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Table of Contents

Table of ContentsInelastic current in a silicon p-n junction

Creating the silicon p-n junctionTransmission calculation without electron-phonon interactions

Setting up the calculationAnalysis of the results

Transmission calculation with electron-phonon interactionsSetting up the dynamical matrix calculationSetting up the Hamiltonian derivatives calculationCalculating the inelastic transmission spectrumAnalysis of the inelastic transmission spectrumCalculation of the current with and without electron-phonon interactions

Speeding up the calculationsThe ‘Phonon energy intervals’ methodUsing the bulk dynamical matrix and Hamiltonian derivatives

References

Downloads & LinksDownloads & Links

PDF version Inelast icTransmissionSpectrum Electron t ransport calculat ions with electron-phonon coupling included via the special thermaldisplacement method - STD-Landauer SpecialThermalDisplacement Si_ pn_ junct ion.py t ransmission_ V-0.4.py dynmat.py dHdR_ V-0.4.py xloe_ V-0.4.py current .py xloe_ pheint_ V-0.4.py dynmat_ bulk.py dHdR_ bulk.py xloe_ V-0.4_ bulk.py

Docs » Tutorials » New or Recent ly Updated Tutorials » Inelast ic current in a silicon p-n junct ion

Inelast ic current in a sil icon p-n junct ionInelast ic current in a sil icon p-n junct ionVersion:Version: 2017.2

Inelast ic scat tering effects due to elect ron-phonon interact ions can play a fundamental role in determiningthe carrier t ransport through a device. In this tutorial, you will invest igate the impact of such effects onthe current through a silicon

- junct ion in reverse bias. In this system, carrier t ransport is dominated by band-to-band tunnelling

between the valence band of the region and the conduct ion band of the region. This process can be enhanced considerably by inelast ic scat tering effects.

Specif ically, you will use the Inelast icTransmissionSpectrum module implemented in AT KAT K to calculate thet ransmission spect rum of the device in the presence of elect ron-phonon interact ions. You will also use theInelast ic T ransmission Spect rum Analyz erInelast ic T ransmission Spect rum Analyz er plugin in Quant umAT KQuant umAT K to perform a detailed analysis ofthe current and determine which phonon mode is the main responsible for the current enhancement .

The methodology implemented in QuantumATK is based on the Lowest Order Expansion (LOE)Lowest Order Expansion (LOE)[FPBJ07] and the eXt ended LOE (XLOE)eXt ended LOE (XLOE) [LCF+14] methods. In this tutorial, you will use the XLOEXLOEmethod, which t reats the f inite energy difference between init ial and f inal states in the t ransit ion

due to elect ron-phonon coupling in a more accurate manner. However, the tutorial can becarried out also using the simpler LOELOE method, which will give qualitat ively similar results. You can f ind moredetails about the theory underlying the two methods in the Notes sect ions of theInelast icTransmissionSpectrum module in the AT K manualAT K manual.

Quant umAT KQuant umAT K

Try it !

QuantumATK

Contact

pn

pn

k → k ± q

Creat ing t he si l icon p- n junct ionCreat ing t he si l icon p- n junct ion

To create the device, you can follow the inst ruct ion in the Create device sect ion of the Silicon p-njunct ion tutorial. However, in the present tutorial you will create a shorter device to speed up thecalculat ions. In the BuilderBuilder, cont ruct the device with the following changes:

When using the Surf ace (Cleave)Surf ace (Cleave) plugin to create a <100>-oriented st ructure, set the t hicknesst hickness to24 layers instead of 52 layers;When using the Device f rom BulkDevice f rom Bulk plugin, choose 5.4306 Å as the elect rode lengt helect rode lengt h;Click the DopingDoping plugin in Miscellaneous ‣ Doping to set the doping of the

and regions. In this case, select the half of device to set the doping region and select the remainder part using the ctrl+i which will select the inversion part to

set the doping region. In both regions, set a doping of

.

You can download the result ing device configurat ion from here: Si_pn_ junct ion.py.

Transm ission calcu lat ion wit hout elect ron - phononTransm ission calcu lat ion wit hout elect ron - phononin t eract ionsin t eract ions

In this sect ion, you will calculate and analyze the elect ronic st ructure of the- junct ion at a reverse bias voltage of

, and the associated t ransmission spect rum without elect ron-phonon interact ions

p

n

p

n

2 ⋅ 1019e/cm3

pnVbias = −0.4 V

included, using the convent ional Landauer formula as implemented in AT KAT K in the TransmissionSpectrummodule.

