# 1.2. Dynamic Conductance

Quantum transport in nanostructures under *AC* bias is a very important and difficult problem in transport theory and has been the subject of a variety of systems including normal quantum dot systems [LL] [OKK] [OFW] as well as normal superconducting hybrid system [CML].
The *AC* response of molecular devices is of fundamental interest because it can probe the charge distribution and its dynamics.
In addition, the frequency introduces another energy scale into the problem.
It is well established that the *DC* conductivity is determined only by resistance.
On the other hand, *AC* conductivity also depends of resistance and additionally on inductive and capacitive effects.

For a system under *AC* bias, two requirements have to be satisfied: the current conservation and the gauge invariance.
In the theoretical treatment, the current is usually defined in terms of conduction current.
The conduction current is a conserved quantity only in the case of *DC*.
Under the *AC* bias, however, the conduction current is not a conserved quantity, due to the charge accumulation in the scattering region [B].
To solve the problem the displacement current due to the long-range Coulomb interaction has to be included, and current-conservation problem can be solved by partitioning the total displacement current.

The current partition at small bias was achieved by Büttiker *et al.* [BPH] using the scattering matrix approach, and was extended to the situation far from equilibrium using the non-equilibrium Green’s function (NEGF) method [WWG]. This formalism ensures both the the current conserving condition as well as the gauge-invariant condition which says that the current of a multi-probe system remains the same if all the bias are shifted by the same amount [WZW].

In this tutorial, we’ll study the *AC* conductance of short carbon chains attached to aluminum (Al) nanowire electrodes using NanoDCAL code.
At the junction, the central scattering region consists of a linear carbon chain with four or five atoms (n=4 and 5).
Moreover, we used four layers of Al wire on each side of junction as buffer to keep the bulk-like electrode characteristics, as shown in Fig. 1.2.1.

This tutorial is based on the following reference [WZW]. This work demonstrated that the *DC* conductance \((G_{dc})\), the real part \((G_{Re})\), and the imaginary part \((G_{Im})\) of dynamic conductance (or *AC* conductance) exhibit oscillatory behaviors for an even-odd number of carbon atoms in a chain at low frequencies.
Moreover, the real part of the *AC* conductance depends only on the parity of the number of carbon atoms *n*, *i.e.*, whether *n* is even or odd. These oscillations of dynamic conduction can be understood by analyzing the average transmission coefficient.

To investigate the *AC* conductance the user should perform the following steps:

Build the structures;

Transport calculation;

Self-consistent calculation of electrodes;

Self-consistent calculation of the central region;

*DC*Conductance calculation;*AC*Conductance calculation.

## 1.2.1. Build the Al-C_{n}-Al

Al nanowire

Firstly, we’ll build an Al nanowire using the Device Studio software creating a new project, as shown in Fig. 1.2.2.

Open Device Studio software ➟

**Create a new Project**➟ File name**Al-Cn-Al**➟ press**Save**;In the menu bar click

**File**➟**Import**➟**Import Local**to pop up the Windows explorer:Go to the Device Studio installation folder ➟

*materials*➟*3D materials*➟*Conductor*➟*Pure_metal*➟ select*Al.hzw*and click**open**.

Select

*Al.hzw*, go to the menu bar and click**Redefine Crystal**to set the parameters;In the cell vector set

**u = (-2a, 2b, 2c)**,**v = (4a, -4b, 4c)**, and**w = (2a, 2b, -2c)**, click**Preview**or**Build**;

On the

*Al_Rede.hzw*file:Delete the left and top layer of atoms ➟ click on

**3D Viewer**➟**xy-View**and delete the left layer of atoms;In the toolbar, click

**Convert to Crystal**to create a vacuum towards the*xz-*axis ➟ go to**Primitive vector**and set (**30.0, 0.00, 0.00**); (**0.00, 14.00, 0.00**); (**00.0, 0.00, 30.0**) ➟ go to toolbar and click on**Center**.

