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 GUI: nextnanomat
 Tool: nextnano³
 Tool: nextnano++
 Tool: nextnano.QCL
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Basics 1
Basics 2
 

Overview

Note: This documentation is nearly 10 years old and might not be related any more to the actual version of the nextnano³ code.
It is still displayed here for historical reasons until a decent manual for nextnano³ will be available.
Interested readers should consult the diploma and PhD theses of S. Hackenbuchner, M. Sabathil, T. Zibold (nextnano++) and T. Andlauer (nextnano++) available from the publications website.

 

What can be calculated with this code? (This is for your motivation!)
What does one need to know to get started?
What are the basic Semiconductor equations?
Introduction
What is the method of the program?
What are the computational schemes of the program?
How successful was nextnano3 in the past?
What kind of input is needed?
What kind of output can be calculated?
What is the structure of the program?
Want to know more?
Additional documentation
   

What can be calculate with this code?

  • nextnano3 is a simulator for calculating, in a consistent manner, the realistic electronic structure of three-dimensional heterostructure quantum devices under bias and its current density close to equilibrium. The electronic structure is calculated fully quantum mechanically, whereas the current is determined by employing a semiclassical concept of local Fermi levels that are calculated self-consistently.

 

What does one need to know to get started?

 

 

   

What are the basic Semiconductor equations?

  • The solving of the Schrödinger, Poisson, current continuity equations for electrons and holes and the current relations for electrons and holes.
  • The basic semiconductor equations:
The Poisson equation:
The current continuity equation for electrons
and holes:
The current relations for electrons
and holes:

 

 

Introduction

  • A realistic simulator of three-dimensional semiconductor nano structures and optoelectronic nano devices should meet two requirements.
    Firstly it should model the electronic structure of any combination of quantum wells, wires, and dots accurately on a length scale from nm to µm.
    Secondly, a device simulator should self-consistently account for the charge redistribution under applied voltage and for the resulting current [compelectron1].
    Recently, several methods have been published that can realistically predict the equilibrium electronic structure of 3D nano structures. they are based on one-band [kumar1], or several-band k.p models [grundmann1, pryor1, pryor2, cusack1], tight binding methods [dicarlo1] or pseudopotential techniques [wang1, dicarlo1]. Some of them include free and bound charge redistributions self-consistently [kumar1, grundmann1]. Most quantum transport methods that include the electronic structure beyond a one-band description are still limited to one spatial dimension [lake1, fischetti1, ferry1]. Thus, a simultaneous realistic treatment of the electronic structure and the quantum transport problem for 3D structures still poses a challenging task.
  • State-of the-art simulators for semiconductor nano structures and optoelectronic nano devices roughly fall into two classes [compelectron1]:
    some models focus on the equilibrium electronic structure. They attempt to predict as accurately as possible, the free and bound charge density as well as optical properties of quantum wells, wires, and dots on a lengths scale that ranges from nm to µm. Several models of this kind have been developed in the last few years that can deal with fully three-dimensional device geometries, and invoke one-band, or several-band k.p models, tight binding methods.
    The second class of models focus on current-voltage characteristics and attempts to solve quantum transport, using nonequilibrium Green functions [lake1, klimeck1], Wigner functions [bordone1, grubin1, ferry1] or the Pauli master equation [fischetti1, fischetti2]. Presently, they are still limited to one spatial dimension and/or put less emphasis on details of the electronic structure and the quantum transport problem for 3D structures still poses a challenging task.
  • We are currently developing a simulator for a wide class of 3D Si and III-V nano structures [hackenbuchner1]. It attempts to bridge the two types of approaches described above, albeit with a stringent limitation that makes it feasible to simulate three-dimensional structures: we solve the electronic structure problem accurately but restrict the current evaluation to situations close to equilibrium where the concept of local quasi-Fermi levels is still justifiable. This approach may be viewed as a low-field approximation to the Pauli master equation [jones1].

What is the method of the program?

  • The nano-device simulator that we have developed so far solves the 8-band-k.p-Schrödinger-Poisson equation for arbitrarily shaped 3D heterostructure device geometries, and for any (III-V and Si/Ge) combination of materials and alloys. It includes band offsets of the minimal and higher band edges, absolute deformation potentials [vandewalle1], local density exchange and correlations (i.e. the Kohn-Sham equations), total elastic strain energy [pryor3, grundmann2] that is minimized for the whole device, the long-range Hartree potential induced by charged impurity distributions, voltage induced charge redistribution, piezo- and pyroelectric charges, as well as surface charges, in a fully self-consistent manner. The charge density is calculated for a given applied voltage by assuming the carriers to be in a local equilibrium that is characterized by energy-band dependent local quasi-Fermi levels EFc(x) for charge carriers of type c, (i.e. in the simplest case, one for holes and one for electrons)

      (1)
  • These local quasi-Fermi levels are determined by global current conservation,



    where the current is assumed to be proportional to the density and to the gradient of the quasi-Fermi level (associated with each band) exactly as in the semi-classical limit (see e.g. [selberherr1]).

