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11.18 Molecular Junctions

In molecular junctions, molecules bridge two metallic electrodes. The conductance and current-voltage relationship of molecular junctions can be calculated using either Landauer or Non-Equilibrium Green’s Functions (NEGF). In both cases, the Green’s function formulation is employed using a chosen level to describe the electronic structure. See Refs. Datta:2005,DiVentra:2008 for further introduction.

In molecular junctions the current-voltage curve depends on the electron transmission function, which can be calculated using the quantum transport code developed by the Dunietz group (Kent State). The scattering-free approach, (Landauer), provides a zero-bias limit, whilst the non-equilibrium approach, (NEGF), iteratively solves for the junction under the effect of the finite-voltage biased system.

This quantum transport utility is invoked by setting the $rem variable TRANS_ENABLE.

TRANS_ENABLE

To invoke the molecular transport code.

TYPE:

INTEGER

DEFAULT:

0	Do not perform transport calculations (default).

OPTIONS:

1	Perform transport calculations.
	Print matrices needed for generating bulk model files.

RECOMMENDATION:

None

Output is provided in the Q-Chem output file and in the following files (for closed shell system in the spin-restricted framework):

• transmission.txt (Transmission function in the requested energy window)

• TDOS.txt (Total density of states)

• current.txt (I-V plot only for the Landauer level)

• FAmat.dat (Hamiltonian matrix for follow up calculations and analysis)

• Smat.dat (Overlap matrix for follow up calculations and analysis)

• IV-NEGF-all.txt (I-V plot obtained by NEGF method)

In the case of unrestricted spin the transmission is calculated for each spin-state as indicated by the A[B] appended to the file names listed above (e.g. transmissionA.txt and transmissionB.txt). (The file name with the letter AB indicates output data including both spin states (e.g. transmissionAB.txt). We note that in the closed-shell spin-restricted case, the transmission.txt corresponds to the spin, where the total transmission due to the spin symmetry is twice the values included in the file. In the NEGF calculation, the above output files are placed in the directories, Vbias1, Vbias2, (the numbers in the directory names are index of bias voltage), where these output files in each directory are of data at the given voltage.

T-Chem requires setting parameters in two transport-specific sections in the input file:

• $trans_model (molecular model regions)

• $trans_method (Specifies the mode to calculated the electrode self-energies)

Figure 11.1: Illustration for the different regions of the molecular junction for Landauer calculation.

Figure 11.2: Illustration for the different regions of the molecular junction for NEGF calculation.

The $trans_model section provides the number of basis functions in the different regions (molecular model partitioning). There are no default values given for these parameters. The different regions are illustrated in Fig. 11.1 with a six carbon atom chain based bridge used as an example of the Landauer calculation. The partitioning as well as the scheme for the NEGF calculation is illustrated in Fig. 11.2. In these systems the electrodes are represented by a chain of Au atoms. In this example of electrode wires, each single atom represents one layer.

The necessary parameters in the $trans_model section are as follows:

• trans_latom: INTEGER, atom index in the $molecule section where the central/bridge region starts (the first atom in junction area).

• trans_ratom: INTEGER, atom index in the $molecule section where the central/bridge region ends (the last atom in junction area).

• trans_lgatom: INTEGER, atom index where the repeat unit of the left electrode starts.

• trans_rgatom: INTEGER, atom index where the repeat unit of the right electrode ends.

See Fig. 11.1 for illustrations of the different regions, and the way the parameters define them. Atoms within numbers trans_lgatom, trans_lgatom+1, .., trans_latom-1 define the repeat unit of the left electrode (similarly for right electrode).

Alternatively, the different regions can be provided with the atomic orbital (AO) index as follows:

• trans_lbasis: INTEGER, the number of basis functions appearing to the left of the device region (the index of the first AO within the central region).

• trans_rbasis: INTEGER, the number of basis functions appearing to the right of the device region right electrode (total number of basis functions minus this number equals last AO that is within the device).