Carbon chain

Next, we’ll build the carbon chain, as shown in Fig. 1.2.3.

In toolbar go to

**File**➟ select the New Crystal;In the lattice tab, set

*a*=**30.0**Å,*b*=**1.3**Å,*c*=**30.0**Å and \(\alpha = \beta = \gamma\) =**90**º ➟ click on plus (+) bottom to add a carbon atom at (**0.5, 0.5, 0.5**), click**Preview**or**Build**.

In the menu bar click

**Build**➟**Redefine Crystal**;In the cell vector set

**u = (1a, 0b, 0c)**,**v = (0a, 5b, 0c)**and**w = (0a, 0b, 1c)**, click**Preview**or**Build**;

Building the scattering region

Now, we’ll couple the Al nanowire to the C_{5} carbon chain, as shown in Fig. 1.2.4.

In the menu bar, click

**Build**➟ select**Device**to pop up the**Device**window . In the**Device**window select**B-C-T structure**;On

**B-C-T**window go to the**Basic**➟ on**Bottom**and**Top electrode**select*Al_Rede.hzw*; on center select**Crystal_Rede.hzw**➟ click on**Match**and**Preview**. In the**D(BC)**and**D(TC)**type**1.2**➟ click on**Preview**and**Build**.

Go to the toolbar click on

**Convert to Crystal**➟**Build**➟ back to the toolbar and click on**Center**.

## 1.2.2. Structural relaxation

The C_{5}carbon chain is attached to the bottom and top Al wire.
Now, we can perform the transport calculation, but before that we have to find the equilibrium position of the carbon atoms.
Let’s generate the relaxation input file:

In the menu bar of Device Studio, click

**Simulator**➟**NanoDCAL**➟**Relax**to pop up the interface**Relax Calculation for Crystal**to set the parameters;Go to the

**Basic settings**on**Electron temperature**➟ set**100 K**;On the

**Relax**tab ➟**Method**and choose**CG**;Go to the

**Interaction control**tab ➟ select**TotalEnergy**and type**1e-04**and click**Generate files**.

The generated *Relax.input* has be edited to relax only the atoms in the carbon chain and one layer of Al wire in each side of carbon chain.
The extra Al layers (buffer) will be constrained, to ensure that the electrodes will perfectly connect to the central region, and keep the bulk characteristics of the device.

The following keywords should be included in the file:

```
calculation.fixCentralCellShape = true
calculation.relaxation.movingAtoms = [1:6, 8, 12, 15, 58, 61, 65, 67]
```

The keywords specify the following:

`calculation.fixCentralCellShape`

If true, the central cell shape will not be changed during the relaxation;`calculation.relaxation.movingAtoms`

The \(3\times n\) coordinates corresponding to the atoms listed in the movingAtoms will be allowed to change during the relaxation, and the others will be fixed.

The generated file for relaxation can be found here: `Relax-Al-C5-linear-Al.input`

.
To perform the relaxation see how to run an input file in the previous tutorial.

After the calculation, the distances between the neighboring carbon atoms in linear configuration found to be \(d_{C-C}\approx 1.27\) and \(\approx 1.36\) Å. These values are in line with the experimental ones for polyyne, where interatomic distances are predicted to be 1.33 Å (C \(-\) C) and 1.23 Å (C \(\equiv\) C) [KOB]. The distance from carbon atom to the Al surface is \(d_{C-Al}\approx 1.04\) Å, and the carbon atom is located at three-fold hollow site.

## 1.2.3. Building the two-probe device

Now, we’ll build the two-probe geometry.
On the unit cell of the central region (Al-C_{5}-Al) the system has a finite cross-section along the *zx-* axis and must be the same as used in both electrodes.
This device can be built using Device Studio modeling as shown in Fig. 1.2.5:

After the relaxation open the

*Atoms_relaxed.xyz*file on Device Studio ➟ go to the toolbar and click**Convert to Crystal**:On

**Bravais lattice**tab set**Cubic: Simple**cell and*a*=**40**Å.

Go to the menu bar click

**Build**➟**Convert to Device**, check**Bottom electrode**and**Top electrode**options, click**Preview**and**Build**the structure.

