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Content Motivation The scattering matrix Properties of the scattering matrix Current operator DC current Summary
The Landauer-Büttiker formalism to transport
phenomena in mesoscopic conducting systems
Alexandra Nagy
September 17, 2015
Alexandra Nagy The Landauer-Büttiker formalism
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Content Motivation The scattering matrix Properties of the scattering matrix Current operator DC current Summary
Content
1 Motivation
2 The scattering matrix
3 Properties of the scattering matrix
4 Current operatorReservoirs and leadsSecond quantizationCurrent operator
5 DC currentDC current and distribution functionsDC current conservation
6 Summary
Alexandra Nagy The Landauer-Büttiker formalism
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Content Motivation The scattering matrix Properties of the scattering matrix Current operator DC current Summary
Motivation
Describing transport phenomena in mesoscopic conducting systems
Scope
a con�ned mesoscopic system coupled to thermal e− reservoirsleads with one conducting sub-bandlow temperature: the phase coherence length (Lφ) � size ofthe sample (L)quantum mechanical scattering problem of e−-s
Alexandra Nagy The Landauer-Büttiker formalism
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Content Motivation The scattering matrix Properties of the scattering matrix Current operator DC current Summary
The scattering matrix
incident electron Ψ(in) → Ψ(out) scattered electron
solution for the ψ(in)α orthonormal basis → scattering state for
an arbitrary initial state
both wave functions can be expanded on a full orthonormal
basis:{ψ
(in)α
},{ψ
(out)β
}Ψ(in) =
∑α aαψ
(in)α Ψ(out) =
∑β bβψ
(out)β
Problem: �nd bβ if the set of aα is known
Alexandra Nagy The Landauer-Büttiker formalism
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Content Motivation The scattering matrix Properties of the scattering matrix Current operator DC current Summary
The scattering matrix
Method
1 Expanding the initial state into a series of ψ(in)α
2 Expanding the scattered state
Ψ(out) =∑
α Ψ(out)α (due to Ψ
(in)α = aαψ
(in)α )
Ψ(out)α = aα
∑β Sβαψ
(out)β
3 Solution
Ψ(out) =∑
α aα∑
β Sβαψ(out)β ≡
∑β bβψ
(out)β
⇓
bβ =∑
α Sβαaα
b = S a
4 Sβα: QM-amplitude to pass from ψ(in)α to ψ
(out)β
Alexandra Nagy The Landauer-Büttiker formalism
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Content Motivation The scattering matrix Properties of the scattering matrix Current operator DC current Summary
Properties - Unitarity
The particle number (�ow) conservation implies the unitarity of S
S†S = S S† = I
Proof - normalized wave function∫d3r |Ψ(in)|2 =
∫d3r
∑α aαψ
(in)α
(∑β a∗βψ
(in)β
)∗=∑
α
∑β aαa
∗β
∫d3rψ
(in)α
(ψ
(in)β
)∗=∑
α
∑β aαa
∗βδαβ
=∑
α |aα|2 = a†a = 1
b†b = a†S†S a = a†a
Alexandra Nagy The Landauer-Büttiker formalism
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Content Motivation The scattering matrix Properties of the scattering matrix Current operator DC current Summary
Properties - Micro-reversibility
The time-reversal symmetry implies
S = ST ⇒ Sαβ = Sβα
i~∂Ψ∂t = HΨ → i~∂(Ψ∗)
∂(−t) = H(Ψ∗)
(Ψout(−t))∗ =(∑
β bβψ(out)β (−t)
)∗=∑
β b∗βψ
(in)β (t)
(Ψin(−t)
)∗=(∑
α aαψ(in)α (−t)
)∗=∑
α a∗αψ
(out)α (t)
⇓
a = S−1b, a∗ = S b∗
Alexandra Nagy The Landauer-Büttiker formalism
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Content Motivation The scattering matrix Properties of the scattering matrix Current operator DC current Summary
Properties - Micro-reversibility
S†S = I
S−1S = I
}⇒ S† = S−1, S∗ = S−1 ⇒ S = ST
Magnetic �eld - H
In addition to the time and momentum reversal, one needs toinverse the direction of the magnetic �eld, H → −H
S(H) = S(−H)T
⇓
Sαβ(H) = Sβα(−H)
Alexandra Nagy The Landauer-Büttiker formalism
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Content Motivation The scattering matrix Properties of the scattering matrix Current operator DC current Summary
Current operator
Application of scattering matrices
single-electron approximation → interaction is described byUeff (t, r)
incident and outgoing e− �ows through the surface∑
elastic, energy conserving scattering ⇒ Lφ >> L
Alexandra Nagy The Landauer-Büttiker formalism
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Content Motivation The scattering matrix Properties of the scattering matrix Current operator DC current Summary
Current operator
Reservoirs and leads
Nr macroscopic contacts as electron reservoirs:Tα, µα → Fermi-distribution
fα(E ) = 1
1+eE−µαkB Tα
Leads:
the eigenwave-functions are the basis functions for the
scattering matrix