Se t t i ng up t he c al c ul at i o nSe t t i ng up t he c al c ul at i o n

From the BuilderBuilder, send the st ructure to the Script generat orScript generat or using the but ton. Add thefollowing blocks and set their parameters as follows:

Calculators ‣ SemiEmpiricalCalculator

In MainMain:

Set the Lef t e lect rode volt ageLef t elect rode volt age to

Set the Right elect rode volt ageRight elect rode volt age to

In Hamilt onianHamilt onian:

Unt ick the ‘No SCF iterat ion‘ opt ion

In Numerical AccuracyNumerical Accuracy , set the k- point Samplingk- point Sampling to:

In It erat ion cont rol paramet ersIt erat ion cont rol paramet ers, set the T oleranceT olerance to ‘4e-05‘

Analysis ‣ ProjectedLocalDensityOfStates

Set the k- point Samplingk- point Sampling to:

Analysis ‣ TransmissionSpectrum

Set the k- point Samplingk- point Sampling to:

Finally, in the Global IOGlobal IO opt ions, change the name of the Def ault out put f ileDef ault out put f ile to transmission_V-0.4.hdf5 .

Send the script to the Job managerJob manager using the but ton, save it as transmission_V-0.4.py and pressthe but ton to run the calculat ion. It will take less than 2 minutes on 4 CPUs. You can also download thefull script from here: t ransmission_V-0.4.py.

Anal ysi s o f t he re sul t sAnal ysi s o f t he re sul t s

In the LabFloorLabFloor, select the Project edLocalDensit yOf St at esProject edLocalDensit yOf St at es object from the f ile

transmission_V-0.4.hdf5 , and click on the Project ed Local Densit y of St at esProject ed Local Densit y of St at es plugin. In the pluginwindow, select Spectral Current from the drop-down menu on the top left , and set the maximum value ofthe Dat a rangeDat a range to 0.1. In the panel on the left -hand side of the screen, use the ZoomZoom but ton to make

Vbias = −0.4 V

−0.2 V

0.2 V

kA = 7kB = 7kC = 101

kA = 21kB = 21

kA = 21kB = 21

sure that the minimum value of the current is about. The features below this value can be associated to numerical noise.

From the left panel of the f igure above, it can be seen that the maximum of the spect ral current occursinside the bias window (

). Furthermore, from the right panel, which shows the density of statesacross the device, it can be seen that within the bias window, there are no elect ronic states in cent ralregion (

) of the device, so that tunnelling between the left and the right elect rodes is the onlypossible t ransport mechanism. This indicates that elect ronic t ransport in the device in reverse bias isdominated by tunnelling.

To analyze the probability of tunnelling in

-space, select the T ransmission Spect rumT ransmission Spect rum object from the same file and click on the

T ransmission Analyz erT ransmission Analyz er plugin. In the plugin window, set the maximum value of the Dat a rangeDat a range to 0.1.

10−24A/eV

−0.2 eV ≤ Energy ≤ 0.2 eV

20 Å ≤ z ≤ 45 Å

k

From the f igure above, it can be seen that the tunnelling probability is st rongly peaked at the-point of the Brilluoin zone at the Fermi level.

Transm ission calcu lat ion wit h elect ron - phononTransm ission calcu lat ion wit h elect ron - phononin t eract ionsin t eract ions

In this sect ion, you will calculate the t ransmission spect rum of the silicon- junct ion at a reverse bias voltage of

including the effect of the elect ron-phonon interact ions within the XLOEXLOE approximat ion[LCF+14].

To calculate the Inelast icTransmissionSpectrum, you f irst need to calculate the DynamicalMat rixDynamicalMat rix andthe Hamilt onianDerivat ivesHamilt onianDerivat ives of the system.

Se t t i ng up t he dynami c al mat r i x c al c ul at i o nSe t t i ng up t he dynami c al mat r i x c al c ul at i o n

In the LabFloorLabFloor, drag and drop the DeviceConf igurat ionDeviceConf igurat ion object contained in the f iletransmission_V-0.4.hdf5 to the Script generat orScript generat or. Remove the DeviceSemiEmpiricalCalculat orDeviceSemiEmpiricalCalculat or

and replace it with a ForceFieldCalculat orForceFieldCalculat or

Select ‘St illingerWeber_Si_1985‘ from the list of available classical potent ials in the Paramet er setParamet er setlist .

