thermionic emission often governs:
• Electron injection into semiconductors from metals.
• Emission of electrons from hot materials into vacuum
(the filament in every cathode ray tube, ion gauge, etc.)
• Partly responsible for emission in thermal field
emission sources (electron microscopy).
Thermionic emission: calculation
semiconductor to metal:
dn( E ) = ν 3d ( E ) f ( E , T )dE
4π (2m* )1/ 2
=
h3
Assume high T
(nondegenerate)
E − Ec exp(−( E − Ec + eVn ) / k BT )dE
Rewrite in terms of velocities (semiclassical picture):
⎛ m*v 2 ⎞
⎛ eVn ⎞
⎛ m* ⎞
⎟⎟(4πv 2 dv)
⎟⎟ exp⎜⎜ −
= 2⎜ ⎟ exp⎜⎜ −
⎝ h ⎠
⎝ 2 k BT ⎠
⎝ k BT ⎠
3
Add up current heading toward barrier, knowing that
minimum velocity to get over barrier at bias V is:
1
m*v02, x = e(Vbi − V )
2
Thermionic emission: calculation
Result is, skipping some algebra,
J s −m
⎛ eφ B ⎞
⎛ eV ⎞
⎛ 4πem*k B2 ⎞ 2
⎟⎟ exp⎜⎜
⎟⎟
⎟⎟T exp⎜⎜ −
= ⎜⎜
3
⎠
⎝ h
⎝ k BT ⎠
⎝ k BT ⎠
A*, Richardson constant.
Heading the other direction, one finds
J m−s
⎛ eφ B ⎞
⎟⎟
= − A*T exp⎜⎜ −
⎝ k BT ⎠
2
Net result:
J tot
⎛ eφ B ⎞ ⎡ ⎛ eV ⎞ ⎤
⎟⎟ ⎢exp⎜⎜
⎟⎟ − 1⎥
= A*T exp⎜⎜ −
⎝ k BT ⎠ ⎣ ⎝ k BT ⎠ ⎦
2
Thermionic emission: details
J tot
⎛ eφ B ⎞ ⎡ ⎛ eV ⎞ ⎤
⎟⎟ ⎢exp⎜⎜
⎟⎟ − 1⎥
= A*T exp⎜⎜ −
⎝ k BT ⎠ ⎣ ⎝ k BT ⎠ ⎦
2
• Temperature dependence is strong.
• Exponential voltage dependence at high bias.
• Other perturbing effects (some tunneling contribution,
for example) can cause measured A* value to deviate from
theory prediction.
• Basically just a classical statistical mechanics effect.
Strongly disordered systems
In strongly disordered systems, the Bloch wave picture breaks down.
Wavefunctions are localized, decaying exponentially away from
trap sites (potential wells).
Conduction is due to hopping from localized site to localized site.
Hopping can be thermally activated (though it can also be due to
tunneling).
Applying a large electric field can affect thermal hopping rates:
Poole-Frenkel behavior
Lowering of hopping barriers in large electric fields is
called Poole-Frenkel behavior.
V (r ) = −
Assume potential well is electrostatic:
e2
4πεε 0 r
Potential maximum occurs when external field and trap
field balance:
e
1/ 2
2
4πεε 0 rmax
2
= eF → rmax
⎛ e ⎞
⎟⎟
= ⎜⎜
⎝ 4πεε 0 F ⎠
Amount barrier potential is lowered:
1/ 2
≈ 2eFrmax
⎛ e ⎞
⎟⎟
= 2e⎜⎜
⎝ 4πεε 0 F ⎠
Poole-Frenkel behavior
If hopping rate is activated,
σ ~ exp[−U 0 / k BT ]
Lowering of barrier means
σ →~ exp[−(e3 F / 4πεε 0 )1/ 2 / k BT ]
Scaling of conductance with T and F like this is the
signature of Poole-Frenkel behavior.
Prevalent in organic semiconductors, amorphous Si, etc.
Tunneling through a rectangular barrier
F
A
B
G
-a
⎛
⎞
=2 ∂2
⎜⎜ −
+ Veff ( z ) ⎟⎟φ ( z ) = Eφ ( z )
2
⎝ 2m* ( z ) ∂z
⎠
⎧ Aeikz + Be − ikz
⎪ γz
φ ( z ) = ⎨ Ce + De −γz
⎪ Feikz + Ge −ikz
⎩
+a
⎛
1 ∂φ ⎞
⎟⎟
⎝ m* ( z ) ∂z ⎠
φ ( z ), ⎜⎜
z < −a
−a < z < a
z>a
Do case where E < V0….
continuous.