The proposed device Fig. 1.2.5 can be divided into two electrodes and a central region. Moreover, the electrodes are semi-infinite with periodic boundary conditions along the \(\pm\) *y*-direction relative to the Al-C_{5}-Al central scattering region.

Note

Since the leads are along the \(\pm\) y-axis. The calculation should be performed considering

topandbottomleads by the convention of NanoDCAL.

Bottom (lead 1): [0 -1 0] means the direction of -

*y*;Top (lead 2): [ 0 1 0] means the direction of +

*y*.

## 1.2.4. Transport calculation

The calculation of a two-probe system typically consists of the following steps:

SCF calculation for the electrodes;

NEGF-SCF calculation for the scattering region.

The required input files for these procedure could be generated using Device Studio with NanoDCAL module simulator. The user can follow the procedure in how generating the device input files from the previous tutorial.

SCF for electrode

Firstly, we have to perform the self-consistent calculation of the electrode. Before that, we’ll briefly present the simulation parameters specified in the generated files.

Top electrode input file

```
%%What quantities should be calculated
calculation.name = scf
%Basic setting
calculation.occupationFunction.temperature = 100
calculation.realspacegrids.E_cutoff = 80 Hartree
calculation.xcFunctional.Type = LDA_PZ81
calculation.k_spacegrids.number = [ 1 100 1 ]'
system.centralCellVectors = [[30 0 0]' [0 4.0495 0]' [0 0 30]']
system.spinType = NoSpin
%Iteration control
calculation.SCF.monitoredVariableName = {'rhoMatrix','hMatrix','totalEnergy','bandEnergy','gridCharge','orbitalCharge'}
calculation.SCF.convergenceCriteria = {1e-06,1e-06,1e-06,[],[],[]}
calculation.SCF.maximumSteps = 200
calculation.SCF.mixMethod = Pulay
calculation.SCF.mixRate = 0.1
calculation.SCF.mixingMode = H
calculation.SCF.startingMode = H
%calculation.SCF.donatorObject = NanodcalObject.mat
%Basic set
system.neutralAtomDataDirectory = '../'
system.atomBlock = 9
AtomType OrbitalType X Y Z
Al LDA-DZP 12.97524999 3.03712500 15.00000000
Al LDA-DZP 17.02475000 3.03712500 15.00000000
Al LDA-DZP 15.00000000 1.01237501 15.00000000
Al LDA-DZP 15.00000000 3.03712500 12.97524999
Al LDA-DZP 15.00000000 3.03712500 17.02475000
Al LDA-DZP 12.97524999 1.01237501 12.97524999
Al LDA-DZP 12.97524999 1.01237501 17.02475000
Al LDA-DZP 17.02475000 1.01237501 12.97524999
Al LDA-DZP 17.02475000 1.01237501 17.02475000
end
```

Bottom electrode input file

```
system.atomBlock = 9
AtomType OrbitalType X Y Z
Al LDA-DZP 12.97524999 1.01237500 15.00000000
Al LDA-DZP 17.02475000 1.01237500 15.00000000
Al LDA-DZP 15.00000000 3.03712500 15.00000000
Al LDA-DZP 15.00000000 1.01237500 12.97524999
Al LDA-DZP 15.00000000 1.01237500 17.02475000
Al LDA-DZP 12.97524999 3.03712500 12.97524999
Al LDA-DZP 12.97524999 3.03712500 17.02475000
Al LDA-DZP 17.02475000 3.03712500 12.97524999
Al LDA-DZP 17.02475000 3.03712500 17.02475000
end
```

The descriptions of keywords were listed in the previous tutorial. The user can see on keywords-scf.