Add the following blocks and set their parameters as follows:

Study Objects ‣ DynamicalMatrix

Set Repet it ionsRepet it ions to ‘Custom‘Set the number of repet it ionsnumber of repet it ions to:

Γ

pnVbias = −0.4 V

nA = 3nB = 3

n = 1

Unt ick the ‘Acooust ic sum rule‘ opt ionTick the ‘Constrain elect rodes‘ opt ionUnt ick the ‘Equivalent bulk‘ opt ion

Analysis ‣ Vibrat ionalMode

Finally, in the Global IOGlobal IO opt ions, change the name of the Def ault out put f ileDef ault out put f ile to dynmat.hdf5 .

Send the script to the Job managerJob manager using the but ton, save it as dynmat.py and press the but ton to run the calculat ion. You can also download the full script from here: dynmat .py.

Not eNot e

Alternat ively, you can also use the special thermal displacement - Landauer method (STD-Landauer) toinclude part of the elect ron-phonon coupling effects in the t ransmission calculat ion at a much lowercomputat ional cost . The STD-Landauer case study t reats a system which is similar to that discussedin this tutorial.

Se t t i ng up t he H ami l t o ni an de r i vat i ve s c al c ul at i o nSe t t i ng up t he H ami l t o ni an de r i vat i ve s c al c ul at i o n

In the LabFloorLabFloor, drag and drop the DeviceConf igurat ionDeviceConf igurat ion object contained in the f iletransmission_V-0.4.hdf5 to the Script generat orScript generat or, and modify the New Calculat orNew Calculat or block as

follows:

In Hamilt onianHamilt onian t ick the ‘No SCF iterat ion‘ opt ion.

Add an Study Objects ‣ HamiltonianDerivat ives block and set the parameters as follows:

Set Repet it ionsRepet it ions to ‘Custom‘Set the number of repet it ionsnumber of repet it ions to:

In Const raint sConst raint s, click AddAdd and constrain the elect rode repet it ions with Fixed constrains. The result inglist of const rained atoms should be: 0,1,2,3,44,45,46,47.Unt ick the ‘Equivalent Bulk‘ opt ion

In the Global IOGlobal IO opt ions, change the name of the Def ault out put f ileDef ault out put f ile to dHdR_V-0.4.hdf5 .

Send the script to the Job managerJob manager using the but ton, save it as dHdR_V-0.4.py and press the but ton to run the calculat ion. The calculat ion will take around 20 minutes on 8 CPUs. You can alsodownload the full script from here: dHdR_V-0.4.py.

Not eNot e

Here, the Hamiltonian derivat ives are calculated non self-consistent ly to speed up the calculat ions.

Cal c ul at i ng t he i ne l ast i c t ransmi ssi o n spe c t rumCal c ul at i ng t he i ne l ast i c t ransmi ssi o n spe c t rum

To calculate the inelast ic part of the current , click the Script generat orScript generat or, and add the following

nC = 1

nA = 3nB = 3nC = 1

blocks:

Analysis from File Analysis ‣ Inelast icTransmissionSpectrum

Open the Analysis from File block, select the transmission_V-0.4.hdf5 f ile and load theDeviceConfigurat ion object contained in the f ile .

You will not ice that two addit ional blocks have also been added:

DynamicalMatrix HamiltonianDerivat ies

In our case, the dynamical matrix and the Hamiltonian derivat ives have already been calculated previouslyand can be re-used. Delete the two blocks, add two more Analysis from File blocks and arrange them asshown in the f igure below:

Click on the second Analysis from File block from the top. Select the f ile dynmat.hdf5 and load theDynamicalMatrix object . Uncheck the DeviceConfigurat ion object included in the same file. The panelshould look like as in the f igure below:

Then, click on the third Analysis from File block from the top. Select the f ile dHdR_V-0.4.hdf5 and loadthe HamiltonianDerivat ives object . The panel should look like as in the f igure below:

Finally, set the parameters for the Analysis ‣ Inelast icTransmissionSpectrum block as shown below:

In the EnergyEnergy part , set :

Points = 25In the k- point Samplingk- point Sampling part :

set Grid t ypeGrid t ype to ‘Regular k-point grid‘

E0 = −0.5 eVE1 = 0.5 eV

set the grid range for and to

set the SamplingSampling to:

In the q- point Samplingq- point Sampling part :

set Grid t ypeGrid t ype to ‘Regular q-point grid‘set the grid range for

and to

set the SamplingSampling to:

In the Global IOGlobal IO opt ions, change the name of the Def ault out put f ileDef ault out put f ile to xloe_V-0.4.hdf5 .