Tunneling through a rectangular barrier
Applying boundary conditions,
Ae − ika + Beika = Ce −γa + Deγa
ik [ Ae −ika − Beika ] = γ [Ce −γa − Deγa ]
⎡ ⎛ ik + γ ⎞ ( ik −γ ) a
⎜
⎟e
⎛ A ⎞ ⎢ ⎝ 2ik ⎠
⎜⎜ ⎟⎟ = ⎢
⎝ B ⎠ ⎢⎛⎜ ik − γ ⎞⎟e −( ik −γ ) a
⎢⎣⎝ 2ik ⎠
⎛ ik − γ ⎞ (ik +γ ) a ⎤
⎜
⎟e
⎥⎛ C ⎞
ik
⎝ 2 ⎠
⎥⎜ ⎟
⎛ ik + γ ⎞ −( ik −γ ) a ⎥⎜⎝ D ⎟⎠
⎜
⎟e
⎥⎦
ik
2
⎝
⎠
Ceγa + De −γa = Feika + Ge − ika
γ [Ceγa − De −γa ] = ik[ Feika − Ge −ika ]
⎡ ⎛ ik + γ ⎞ ( ik −γ ) a
⎟⎟e
⎢ ⎜⎜
C
⎛ ⎞ ⎢ ⎝ 2γ ⎠
⎜⎜ ⎟⎟ =
⎝ D ⎠ ⎢− ⎛⎜ ik − γ ⎞⎟e (ik +γ ) a
⎢ ⎜
⎟
⎣ ⎝ 2γ ⎠
⎛ ik − γ ⎞ −( ik +γ ) a ⎤
⎟⎟e
− ⎜⎜
⎥ F
γ
2
⎝
⎠
⎥⎛⎜ ⎞⎟
⎛ ik + γ ⎞ −(ik −γ ) a ⎥⎜⎝ G ⎟⎠
⎜⎜
⎟⎟e
⎥
2
γ
⎝
⎠
⎦
Tunneling through a rectangular barrier
Combining, one gets
⎛ A ⎞ ⎡ M 11
⎜⎜ ⎟⎟ = ⎢
⎝ B ⎠ ⎣ M 21
Incident flux:
M 12 ⎤⎛ F ⎞
⎜⎜ ⎟⎟
⎥
M 22 ⎦⎝ G ⎠
f inc = A
2
=k
m*
where elements come from
multiplying matrices on prev. slide.
f tran = F
transmitted flux:
2
Transmission coefficient:
F
f
1
T ( E ) = tran = 2 =
2
f inc
A
M 11
=
1
1+
( ) sinh (2γa)
k 2 +γ 2
2 kγ
2
2
2
Limit of wide barrier:
⎛ 4kγ ⎞ − 4γa
−4a
⎟ e
T ( E ) → ⎜⎜ 2
∝e
2 ⎟
⎝ k +γ ⎠
2 m* (V0 − E ) / =
2
=k
m*
Tunneling through a rectangular barrier
Reflection coefficient = 1 - T, unsurprisingly.
Can also do case where E > V0 :
T (E) =
1
1+
( ) sin (2k ' a)
k 2 − k '2
2 kk '
2
Combined picture:
image from Ferry
2
where
k ' = 2m* ( E − V0 )
Tunneling through a generic barrier:
the WKB approximation
classically
forbidden
x2
x1
• define a local effective wave vector - imaginary
in classically forbidden region.
⎤
⎡ 2 x2
1/ 2
T ( E ) = exp ⎢− ∫ {2m* (V ( x) − E )} dx ⎥
⎣ = x1
⎦
• valid when turning points are far apart (many wavelengths).
• valid when V(x) varies slowly (compared to wavelength).
Realistic tunneling probabilities
As we saw from rectangular
barrier case, tunneling only
occurs with significant probability
at very short length scales (very
narrow barriers).
Tunneling is exponentially
suppressed with barrier width.
Factoid: metal-metal tunneling current in vacuum (or air) decays
with distance roughly like 1 order of magnitude per Angstrom (!).
One of your homework problems: what is the probability of
a Volkswagon Beetle tunneling through a speed bump?
A triangular barrier: Fowler-Nordheim tunneling
Can pull barrier down by
applying electric field.
