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Nanowire structure
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Landauer-Buttikerformalism
Electrodes: macroscopic reservoirs (L>>Lphi); experience
inelasticscattering; thermalize and reach an equilibrium
distribution function(Fermi distribution function)
Channel (mesoscopic system): L
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Process (secondquantization)
In second quantization, the hamiltonian operator can be
written
Single-particle first quantization hamiltonian for aChannel with
one impurity
Time evolution of the field operator
Equation to be solved
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Process (secondquantization)
Introducing Energy instead of momentum k, and integrating over
all energies range
The general solution of this differential equation is
a's and b's are called amplitude operators.
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Scattering formalism
Scattering matrix
Transmission and Reflection coefficients
Scattering matrix coefficients depend solely on the channel
andare computed imposing boundary conditions to thedifferential
equation.
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Examples
For one impurityin the channel
For two impuritiesin the channel
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Current operator
In second-quantized form
Where we can use the field operator we just found (usually we
compute current in the leads)
In the left lead
After linearization around Ef
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Current calculation
To compute the current we average over many samples (canonical
ensemble)
Where and
From scattering matrixEnergy Density operator inSecond
quantization
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Landauer-Buttiker formula
Finally one gets the Landauer-Buttiker formula for the current
in the leads
As simple results one can compute the linear conductance of a
sample
Even if there are no impurities in the channel (T=1), a
resistance arises anyway.It is related to the coupling of the
mesoscopic (ballistic) system with the macroscopiccontacts.
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Self-consistent field
Source and drain are macroscopic contacts with fermi
distribution functionsf1(E) and f2(E).
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Effect of Vg on the channel
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What makes current flow?
When a voltage V is applied between source and drain
An energy level E inside the channel sees two differentCharge
flows: Source would like to fill it with electrons to see f1(E);
Drain would like to empty it to see f2(E).
Equilibrium is never reached.
Current from the source Current from the drain
At steady-state
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Broadening of the level
Part of the energy level spreads outside the energy range
between the electrochemicalPotentials where current flows.The
current is then reduced by a factor
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Density of level D(E)
Considering the density of level D(E) inside the channel, with
broadening becomes
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Current and number ofelectrons
To compute the current in presence of broadening, we have to
integrate over all theEnergy range
Similarly, the number of electrons N at steady-state becomes
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Potential profile
To inlcude the effect of gate voltage, potential inside the
channel must be computed.
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Iterative procedure
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A simple comparison
From Landauer-Buttiker theory
Datta