• trans_lgbasis: INTEGER, the number of basis functions of the repeat unit of the left electrode.

• trans_rgbasis: INTEGER, the number of basis functions of the repeat unit of the right electrode.

For the NEGF calculation, trans_lbasis and trans_lgbasis must be the same number as shown in Fig. 11.2 (as for trans_rbasis and trans_rgbasis.

Note: The assignment of trans_latom etc. has priority. If trans_latom is specified, then trans_lbasis is ignored. Similarly for trans_latom and trans_lbasis.

For the example in Fig. 11.1, if there are a total of 18 atoms and the Au and Ag basis sets each contain 22 basis functions per atom and the repeat unit includes a pair of Au Ag atoms, then the parameters should be given as follows (only the first two columns are required, the rest are included for explanation):

Or use the atom numbers corresponding to their position in the $molecule section:

In this example, we have used for the same repeat unit for the left and right electrodes; this symmetry is not required.

Note: The order of atoms in the $molecule section is important and requires to following:

The atoms are provided first by the leftmost repeat unit with the left electrode then proceeds to the next repeat unit up to the surface unit. Next the bridge atoms are provided followed by the right surface unit and the right electrode region. The right electrode region starts with the surface layer and ends with the most distant layer within the bulk. The atoms order within each electrode layer (the repeat units) must be consistent. The atoms order within the bridge region (excluding the electrode repeat unit atoms) is arbitrary.

That is, the order of atoms in the molecule section has to adhere to the following:

2. atoms of the leftmost repeat unit

3. atoms of the next repeat unit

4. atoms of the left surface unit device

5. bridge atoms

6. atoms of the right surface unit

7. atoms of the next right electrode unit

8. atoms of the rightmost repeat unit

T-Chem allows for complete flexibility in determining the different regions of the electrode models. As a consequence, incorrect setting of regions is not caught by the program and may produce transmission functions that are unphysical (e.g. large values or even negative). Such errors can occur where the cluster model is partitioned (by mistake) within the orbital space of an atom. Regions must always be defined between atomic layers. Each repeat unit atoms should be always provided with the same internal order.

Note: At least a single repeat unit of the electrodes should be included in the bridge region. With the Landauer model, if trans_readhs == 0, then at least one layer beyond the bridge region has to be included, an additional layer (total of two or more) is required when trans_method != 0.

The necessary parameters in $trans_method section are listed as follows,

trans_mode

Mode of calculation.

INPUT SECTION: $trans_method
TYPE:

INTEGER

DEFAULT:

1	Landauer level.

OPTIONS:

3	A self-consistent Green’s function calculation with zero bias voltage (i.e. NEGF with zero bias, which is used for preparation of full NEGF).
4	Full NEGF level

RECOMMENDATION:

For modes 3 and 4 SCF_ALGORITHM = NEGF must be set in in the $rem section.

trans_spin

Spin coupling scheme.

INPUT SECTION: $trans_method
TYPE:

INTEGER

DEFAULT:

0

OPTIONS:

0	For restricted spin calculations or closed-shell singlet states.
3	For unrestricted spin calculations or open-shell systems

RECOMMENDATION:

None

trans_method

Electrode surface GFs model.

INPUT SECTION: $trans_method
TYPE:

INTEGER

DEFAULT:

0

OPTIONS:

0	A wide band limit with a constant parameter trans_greens (default)
1	WBL using the Ke-Baranger-Yang TB at the Fermi energy.
2	WBL using the Lopez-Sancho TB at the Fermi energy.
3	Tight-binding (TB) following the procedure proposed by Ke-Baranger-Yang
4	TB following the procedure proposed by Lopez-Sancho (decimation).

RECOMMENDATION:

Only option 0 is available for the NEGF calculation at the current version.

trans_npoint

Number of grid points within the energy window of the transmission spectra calculation.

INPUT SECTION: $trans_method
TYPE:

INTEGER

DEFAULT:

300

OPTIONS:

User-specified number of points

RECOMMENDATION:

None

trans_readhs

Flag to read the Hamiltonian and overlap matrices for the bulk model.