Fowler-Nordheim limit:
barrier gets thin enough to
allow tunneling.
φbe
EF
Analytical expression can be obtained using WKB, as long
as barrier doesn’t get too thin:
(V ( x) − E ) = eφ B − eFx
⎤
⎡ 2 φ B / eF
1/ 2
{
2m* (eφ B − eFx)} dx ⎥
T ( E ) = exp ⎢− ∫
⎦
⎣ = 0
⎡ 4 (2m*e)1/ 2 φ B3 / 2 ⎤
= exp ⎢−
⎥
3
=
F
⎣
⎦
-eFx
Double barriers
A
B
F
G
A’
B’
We can easily consider the case of two barriers in a row.
This case gets particularly interesting if the barriers are
sufficiently close that there are “bound” states in the space
between the barriers that have energies close to those of incident
electrons.
Matrix-wise, we know we can combine the two single-barrier
problems.
F’
G’
Double barriers
A
F
B
G
A’
F’
G’
B’
-2a
b
z=0
Need to relate phases of A’, B’ to F, G:
− ikb
F
⎛
e
⎞
⎛
Define transfer matrix: ⎜ ⎟ = ⎜
⎜G⎟ ⎜ 0
⎝ ⎠ ⎝
Combined system:
A' = Feikb ,
B' = Ge − ikb
0 ⎞⎛ A' ⎞
⎛ A' ⎞
⎟
⎜ ⎟⎟ = MW ⎜⎜ ⎟⎟
ikb ⎟⎜
e ⎠⎝ B' ⎠
⎝ B' ⎠
⎛ A⎞
⎛ F '⎞
⎜⎜ ⎟⎟ = M L MW M R ⎜⎜ ⎟⎟
⎝ B⎠
⎝ G'⎠
where the ML and MR are matrices for each barrier individually.
Resonant tunneling
Write the combined matrix as Mtot=MLMWMR, and write the
matrix elements as a magnitude and a phase. Final result:
Ttot ( E ) =
1
M tot11
2
T12
= 2
T1 + 4 R1 cos 2 (kb − θ )
⎡⎛ k 2 − γ 2 ⎞
⎤
⎟⎟ tanh(2γa)⎥ + 2ka
θ = − arctan ⎢⎜⎜
⎣⎝ 2kγ ⎠
⎦
Transmission through the whole structure has resonances!
These correspond to cases when the energy of the incident
particle coincides with “bound state” energies of the well.
Incident + reflected waves interfere constructively in well!
At resonance, Ttot = 1.
Minimum transmission: Ttot ~ T12/4.
Resonant tunneling
What about asymmetric case? Relevant at finite bias:
Expect to see peak in transmission
when bias tilt aligns bound level with
source or drain energy levels.
At higher bias, expect transmission to
drop again as resonance condition
vanishes.
• Predicts NDR!
Resonant tunneling
Algebra gets a bit messy….
image from Ferry
Resonant tunneling diode
Final result:
Ttot ( E ) =
T1T2
(1 − R1 R2 ) 2 + 4 R1 R2 cos 2 (Φ )
Φ = k1b +
θ L12 + θ R 21 − θ L11 − θ R11
2
Again, at resonance
Tres =
4T1T2
≈1
2
(T1 + T2 )
Off resonance
Toff
T1T2
≈
4
for T1=T2.
Resonant tunneling diode
Does this actually work?
GaAs RTD, AlGaAs barriers, a = b = 5 nm.
image from Ferry
A very subtle question
A philosophy question worth pondering, even though
it’s not directly germane to the course:
How long does the tunneling process take?
That is, for particles that successfully traverse barrier,
how long are they “in” the classically forbidden region?
Consider incident Gaussian wavepacket + transmitted
Gaussian wavepacket.
Surprising answer: measuring positions of peaks of
wavepacket, tunneling “velocity” can greatly exceed c !
For more information, read Landauer and Martin, RMP 66,
217 (1994).
To summarize:
• Thermionic emission = classical thermal over-barrier
• Poole-Frenkel = field-assisted classical thermal hopping
• Single-barrier tunneling is straightforward.
• Generic barriers: WKB approximation
• Fowler-Nordheim = field-assisted tunneling
• Double barriers: must account for interference
• Result: resonant tunneling diodes w/ NDR
• Nontrivial interpretation issues associated with tunneling!
Next time:
• Scattering matrices
• Landauer-Buttiker formalism - conductance as
transmission.