Send the script to the Job managerJob manager using the but ton, save it as xloe_V-0.4.py and press the but ton to run the calculat ion. The calculat ion will take around 1.5 hour on 24 CPUs. You can also download

kA

kB

[−0.2 : 0.2]

kA = 3kB = 3

qA

qB

[−0.5 : 0.5]

qA = 9qB = 9

the full script from here: xloe_V-0.4.py.

Anal ysi s o f t he i ne l ast i c t ransmi ssi o n spe c t rumAnal ysi s o f t he i ne l ast i c t ransmi ssi o n spe c t rum

The results of the inelast ic t ransmission spect rum can be analyzed in detail using the Inelast icInelast icT ransmission Spect rum Analyz erT ransmission Spect rum Analyz er.

In the LabFloorLabFloor, select the Inelast icT ransmissionSpect rumInelast icT ransmissionSpect rum object contained in the f ile

xloe_V-0.4.hdf5 , and click on the Inelast ic T ransmission Spect rum Analyz erInelast ic T ransmission Spect rum Analyz er plugin in the plugins panelon the right -hand side of the screen.

First of all, you will analyze the- and-dependency of the current . In the main window of the analyzer, set the following parameters:

Plot t ypePlot t ype : ‘Current vs. k-points‘Bias volt ageBias volt age :

You will obtain a f igure similar to the one below, showing the-dependency of the current within the sampled range of k-points. From the f igure, it is evident that the

main contribut ion to the current comes from the-point of the two-dimensional Brillouin zone defined by

and, with a non-negligible contribut ion from finite

-vectors.

Next , change the Plot t ypePlot t ype to ‘Current vs. q-points‘. You will obtain a f igure similar to the one below,showing the

-dependency of the current . From the f igure, it is evident that also in this case the main contribut ion tothe current comes from the

-point of the two-dimensional Brillouin zone defined by and

kq

−0.4 V

k

ΓkAkBk

q

ΓqA

, , with a non-negligible contribut ion from finite-vectors.

The two f igures above indicate that the current is mainly associated with t ransit ions occurring at the-point of the two-dimensional Brillouin zone, both in-space and in-space. To analyze which of the phonon modes at the-point is the one that contributes the most , set the following parameters in the main window of the

analyzer:

Plot t ypePlot t ype : ‘Current vs. phonon mode‘k- point sk- point s : ‘(0.00, 0.00)‘q- point sq- point s : ‘(0.00, 0.00)‘

qBq

ΓkqΓ

From the f igure, it can be seen that the phonon that contributes the most to the current is that with index, and energy

.

This phonon mode can be visualized by select ing the Vibrat ionalModeVibrat ionalMode object in the f ile dynmat.hdf5calculated in Sect ion Set t ing up the dynamical matrix calculat ion and using the Vibrat ion Visualiz erVibrat ion Visualiz erplugin:

66ℏω = 63.01 meV

Not eNot e

In the f igures presented above, the values of the current are negat ive, because the simulat ions areperformed in reverse bias condit ions.

Cal c ul at i o n o f t he c ur re nt wi t h and wi t ho ut e l e c t ro n- pho no nCal c ul at i o n o f t he c ur re nt wi t h and wi t ho ut e l e c t ro n- pho no ni nt e rac t i o nsi nt e rac t i o ns

In order to calculate the current without and with elect ron-phonon interact ions, download the script current .py and run it from the terminal as:

atkpython current.py transmission_V-0.4.hdf5 xloe_V-0.4.hdf5 .

Not eNot e

On windows, it is not convenient to run current.py script from the terminal. In this script , you canreplace f ilename1 and f ilename2 to the transmission_V-0.4.hdf5 and xloe-V-0.4.hdf5 . In order to run it ,drag it on the Job managerJob manager.