INPUT SECTION: $trans_method
TYPE:

INTEGER

DEFAULT:

0	Use the current Hamiltonian and overlap matrices to parse the electrode integrals.

OPTIONS:

1	Use pre-calculated electrode Hamiltonian and overlap matrices.

RECOMMENDATION:

If set to 1, the following files are requred: FAmat2l.dat and Smat2l.dat (for left electrode model), FAmat2r.dat and Smat2r.dat (for right electrode model). If both electrodes are of the same type, may use symbolic links of these files to the same matrices. (For unrestricted spin model, FBmat2l.dat and FBmat2r.dat are also necessary)

Note: NEGF requires trans_readhs to be set!

trans_htype

Determines the TB property on the relevant coupling terms.

INPUT SECTION: $trans_method
TYPE:

INTEGER

DEFAULT:

0

OPTIONS:

When trans_readhs = 0:
0	All coupling integrals between the junction and electrode functions are allowed as determined at the cluster model level (no screening imposed).
1	Only coupling between neighboring repeating units of the electrode model are allowed – all terms of the repeating units elements that are beyond the neighboring unit are set to zero.
2	Force both the TB coupling terms and the self-energy coupling terms to the same value as determined by the electrode model. If trans_method != 0, the inequalities trans_lbasis trans_lgbasis and trans_rbasis trans_rgbasis must be satisfied.
When trans_readhs = 1:
2	Available using the pre-calculated electrode data.
3	The same way as 1 but using the pre-calculated electrode data. This is only for NEGF calculation.

RECOMMENDATION:

None

Further options are summarized below:

trans_printdos (Integer):
Controls the printout of TDOS.

trans_printiv (Integer):
Controls printout of calculated current.

trans_ipoints (Integer):
Number of points for current calculation.

trans_adjustefermi (Integer):
Flag to adjust the Fermi energy (FE) for trans_mode = 3 or 4

Options 1, 2, and 3 use the same FE for and spins. -1, -2, -3 are the same as 1, 2, 3, respectively, but allow for different FEs for and spins.

trans_nvbias (Integer):
(Only for trans_mode = 4), number of points of bias voltage.

The bias voltage values to be calculated are defined by dividing the range between trans_vstart and trans_vmax with this number. For example, when trans_vstart = 0.0, trans_vmax = 1.0, and trans_nvbias = 5, the voltages are 0.0, 0.25, 0.50, 0.75, and 1.0. For the case of trans_nvbias = 1, the voltage to be calculated is trans_vmax.

trans_updatedmatlr (Integer):
(Only for trans_mode = 3), flag to update L, R, LC, RC blocks (i.e. except for center block) of density matrix during SCF with zero bias. This is to prepare consistent density matrix with modified Hamiltonian matrix forced by read-in bulk electrode data. Note that it may make it difficult or slow to converge.

The following are parameters are set to a double precision value. The allowed values are set using trans_itodfac as follows:

trans_itodfac (Integer):
Controls the accuracy for input parameters.

trans_vstart (Double):
(Only for trans_mode = 4). Starting voltage bias (V). The bias voltage increases from trans_vstart to trans_vmax in the NEGF calculation.

trans_devsmear (Double):
Imaginary smearing (in eV) added to the real Hamiltonian in central region retarded GF evaluation.

trans_bulksmear (Double):
Imaginary smearing (in eV) added to the real Hamiltonian in electrodes GF evaluation.

trans_greens (Double):
Imaginary smearing/Broadening (in eV) added to the Green’s function.

trans_efermi (Double):
Fermi energy of the electrode (for spin) (in eV) used for defining energy range in calculating current for T(E).

trans_efermib (Double):
Fermi energy for spin (in eV). If this is not given, the same value of spin is used.

trans_vmax (Double):
Maximum voltage bias (V).

trans_gridoffset (Double):
(Only for trans_mode = 4), Offset distance (in Å) to define the grid box region for bias potential energy.

