Contact block reduction method for ballistic transport and carrier densities of open nanostructures

A method is presented for quantum-mechanical ballistic transport calculations of realistic two- and three-dimensional open devices that may have any shape and any number of leads. Observables of the open system can be calculated with an effort comparable to a single calculation of a suitably defined closed system. The method is based on a previously developed scheme for calculating transmission functions, the contact block reduction method, and is shown to be applicable to the density matrix, the density of states, and the local carrier density. The electronic system may be characterized by a single or multiband Hamiltonian. We illustrate the method for the four-band GaAs hole transport through a two-dimensional three-terminal T-junction device and for the electron tunneling through a three-dimensional InAs quantum dot molecule embedded into an InP heterostructure.

Figure 1

Schematic picture of local coordinate system of an individual semi-infinite lead attached to the device. The dots indicate the two- or three-dimensional discrete grid points. Along the lead axis $z$, the lattice spacing is $a$. The origin of the $z$ axis lies at the contact boundary.

Figure 2

Contour plot of the difference between the exact and mode truncated transmission function for the device shown in the inset. The parameters used are $W_D=5\mathrm{nm},$ $W_L=10\mathrm{nm},$ $L=10\mathrm{nm},$ $m^{*}=0.3m_0$, and a constant grid spacing of 0.167 nm. The total number of modes is $M=61$.

Figure 3

Sketch of the T-junction device used in the calculation. The leads are indicated by the patterned regions.

Figure 4

Lower box, subband structure of the lead $L_1$. Energy is in eV, the wave vector $k$ lies along the [011] direction. Upper box, corresponding group velocity in ${10}^5\mathrm{cm}∕\mathrm{s}$ as a function of energy in each band. The perpendicular wave vector along $\left[01\overline{1}\right]$ has been set equal to zero.

Figure 5

Lower box, subband structure of the lead $L_2$. Energy is in eV, the wave vector $k$ lies along the [100] direction. Upper box, corresponding group velocity in ${10}^5\mathrm{cm}∕\mathrm{s}$ as a function of energy in each band. The perpendicular wave vector along $\left[01\overline{1}\right]$ has been set equal to zero.

Figure 6

The transmission functions $T_{12}\left(E\right)=T_{13}\left(E\right)$ and $T_{23}\left(E\right)$ of the T-junction, plotted as a function of energy in eV. The perpendicular wave vector along $\left[01\overline{1}\right]$ has been set equal to zero. The exact calculations (solid curves) are compared to results with a reduced set of eigenstates, either using 25% (dashed curves) or using 10% (dotted curves) of all eigenstates.

Figure 7

Density of states of the T-junction in units of ${10}^3{\mathrm{eV}}^{-1}$ as a function of energy in eV. The perpendicular wave vector along $\left[01\overline{1}\right]$ has been set equal to zero. The exact calculations (solid curves) are compared to results with a reduced set of eigenstates, either using 25% (dashed curves) or using 10% (dotted curves) of all eigenstates.

Figure 8

The total conductance is calculated via the integration of the transmission in $k$ space. The exact result (solid curves) is compared to the transmission obtained using only the lowest 10% of all eigenstates (dashed curves).

Figure 9

Contour plot of the hole density in units of ${10}^{18}{\mathrm{cm}}^{-3}$ for the T-junction at zero temperature. The Fermi level is set to 0.05 eV.

Figure 10

Contour plot of the lateral bound state for holes. The density is given in units of ${10}^{17}{\mathrm{cm}}^{-3}$, the other parameters are the same as in the preceding figure.

Figure 11

The wave vector resolved density of states of the T-junction as a function of energy in eV and wave vector $k_{\left[01\overline{1}\right]}$ in units of ${\mathrm{\AA }}^{-1}$. Dark regions indicate a high density of states on a linear scale. The open circles and squares indicate the subband edges of the isolated 6 nm and 4 nm quantum wells, respectively. The solid squares depict the location of the bound states.

Figure 12

The two InAs quantum dots (QDs) on top of the InAs wetting layers (WLs) are embedded within InP barrier material. The two faces in growth direction act as source and drain contacts and consist of doped ${\mathrm{In}}_{0.6}{\mathrm{Ga}}_{0.4}\mathrm{As}$.

Figure 13

Current in units of pA for the quantum dot molecule based resonant tunneling structure as a function of the applied bias in V. The scale for the experimentally observed and the calculated current is shown on the left-hand and right-hand vertical axis, respectively.