# cbr{}¶

Specifications that define CBR (**C**ontact **B**lock **R**eduction method) calculation, i.e. ballistic current calculations.

This method is based on the following publications: [BirnerCBR2009], [MamaluyCBR2003]

At a glance: CBR current calculation

full 1D, 2D and 3D calculation of quantum mechanical ballistic transmission probabilities for open systems with scattering boundary conditions

**C**ontact**B**lock**R**eduction method:only incomplete set of quantum states needed (~ 100)

reduction of matrix sizes from \(O(N^3)\) to \(O(N^2)\)

ballistic current according to Landauer-Büttiker formalism

The CBR method is an efficient method that uses a limited set of eigenstates of the decoupled device and a few propagating lead modes to calculate the retarded Green’s function of the device coupled to external contacts. From this Green’s function, the density and the current is obtained in the ballistic limit using Landauer’s formula with fixed Fermi levels for the leads.

It is important to note that the efficiency of the calculation and also the convergence of the results are strongly dependent on the cutoff energies for the eigenstates and modes. Thus it is important to check during the calculation if the specified number of states and modes is sufficient for the applied voltages. To summarize, the code may do its job very efficiently but is far away from being a black box tool.

cbr{ name = "qr" # CBR quantum region lead{ name = "lead_1" } # lead quantum region rel_min_energy = -0.01 # lower boundary (relative) rel_max_energy = 0.3 # upper boundary (relative) abs_min_energy = 2.5 # lower boundary (absolute) abs_max_energy = 2.6 # upper boundary (absolute) delta_energy = 1e-6 # energy grid resolution ildos = yes # outputs integrated LDOS options = [1, 0, 0] # for one particle model }

## name¶

- value
“string”

- example

`qr_device`

refers to quantum region to which CBR method will be applied (\(d\)-dimensional)

## lead{}¶

- name

- value
“string”

- example

`qr_lead1`

Provides the name of the quantum region of the lead.

Refers to lead quantum region. Make sure that the lead region specified here has dimension \(d-1\).

## rel_min_energy¶

- value
double

- default

`-0.01`

#`[eV]`

Lower boundary for transmission energy interval relative to lowest eigenvalue

## rel_max_energy¶

- value
double

- default

`1.01`

#`[eV]`

Upper boundary for transmission energy interval relative to highest eigenvalue

## abs_min_energy¶

- value
double

- default

`0.0`

#`[eV]`

Lower boundary for transmission energy interval on an absolute energy scale

## abs_max_energy¶

- value
double

- default

`0.0`

#`[eV]`

Upper boundary for transmission energy interval on an absolute energy scale

Note

- Specify either
`rel_min_energy`

and`rel_max_energy`

or`abs_min_energy`

and`abs_max_energy`

.

## delta_energy¶

- value
double

- default

`1e-4`

#`[eV]`

This value determines the resolution of the transmission curve \(T(E)\).

## options¶

- value
double array

- default

`[1, 0, 0]`

3 values for one-particle model

`[1, 0, 0]`

11 values for two-particle model

`[., ..., .]`

numStates2_ = (int)options_[0]; const double epsRel = options_[1]; const DVector3 r1(options_[2]*uNanometer,options_[3]*uNanometer,options_[4]*uNanometer); const DVector3 r2(options_[5]*uNanometer,options_[6]*uNanometer,options_[7]*uNanometer); const double delta = options_[8]*uEVolt; // splitting const double z = options_[9]*uEVolt; // tunneling // [prefactor] = Q^2/[cEps0], [cEps0] = Q/L*V => [prefactor] = Q L V = eV L const double prefactor = options_[10] * sqr(cEcharge)/(4*Pi*epsRel*cEps0);

- Example
The following figure shows the calculated transmission from lead 1 to lead 3 as a function of energy \(T_{13}(E)\). Full line: All eigenfunctions of the decoupled device are taken into account. Dashed line: Only the lowest 7% of the eigenfunctions are included. Here, Neumann boundary conditions are used for the propagation direction. The vertical line indicates the cutoff energy, i.e. the highest eigenvalue that is taken into account.

- Additional notes
Special boundary conditions for CBR method

Along propagation direction

*Neumann*boundary conditions are applied to the Schrödinger equation.Perpendicular to the propagation direction

*Dirichlet*boundary conditions are applied to the Schrödinger equation.

Note

Physically speaking, the lead quantum cluster must be a two-dimensional surface in a 3D simulation, a one-dimensional line in a 2D simulation and a zero-dimensional point in a 1D simulation.