The box size is defined by adding the offset distance with maximum and minimum , and atomic coordinates for each direction. The bias voltage on the grid points in the box is used for calculating correction term for the Fock matrix (i.e. ). Note that this box is not for correcting electrostatic potential by solving Poisson equation. The grid box region and its grid size for the Poisson equation is given by $plots block keyword (see also example later). The same grid size is used for both boxes.

The following are parameters for when trans_mode = 3, 4:

trans_mixing (Double):
Mixing ratio of DIIS mixing method for updating the central block of density matrix in the NEGF iteration.

trans_mixhistory (Integer):
The number of NEGF iteration steps in which the history of density matrix is stocked for the DIIS method.

trans_dehcir (Double):
Grid size, dE (in eV), for integrating the Greens function on the half circle path on imaginary plane.

trans_delpart (Double):
Grid size, dE (in eV), for integrating the Greens function on the path of the linear part on imaginary plane.

trans_debwin (Double):
(For trans_mode = 4). Grid size, dE (in eV), for integration on the non-equilibrium term.

trans_numres (Integer):
The number of poles at Fermi energy enclosed by closed contour on the imaginary plane.

trans_peconv (Integer):
The convergence criteria of the iteration of the Poisson equation. The threshold is 10 hartree of maximum energy difference over the all grid points.

trans_pemaxite (Integer):
Maximum iteration number of the Poisson equation.

trans_readesp (Integer):
Flag of read-in electrostatic potential energy. The data is printed in "ReadInESP/" directory with option of -1 or 0, and read from the same directory name with option of 0 or 1.

trans_restart (Integer):
Flag to restart reading the density matrix files, DAmat.dat (and DBmat.dat) from the "TransRestart/" directory.

Note: The default energy window for transmission and current calculations is defined as:
trans_emin = trans_efermi - trans_vmax/2 and trans_emax = trans_efermi + trans_vmax/2
if trans_emin and trans_emax are not given.

The trans_emin and trans_emax values can be set to determine the energy window for calculating the transmission function. If specified, these values will override the window defined by trans_efermi and trans_vmax values.

If higher accuracy parameters are set, trans_itodfac must be increased. For example for three digits accuracy (e.g. trans_efermi = -5.341), then trans_itodfac = 1000 or higher must be used. (This parameter applies to all double precision parameters.)

For trans_readhs = 1, the following parameters to parse precalculated Hamiltonian and overlap matrices have to be provided:

• trans_totorb2 (Integer):
  Total number of basis functions in the electrode models (if set, then same size is assumed for both electrodes) (no default value).

• trans_totorb2l: Integer number, total number of basis functions in left electrode model (no default value).

• trans_totorb2r: Integer number, total number of basis functions in right electrode model (no default value).

• trans_startpoint: Integer number, start point (basis number) for reading the TB integrals. Note, the basis number, that is, index number of basis function starts from 0. (if set, then same size is assumed for both electrodes) (default value 0).

• trans_startpointl: given as integer number, left start point (basis number) for reading the TB integrals. Note, the basis number, that is, index number of basis function starts from 0. (default value 0).

• trans_startpointr: given as integer number, right start point (basis number) for reading the TB integrals. Note, the basis number, that is, index number of basis function starts from 0. (no default value).

As an example of the Landauer calculation, the sample Q-Chem input is given below.

Example 11.277  Quantum transport Landauer calculation applied to C between two gold electrodes.