     

    Recombination and generation processes are included additionally. The carrier wave functions Psiic and energies Eic are calculated by solving the multi-band Schrödinger-Poisson equation. The open system is mimicked by using mixed Dirichlet and Neumann boundary conditions [fischetti1, lent1, frensley1] at ohmic contacts. The charge density at these contacts is assumed to be equal to the bulk equilibrium density. Thus, the quasi-Fermi levels and the potential in the contact region are fixed according to the applied voltage. Our method leads to globally orthogonal eigenstates including valence (split-off, light and heavy hole) and conduction band states. Further, it automatically includes tunneling, and yields optical transition energies and as well as optical matrix elements.
  • According to equation (1), one and the same bound state Psiic(x) may get occupied differently at different positions according to the spatial dependence of EFc(x). This is a consequence of invoking the well-defined but semiclassical concept of local Fermi levels together with nonlocal quantum mechanics. Fortunately, no conflict arises for situations close to equilibrium since the spatial variation of the occupancy of any given eigenstate turns out to be negligible for three reasons:
    (i) Deeply bound states do not contribute to the current and thus do not lead to a gradient of the Fermi level.
    (ii) The Fermi level has the largest variation in regions where the density is very low (within barriers, for example).
    (iii) Very extended states that are treated formally as bound states in our method are either not occupied because of their high energy, or occur in regions of high density (near contacts, for example) where the quasi-Fermi level is nearly constant.
  • The computational methods [see computational scheme for more details] solve the Kohn-Sham-Schrödinger, Poisson and current continuity equations iteratively using conjugate gradient, inverse-iteration and predictor-corrector methods [trellakis1] in an inhomogeneous finite difference framework.

 

What are the computational schemes of the program?

  • For a given nano-structure, the computations start by globally minimizing the total elastic energy [pryor3, grundmann2] using a conjugate gradient method. This determines the piezo-induced charge distributions, the deformation potentials and band offsets. Subsequently, the 8-band-Schrödinger, Poisson, and current continuity equations are solved iteratively. All equations are discretized according to the finite difference method invoking the box integration scheme [kumar1, selberherr1]. The irregular rectilinear mesh is kept fixed during the calculations. As a preparatory step, the built-in potential is calculated for zero applied bias by solving the Schrödinger and Poisson equation self-consistently employing a predictor-corrector approach [trellakis1] and setting to zero the electric field at ohmic contacts. For applied bias, the Fermi level and the potential at the contacts are then shifted according to the applied potential which fixes the boundary conditions. The main iteration scheme itself consists of two parts.
    In the fist part, the wave functions and potential are kept fixed and the quasi-Fermi levels are calculated self-consistently from the current continuity equations, employing a conjugate gradient method and a simple relaxation scheme.
    In the second part, the quasi-Fermi levels are kept constant, and the density and the potential are calculated self-consistently from the Schrödinger and Poisson equation. The discrete 8-band Schrödinger equation represents a huge sparse matrix (typically of dimension 105 for 3D structures) and is diagonalized using the iterative methods that yields the required inner eigenvalues and eigenfunctions close to the energy gap. We very slightly shift the spin-up and spin-down diagonal Hamiltonian matrix elements with respect to each other in order to avoid degeneracies and guarantee orthogonal eigenstates automatically. To reduce the number of necessary diagonalizations, we employ an efficient predictor-corrector approach [trellakis1] to calculate the potential from the nonlinear Poisson equation. In this approach, the wave functions are kept fixed within one iteration and the density is calculated perturbatively from the wave functions of the previous iteration [trellakis1]. The nonlinear Poisson equation is solved using a modified Newton method, employing a conjugate gradient method and line minimizations. The code is written in Fortran 2003 and consists of some 200.000 lines by now.

How successful was nextnano3 in the past?

  • Piezoelectric fields and electron-hole localization in quantum dots
    We have applied our simulator to study theoretically single quantum-dot photodiodes [findeis1, frey1, itskevich1, kapteyn1, hackenbuchner1] consisting of self-assembled InGaAs quantum dots with a diameter of 30-40 nm and heights of 4-8 nm that are embedded in the intrinsic region of a Schottky diode.
     

    Towards fully quantum mechanical 3D device simulation
    M. Sabathil, S. Hackenbuchner, J.A. Majweski, G. Zandler, P. Vogl
    submitted to Journal of Computational Electronics (2001)

    Nonequilibrium band structure of nano-devices
    S. Hackenbuchner, M. Sabathil, J.A. Majweski, G. Zandler, P. Vogl, E. Beham, A. Zrenner, P. Lugli
    to appear in Physica B (2001)

     

What kind of input is needed?

  • dimension of sample (1D, 2D or 3D)
  • materials and shape of the heterostructure
  • applied bias if any
  • doping if any
  • Where are the quantum regions, where should be calculated classically?
  • specification of desired output

What kind of output can be calculated?

  • band structure
  • strain
  • piezo- and pyroelectric charges
  • electron/hole densities (space charge)
  • electrostatic potential
  • current
  • wave functions

What is the structure ("Setup") of the program?

  • The input file is processed, i.e. the material data will be read in from the data base and the geometry will be mapped on the grid.
  • The strain is calculated.
  • The band edges will be calculated by taking account of the van-de-Walle model and the strain. This can possibly lead to a splitting of degenerate energy states.
  • The piezoelectric and pyroelectric  polarization charges will be determined.
  • The program sets up quantum regions and allocates quantum states, and all other relevant variables that contain the physical solutions.
  • The main program starts.
  • A starting value for the potential is determined.
  • The nonlinear Poisson equation in thermodynamic equilibrium will be determined leading to the built-in potential.
  • Then the program can continue differently which will be discussed later.
  • Eventually the results will be written into the specified files.
  • Finally, here is a more detailed explanation of the structure of the program ("Setup").

Want to know more?

Go to Basics and you will get a more deeper understanding about how all fits together...

Additional documentation

(pdf/dvi/doc files)