Warning

For the Al-C_{5}-Cl the top and bottom electrodes are not equal.
The stacking order that consists of alternating type A and type B (i.e., ABABAB \(\cdots\)) has to be preserved.
To avoid stacking fault we have to perform the calculation for both of them.

The electrode Hamiltonians will be stored in **NanodcalObject.mat** files that are required for the transport calculation.

Now, just run the SCF calculation for top and bottom electrodes following how to run an input file

Two-probe SCF-NEGF calculation

After calculating the electrodes, we can move forward to carry out the two-probe transport calculation.

Scattering region input file

```
%%What quantities should be calculated
calculation.name = scf
%Basic setting
calculation.occupationFunction.temperature = 100
calculation.realspacegrids.E_cutoff = 80 Hartree
calculation.xcFunctional.Type = LDA_PZ81
calculation.k_spacegrids.number = [ 1 1 1 ]'
%Description of electrode
system.numberOfLeads = 2
system.typeOfLead1 = bottom
system.voltageOfLead1 = 0
system.objectOfLead1 = ../BottomElectrode/NanodcalObject.mat
system.typeOfLead2 = top
system.voltageOfLead2 = 0
system.objectOfLead2 = ../TopElectrode/NanodcalObject.mat
%Contour integral
%calculation.complexEcontour.lowestEnergyPoint = 1.5 Hartree
calculation.complexEcontour.numberOfPoints = 40
calculation.realEcontour.interval = 0.0272114
calculation.realEcontour.eta = 0.0272114
system.centralCellVectors = [[30 0 0]' [0 24.2705 0]' [0 0 30]']
system.spinType = NoSpin
%Iteration control
calculation.SCF.monitoredVariableName = {'rhoMatrix','hMatrix','totalEnergy','bandEnergy','gridCharge','orbitalCharge'}
calculation.SCF.convergenceCriteria = {1e-06,1e-06,1e-06,[],[],[]}
calculation.SCF.maximumSteps = 200
calculation.SCF.mixMethod = Pulay
calculation.SCF.mixRate = 0.1
calculation.SCF.mixingMode = H
calculation.SCF.startingMode = H
%calculation.SCF.donatorObject = NanodcalObject.mat
%Basic set
system.neutralAtomDataDirectory = '../'
system.atomBlock = 48
AtomType OrbitalType X Y Z
Al LDA-DZP 14.97174181 15.15922403 12.95819733
Al LDA-DZP 12.94701418 17.18393782 12.95819733
Al LDA-DZP 16.99646872 17.18393782 12.95819733
Al LDA-DZP 14.97174181 19.20865161 12.95819733
Al LDA-DZP 12.94701418 21.23336540 12.95819733
Al LDA-DZP 16.99646872 21.23336540 12.95819733
Al LDA-DZP 14.97174181 23.25807920 12.95819733
Al LDA-DZP 15.02825819 1.012384480 12.99234741
Al LDA-DZP 13.00353056 3.037097390 12.99234741
Al LDA-DZP 17.05298582 3.037097390 12.99234741
Al LDA-DZP 15.02825819 5.061811180 12.99234741
Al LDA-DZP 13.00353056 7.086524970 12.99234741
Al LDA-DZP 17.05298582 7.086524970 12.99234741
Al LDA-DZP 15.02825819 9.111238760 12.99234741
Al LDA-DZP 12.94701418 15.15922403 14.98292496
Al LDA-DZP 16.99646872 15.15922403 14.98292496
Al LDA-DZP 14.97174181 17.18393782 14.98292496
Al LDA-DZP 12.94701418 19.20865161 14.98292496
Al LDA-DZP 16.99646872 19.20865161 14.98292496
Al LDA-DZP 14.97174181 21.23336540 14.98292496
Al LDA-DZP 12.94701418 23.25807920 14.98292496
Al LDA-DZP 16.99646872 23.25807920 14.98292496
C LDA-DZP 14.95202096 14.11357851 14.99086132
C LDA-DZP 14.97462808 12.79210510 14.99903779
C LDA-DZP 14.99458904 11.51878238 15.00601230
C LDA-DZP 15.00853734 10.15688605 15.00913868
Al LDA-DZP 13.00353056 1.012384480 15.01707504
Al LDA-DZP 17.05298582 1.012384480 15.01707504
Al LDA-DZP 15.02825819 3.037097390 15.01707504
Al LDA-DZP 13.00353056 5.061811180 15.01707504
Al LDA-DZP 17.05298582 5.061811180 15.01707504
Al LDA-DZP 15.02825819 7.086524970 15.01707504
Al LDA-DZP 13.00353056 9.111238760 15.01707504
Al LDA-DZP 17.05298582 9.111238760 15.01707504
Al LDA-DZP 14.97174181 15.15922403 17.00765187
Al LDA-DZP 12.94701418 17.18393782 17.00765187
Al LDA-DZP 16.99646872 17.18393782 17.00765187
Al LDA-DZP 14.97174181 19.20865161 17.00765187
Al LDA-DZP 12.94701418 21.23336540 17.00765187
Al LDA-DZP 16.99646872 21.23336540 17.00765187
Al LDA-DZP 14.97174181 23.25807920 17.00765187
Al LDA-DZP 15.02825819 1.012384480 17.04180267
Al LDA-DZP 13.00353056 3.037097390 17.04180267
Al LDA-DZP 17.05298582 3.037097390 17.04180267
Al LDA-DZP 15.02825819 5.061811180 17.04180267
Al LDA-DZP 13.00353056 7.086524970 17.04180267
Al LDA-DZP 17.05298582 7.086524970 17.04180267
Al LDA-DZP 15.02825819 9.111238760 17.04180267
end
```