$molecule
   0 1
   Ag  -11.0  0  0
   Au   -8.3  0  0
   Ag   -5.6  0  0
   Au   -2.9  0  0
   Ag   -0.2  0  0
   Au    2.5  0  0
   C     4.8  0  0
   C     6.5  0  0
   C     8.2  0  0
   C     9.9  0  0
   C    11.6  0  0
   C    13.3  0  0
   Au   15.6  0  0
   Ag   18.3  0  0
   Au   21.0  0  0
   Ag   23.7  0  0
   Au   26.4  0  0
   Ag   29.1  0  0
$end

$rem
   METHOD            B3LYP
   BASIS             lanl2dz
   ECP               lanl2dz
   GEOM_OPT_MAXCYC   200
   INCDFT            FALSE
   MEM_STATIC        8000
   MAX_SCF_CYCLES    400
   MEM_TOTAL         32000
   MOLDEN_FORMAT     TRUE
   SCF_CONVERGENCE   10
   SCF_ALGORITHM     diis
   TRANS_ENABLE      1
$end

$trans-method
   trans_spin        0
   trans_npoints     300
   trans_method      0
   trans_readhs      0
   trans_printdos    1
   trans_efermi      -6.50
   trans_vmax        4.00
$end

$trans-model
   trans_lgatom      3
   trans_latom       5
   trans_ratom       14
   trans_rgatom      16
$end

A sample for unrestricted spin calculation can be found in the $QC/samples/tchem directory.

For NEGF calculations, note the followings concerning input etc:

• SCF_ALGORITHM = NEGF in $rem is necessary for NEGF calculation.

• The same numbers for trans_lgbasis and trans_lbasis, and also for trans_rgbasis and trans_rbasis must be used.

• Only WBL method for evaluating self-energy is available for the NEGF in the current version (trans_method = 0)

• trans_htype = 3 and trans_readhs = 1 are required

• SYM_IGNORE = TRUE to keep the original input coordinates.

• MEM_TOTAL and MEM_STATIC may need to be set to ensure enough memory is available.

• The bias voltage of is added on the left electrode, and on the right electrode, where the bias potential slope is along -axis, i.e. the system structure must be built along direction.

• In the $plots section (when defining a grid box for the Poisson equation solving), the grid box region must cover all atoms except for the left and right electrode parts defined in $trans_model. All integer index flags in $plots can be 0.

• For calculations on bulk electrode, the same structure with the same orientation in -Cartesian coordinate system (but different size) must be used so as to reproduce the same overlap matrix at the used part.

• For better calculations, update of electrode relevant (non-center) blocks in the density matrix (trans_updatedmatlr = 1) and Fermi energy to satisfy charge neutrality (trans_adjustefermi = 2 or 3) are recommended at zero bias (trans_opt = 3) before performing a full NEGF calculation. However, these options may make it difficult or slow to reach convergence.

• The criterion for convergence in the NEGF iterations is the maximum difference in density matrix elements, and is not a energy threshold.

NEGF calculations depend on the following pre-calculated properties:

Step 1: Pre-calculations:
A: Hamiltonian and overlap matrices for the left and right bulk electrodes (required)
B: A converged junction electronic state by standard DFT (recommended)
C: Electrostatic potential of large electrode region (optional)

Step 2: Self-consistent Greens function calculation with zero bias:
Non zero bias cases should be calculated using density matrices calculated with zero bias voltage to obtain converged density matrix. It is also recommend to evaluate the Fermi energy level within the NEGF scheme.

Step 3: NEGF calculations should be obtained by increasing the bias sequentially.

Examples of these steps applied to C between two aluminium electrodes are given below.

Example 11.278  Step 1-A of the NEGF calculation, the pre-calculation of the bulk electrode. Flag keyword of printing matrices must be set (i.e. trans_enable != 0)

$molecule
   0 1
   Al    -15.04250    0.0   0.0
   Al    -12.30750    0.0   0.0
   Al    -9.572500    0.0   0.0
   Al    -6.837500    0.0   0.0
   Al    -4.102500    0.0   0.0
   Al    -1.367500    0.0   0.0
   Al     1.367500    0.0   0.0
   Al     4.102500    0.0   0.0
   Al     6.837500    0.0   0.0
   Al     9.572500    0.0   0.0
   Al     12.30750    0.0   0.0
   Al     15.04250    0.0   0.0
$end

