cbr{}

Specifications that define CBR (Contact Block Reduction 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

  • Contact Block Reduction 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 \(T(E)\).

ildos

value

yes or no

default

no

Outputs integrated local density of states.

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.

Transmission using the CBR method

The transmission calculated with the CBR method using all eigenstates and only 7% of the eigenstates. In the latter case, the transmission is still very accurate for the lower energies.

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.