Hα = 1
2m∗ p2
xα + 1
2m∗ p2
⊥α+ U(r⊥)
assuming one conducting sub-band
the solution is the product of transverse and longitudinal terms
Alexandra Nagy The Landauer-Büttiker formalism
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Content Motivation The scattering matrix Properties of the scattering matrix Current operator DC current Summary
Current operator
Second quantization
operators creating/annihilating particles in quantum states
a†α(E )/aα(E )→ ψ(in)α (E )/
√vα(E )
b†α(E )/bα(E )→ ψ(out)α (E )/
√vα(E )
a†α(E)aβ(E ′) + aβ(E ′)a†α(E) = δαβδ(E − E ′)
b†α(E)bβ(E ′) + bβ(E ′)b†α(E) = δαβδ(E − E ′)
Ψα(t, r) =1√
h
∫ ∞0
dE e−i E~ t
{aα(E)
ψ(in)α (E , r)√
vα(E)+ bα(E)
ψ(out)α (E , r)√
vα(E)
}
Ψ†α(t, r) =1√
h
∫ ∞0
dE e i E~ t
{a†α(E)
ψ(in)∗α (E , r)√
vα(E)+ b†α(E)
ψ(out)∗α (E , r)√
vα(E)
}
Alexandra Nagy The Landauer-Büttiker formalism
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Content Motivation The scattering matrix Properties of the scattering matrix Current operator DC current Summary
Current operator
Iα(t, x) =i~e2m
∫dr⊥
{∂Ψα†(t, r)
∂xΨα(t, r)− Ψα†(t, r)
∂Ψα(t, r)
∂x
}
The basis wave functions
ψ(in)(E , r) = ξE (r⊥)e−ik(E)x
ψ(out)(E , r) = ξE (r⊥)e ik(E)x
Much smaller bias than the Fermi-energy, µ0
|E − E ′| << E ∼ µ0
⇓
Iα(t) =
∫ ∫dE dE ′e i E−E′
~ t{b†α(E)bα(E ′)− a†α(E)aα(E ′)
}Alexandra Nagy The Landauer-Büttiker formalism
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Content Motivation The scattering matrix Properties of the scattering matrix Current operator DC current Summary
Measurable current - 〈Iα〉
Propagating from reservoir - a†α(E )/aα(E )
equilibrium in the reservoirs
quantum-statistical average ⇒ Fermi-distribution
〈a†α(E)aβ(E ′)〉 = δαβδ(E − E ′)fα(E)
〈aα(E)a†β(E ′)〉 = δαβδ(E − E ′) {1− fα(E)}
Scattered particles - b†α(E )/bα(E )
non-equilibrium particles
calculate them from the in-coming particle operators
Ψ(in) =∑Nrα=1 aα
ψ(in)α√vα
Ψ(out) =∑Nrβ=1 bβ
ψ(out)β√vβ
⇓
bα =∑Nrβ=1 Sαβ aβ b†α =
∑Nrβ=1 S
∗αβ a†β
Alexandra Nagy The Landauer-Büttiker formalism
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Content Motivation The scattering matrix Properties of the scattering matrix Current operator DC current Summary
DC current and the distribution functions
Scope
under the DC bias: ∆Vαβ = Vα − Vβ
the chemical potentials: µα = µ0 + eVα
energy: E = Ekin + Epot is conserved (in the stationary case)
Distribution functions
Averaging:
〈a†α(E)aα(E ′)〉 = δ(E − E ′)f (in)α (E)
〈b†α(E)bα(E ′)〉 = δ(E − E ′)f (out)α (E)
Average number of e−-s: dEhf
(in/out)α (E)
For in-coming electrons: f(in)α (E) = fα(E)
Iα = eh
∫dE{f
(out)α (E )− f
(in)α (E )
}Alexandra Nagy The Landauer-Büttiker formalism
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Content Motivation The scattering matrix Properties of the scattering matrix Current operator DC current Summary
Distribution for scattered electrons - f(out)α (E )
δ(E − E ′)f(out)α (E) ≡ 〈b†α(E)bα(E ′)〉 =
=
Nr∑β=1
Nr∑γ=1
S∗αβ(E)Sαγ(E ′)〈a†β(E)aγ(E ′)〉 =
=
Nr∑β=1
Nr∑γ=1
S∗αβ(E)Sαγ(E ′)δ(E − E ′)δβγ fβ(E)
⇓
f (out)α (E ) =
Nr∑β=1
|Sαβ(E )|2fβ(E )
Iα =e
h
∫dE
Nr∑β=1
|Sαβ(E )|2 {fβ(E )− fα(E )}
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Content Motivation The scattering matrix Properties of the scattering matrix Current operator DC current Summary
DC current conservation
Current conservation
Electrical charge continuity equation:
divj + ∂ρ∂t = 0∫
it over the volume enclosed by the surface∑
Nr∑α=1
Iα(t) +∂Q
∂t= 0
⇒ both in stationary and non-stationary case:
Nr∑α=1
Iα = 0
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Content Motivation The scattering matrix Properties of the scattering matrix Current operator DC current Summary
DC current conservation
Does Iα satisfy the conservation law?
The unitarity of the scattering matrix:
S†S = I ⇒Nr∑α=1
|Sαβ(E)|2 = 1
Substituting back:
Nr∑α=1
Iα =e
h
∫dE
Nr∑α=1
Nr∑β=1
|Sαβ(E)|2 {fβ(E)− fα(E)} =
=e
h
∫dE
Nr∑β=1
fβ(E)
Nr∑α=1
|Sαβ(E)|2 −Nr∑α=1
fα(E)
Nr∑β=1
|Sαβ(E)|2 =
=e
h
∫dE
Nr∑β=1
fβ(E)−Nr∑α=1
fα(E)
= 0
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Content Motivation The scattering matrix Properties of the scattering matrix Current operator DC current Summary
Summary
Derivation of the scattering matrix method
Problem of transport phenomena in macroscopic sample
De�nition of current operator in 2nd quantization
Measurable current for DC bias
DC current conservation
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Content Motivation The scattering matrix Properties of the scattering matrix Current operator DC current Summary
Thank you for your attention!
Alexandra Nagy The Landauer-Büttiker formalism