$rem
   UNRESTRICTED        true
   SYM_IGNORE          true
   MAX_SCF_CYCLES       500
   EXCHANGE              hf
   CORRELATION         none
   ECP                 hwmb
   BASIS               hwmb
   SCF_CONVERGENCE        4
$end

@@@

$molecule
   read
$end

$rem
   UNRESTRICTED        true
   SYM_IGNORE          true
   MAX_SCF_CYCLES       500
   EXCHANGE           b3lyp
   CORRELATION         none
   ECP                 hwmb
   BASIS               hwmb
   SCF_GUESS           read
   SCF_GUESS_MIX          3
   SCF_CONVERGENCE        4
$end

@@@

$molecule
   read
$end

$rem
   UNRESTRICTED        true
   SYM_IGNORE          true
   MAX_SCF_CYCLES       500
   EXCHANGE           b3lyp
   CORRELATION         none
   ECP                 hwmb
   BASIS               hwmb
   SCF_GUESS           read
   SCF_GUESS_MIX          3
   SCF_CONVERGENCE        7
   TRANS_ENABLE          -1
$end

$trans-method
   trans_spin             2
$end

Example 11.279  Step 1-B of the NEGF calculation - the pre-calculation by the standard DFT. The same molecular structure with the step 2 must be used

$molecule
   0 1
   Al  -13.615   0.0   0.0
   Al  -10.880   0.0   0.0
   Al   -8.145   0.0   0.0
   Al   -5.410   0.0   0.0
   Al   -2.675   0.0   0.0
   C    -0.627   0.0   0.0
   C     0.627   0.0   0.0
   Al    2.675   0.0   0.0
   Al    5.410   0.0   0.0
   Al    8.145   0.0   0.0
   Al   10.880   0.0   0.0
   Al   13.615   0.0   0.0
$end

$rem
   UNRESTRICTED          true
   SYM_IGNORE            true
   MAX_SCF_CYCLES        400
   EXCHANGE              b3lyp
   CORRELATION           none
   BASIS                 hwmb
   ECP                   hwmb
   SCF_CONVERGENCE       5
$end

@@@

$molecule
   read
$end

$rem
   UNRESTRICTED          true
   SYM_IGNORE            true
   EXCHANGE              b3lyp
   CORRELATION           none
   BASIS                 hwmb
   ECP                   hwmb
   MAX_SCF_CYCLES        400
   SCF_CONVERGENCE       6
   SCF_GUESS             read
   SCF_GUESS_MIX         3
$end

Example 11.280  Step 2 of the NEGF calculation. As the preparation of this step, followings are necessary:

3. FAmat2l.dat, FAmat2r.dat, Smat2l.dat, and Smat2r.dat (also FBmat2l.dat and FBmat2r.dat if the calculation is spin-unrestricted ) must be placed in the same directory of the Q-Chem input file by coping or linking the output files of the step 1-A.

4. Restart directory of the standard DFT obtained in the step 1-B must be copied to here.

5. (Optional) Read-in electrostatic potential data in "ReadInESP/" directory must be placed if this option is used (for trans_readesp = 1). This option can provide more bulk electrode electrostatic environment as the boundary condition of Poisson equation solving ( see the sample files in $QC/samples/tchem/) for more details).

Example 11.281  Step 3 of the NEGF calculation. As the preparation of this step, followings are necessary:

3. In the same way as step 2, FAmat2l.dat, FAmat2r.dat, Smat2l.dat, and Smat2r.dat (also FBmat2l.dat and FBmat2r.dat for spin-unrestricted calculations) must be placed.

4. Restart directory for Q-Chem generated in the step 2 must be copied to here (only coordinates are used).

5. Restart directory for density matrix "TransRestart/" must be created and DAmat.dat (and DBmat.dat for spin-unrestricted) generated in the step 2 must be copied or linked in the directory.

6. Read-in electrostatic potential data in "ReadInESP/" directory used in the step 2 must be copied to here.

7. Put Fermi energy obtained in Step 2 (recommended).