The keywords for central region were already explained. The user can see on keywords for central region.

At this point, just run the SCF-NEGF calculation.
The input files for Al-C_{5}-Al two-probe device are found bellow:

Bottom electrode

*scf.input*(lead1) (`Bottom-elec-AlC5-scf.input`

);Top electrode

*scf.input*(lead2) (`Top-elec-AlC5-scf.input`

);Central region

*scf.input*(Al-C_{5}-Al) (`Central-region-Al-C5-Al-scf.input`

).

Tip

Based on the knowledge gained from this C_{5} linear chain, the user should be able to build and study a linear carbon chain with *n* carbon atoms between Al nanowires.

The carbon chain with an even number of atoms will be important for comparison with the odd chain (Al-C_{5}-Al), and to verify the oscillatory behavior [WZW].
Thus the Al-C_{4}-Al device input files are provided bellow:

Bottom electrode

*scf.input*(lead1) (`Bottom-elec-AlC4-scf.input`

);Top electrode

*scf.input*(lead2) (`Top-elec-AlC4-scf.input`

);Central region

*scf.input*(Al-C_{4}-Al) (`Central-region-Al-C4-Al-scf.input`

);

## 1.2.5. *DC* Conductance calculation

After completing the self-consistent calculation, it is time to compute the quantum transport properties.

The *conductance.input* file can be generated using Device Studio, following the steps below:

Select the Al-C

_{5}-Al device, go the the menu bar, click on**Simulator**➟**NanoDCAL**➟**Analysis**.In the

**Analysis**window, on the top left choose the**Device**option;Navigate to the

**Analysis**panel (left side) ➟ select**Conductance**➟ click on the*Right arrow*button to transfer**Conductance**calculator to the**Calculation Selected**panel.

The

**Conductance**options are shown in the right side of the**Analysis**window. Edit the parameters to fit the studied system:Set

**Number of energy points**to**10**.

Before starting the calculation, we’ll briefly present the simulation parameters specified in the generated file.

Conductance input file

```
system.object = NanodcalObject.mat
calculation.name = conductance
calculation.conductance.method = 'GreenFunction'
calculation.conductance.kSpaceGridNumber = [1 1 1]'
calculation.conductance.numberOfEnergyPoints = 10
calculation.control.xml = true
```

The keywords specify the following:

`calculation.conductance.method`

Method used for calculating conductance, the possible values are GreenFunction or WaveFunction;

`calculation.conductance.kSpaceGridNumber`

Number of small k-space grids in each direction which, together with kSpaceGridShift, are used to produce the parameter kSpacePoints.

`calculation.conductance.numberOfEnergyPoints`

The number of energy points used in the energy space integration. This is only used when the energyPoints and energyPointWeights are not given explicitly.

Run the conductance simulation with the file `Conductance-Al-C5-Al.input`

.

After the calculation finish, the following output files are generated: *CalculatedResults.mat*, *log.txt*, and *ConductanceAndCurrent.mat*.
The *.mat* results can be loaded using the *MATLAB* platform.

DC Conductance analysis

The output file has the important information:

```
leadPairs ➟ [1,2] ; [2,1]
conductance ➟ [0.792652625690863 , 0.792652625690869]
current ➟ [0 , 0]
appliedVoltages ➟ [0 , 0]
```

The conductance and current are calculated for each given lead pair (i.e. the portion of the current which is caused by the probability flux of the electronic waves, coming from the first lead, transmitted into the second lead). The the direction of the current (lead) is from the lead to the device. The calculated conductance unit is \(G_{0}=2e^{2}/h\), and the current is expressed in the unit of ampere (A).

The available conductance for this polyyne with five carbon atoms (*n* = 5) between the Al nanowires is \(G_{DC}\approx 0.79 G_{0}\).
In addition, the evaluated the conductance for *n* = 4 to be \(G_{DC}\approx 1.98 G_{0}\).
This behavior is in good agreement with a recent experiment, which observed that in polyyne chains the **DC** conductance decays with molecular length [ZFZ].

## 1.2.6. *AC* Conductance calculation

The required input file for the *AC* conductance calculation of Al-C_{5}-Al is writhed as follows:

Conductance input file

```
system.object = NanodcalObject.mat
calculation.name = acConductance
calculation.acConductance.maximumFrequency = 50000000
calculation.acConductance.numberOfFrequencyPoints = 101
calculation.acConductance.kSpaceGridNumber = [1 1 1]'
calculation.acConductance.temperature = 100
calculation.acConductance.numberOfEnergyPoints = 10
calculation.acConductance.energyInterval = 1e-3
calculation.conductance.eta = 1e-4
calculation.acConductance.etaSigma = 1e-4
```

`calculation.acConductance.maximumFrequency`

TheACconductance is calculated as a function of frequency from zero up to this value. The unit is MHz.

`calculation.acConductance.numberOfFrequencyPoints`

Number of the frequency points at which theACconductance is calculated.

`calculation.acConductance.kSpaceGridNumber`

The small k-space grid number in each direction which, together with kSpaceGridShift, are used to produce the parameter kSpacePoints.

`calculation.acConductance.temperature`

Temperature used in the Fermi function when calculating the acConductance, in unit of Kelvin.Note: the Boltzmann constant \(k= 8.617342 \times 10^{-5}(eV/K)=3.1668151 \times 10^{-6} (Ha/K)\).

`calculation.acConductance.numberOfEnergyPoints`

The number of energy points used in the energy space integration.

`calculation.acConductance.energyInterval`

Energy interval used to determine the parameter numberOfEnergyPoints.

`calculation.conductance.eta`

The small eta used in the calculation of self-energy and/or Green’s function when the GreenFunction method is chosen. This parameter is only used when the parameter conductance.etaSigma and/or conductance.etaGF is not given.

`calculation.acConductance.etaSigma`

The small eta used in the calculation of self-energy when the GreenFunction method is chosen

Run the conductance simulation with `ac-Conductance-Al-C5-Al.input`

file.

After the calculation finishes, the following output files are generated: *CalculatedResults.mat*, *log.txt*, and *AcConductance.mat*.
The *.mat* results can be loaded using the *MATLAB* platform.

AC Conductance analysis

The output file has the following information:

```
calculationMethod ➟ GreenFunction
frequencies ➟ 0 to 50000000
conductance ➟ val(:,:,1) , val(:,:,2)
```

In the output file, the user will find at first the information on the method chosen to perform the calculation.
Here was set as *GreenFunction*.

Then, an array (\(1\times101\)) where frequency values are distributed from 0 to \(5\times10^{7}\).

Next, the *AC* conductance is calculated for each lead pair (:,:,1) and (:,:,2), which includes contributions from both particle current and displacement current.

On the *data.conductance* file the real and imaginary part of the *AC* conductance (\(G_{Re}\) and \(G_{Im}\)) are disposed on the first and second columns, respectively.
The frequency is expressed in the unit of MHz, and the calculated *AC* conductance in the unit of \(G_{0}=2e^{2}/h\) or \(G_{0}=77.4809169\mu S\).

The real part of AC conductance

In order to further analyze the *AC* conductance on Al-C_{n}-Al (*n* = 4 and 5), we plot \(G_{Re}-G_{DC}\) as a function of frequency, as one can see in Fig. 1.2.6.

In this case, \(G_{Re}\) is the real part of *AC* conductance and \(G_{DC}\) is the *DC* conductance (Conductance analysis).
The \(G_{Re}-G_{DC}\) shows quadratic dependence on frequencies.
In addition, the increasing rate of \(G_{Re}-G_{DC}\) for the even number of carbon atoms is a little larger than that for the odd number of carbon atoms.

The imaginary part of AC conductance

The Fig. 1.2.7 shows the imaginary part (\(G_{Im}\)) of *AC* conductance as a function of frequency for the carbon chain with number of atoms *n* = 4 and 5.
For both even and odd carbon wires, the \(G_{Im}\) are negative and decrease linearly in frequency than that of odd number carbon wires.

Importantly, it shows that the imaginary part of *AC* conductance approximately obeys a linear relation.
At relatively low frequencies such as \(5\times10^{7}\) MHz, the imaginary part of the *AC* conductance is largely contributed by the emittance.

Note

The emittance is defined as

Here \(dn_{\alpha\beta}/dE\) is the partial density of states defined as [BT]

Here \(dn_{\alpha}/dE=\sum_{\beta}dn_{\alpha\beta}/dE\) is the injectivity describing the local DOS when an incoming electron is injected from the \(\alpha\) electrode and \(dn/dE=\sum_{\alpha}dn_{\alpha}/dE\) is the total local DOS. In the case of single-channel transmission, the sign of \(E_{\alpha\beta}\) determines the dynamic response of the system (either capacitivelike or inductivelike). For more information please refer to [WZW] and [BT].

The AC conductance for the zigzag carbon chain

As an additional task we can explore the behavior of *AC* conductance in a zigzag carbon chain (Fig. 1.2.8) and reproduce the result of reference [WZW].
At this point, the user is able to build, relax and reproduce all the steps of this tutorial.

Tip

For zigzag carbon chain the bonds alternates between the single (C \(-\) C) and double (C \(=\) C) bonds.

On NanoDCAL, these bond lengths values was found to be 1.30 Å and 1.40 Å, respectively.

These bond lengths are found to be in good agreement with previous experimental values of 1.38 Å and 1.41 Å [WSG]. The C \(-\) H bond lengths were fairly uniform around 1.11 Å.

The \(G_{Re}-G_{DC}\) as a function of frequency shows that the linear and zigzag carbon chains for *n* = 4 and 5 exhibits similar quadratic dependence. However, the \(G_{Re}-G_{DC}\) values are greater for linear carbon chains than zigzag carbon chains, as one can see in Fig. 1.2.9.

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