The script will calculate the elast ic () and inelast ic (

) components of the current . The total current without and with elect ron-phonon interact ions will thenbe calculated as:

The output of the calculat ion will show the name of the f iles used as input , as well as the values of thecalculated currents (highlighted in yellow):

IelIinel

I(w/o interactions) = Iel

I(w interactions) = Iel + Iinel

+------------------------------------------------------------------------------+ | | | QuantumATK 2017.2 [Build 997d195] | | | +------------------------------------------------------------------------------+ -------------------------------------------------------------------------------- TransmissionSpectrum read from file : transmission_V-0.4.hdf5 InelasticTransmissionSpectrum read from file : xloe_V-0.4.hdf5 -------------------------------------------------------------------------------- Current (w/o interactions) = -3.71e-09 nA Current (w interactions) = -9.03e-06 nA --------------------------------------------------------------------------------

Timing: Total Per Step %

--------------------------------------------------------------------------------

Loading Modules + MPI : 1.33 s 1.33 s 2.58% |=============| -------------------------------------------------------------------------------- Total : 51.58 s

Not eNot e

The above calculated currents are different with the currents before analysis of theInelast icTransmissionSpectrum because of different k-points samplings. Also in this case, the value ofthe current is negat ive, because the simulat ions are performed in reverse bias condit ions.

It can be seen that elect ron-phonon scat tering leads to an increase in the reverse bias current of aboutthree orders of magnitude!

Speeding up t he calcu lat ionsSpeeding up t he calcu lat ions

AT KAT K implements several methods which can be used to speed up the calculat ions considerably and enablethe calculat ion of the inelast ic t ransmission for devices comprised of thousands of atoms. Thesemethods are:

The ‘Phonon energy intervals’ methodThe possibility of using the bulk dynamical matrix and Hamiltonian derivat ives

For a more detailed descript ion of these methods, see the sect ion Large Device Calculat ions of theInelast icTransmissionSpectrum module in the AT K manualAT K manual.

T he ‘P ho no n e ne rgy i nt e rval s’ me t ho dT he ‘P ho no n e ne rgy i nt e rval s’ me t ho d

This method allows us to group phonon modes of a device with

vibrat ing atoms into energy intervals to form new effect ive phonon modes, with

. The inelast ic t ransmission spect rum will therefore be calculated only for these effect ive phonon modes, great ly reducing the computat ional cost of the calculat ion.

In the LabFloorLabFloor, drag and drop the DeviceConf igurat ionDeviceConf igurat ion object contained in the f iletransmission_V-0.4.hdf5 to the Script generat orScript generat or. Setup the calculat ion of the inelast ic t ransmission

spectrum as in the Calculat ing the inelast ic t ransmission spect rum Sect ion, but in this case set thePhonon Met hodPhonon Met hod in the PhononsPhonons part to ‘Phonon energy intervals‘.

3NNMMM << 3NM

Not eNot e

The default parameters of the Phonon energy intervals are f ine for the present calculat ion, but ingeneral one should ensure that the energy range

spans that of the phonon modes of the calculated dynamical matrix.

In the Global IOGlobal IO opt ions, change the name of the Def ault out put f ileDef ault out put f ile to xloe_pheint_V-0.4.hdf5 .

Send the script to the Job managerJob manager using the but ton, save it as xloe_pheint_V-0.4.py and pressthe but ton to run the calculat ion. The calculat ion will take only 20 minutes on 24 CPUs. You can alsodownload the full script from here: xloe_pheint_V-0.4.py.

Once the calculat ion is f inished, run the script current .py as:

atkpython current.py transmission_V-0.4.hdf5 xloe_pheint_V-0.4.hdf5 .

If you are a Windows user, see how to run the script .

The result ing output will be:

[E0 : E1]

+------------------------------------------------------------------------------+ | | | QuantumATK 2017.2 [Build 997d195] | | | +------------------------------------------------------------------------------+ -------------------------------------------------------------------------------- TransmissionSpectrum read from file : transmission_V-0.4.hdf5 InelasticTransmissionSpectrum read from file : xloe_pheint_V-0.4.hdf5 -------------------------------------------------------------------------------- Current (w/o interactions) = -3.71e-09 nA Current (w interactions) = -9.25e-06 nA --------------------------------------------------------------------------------

Timing: Total Per Step %

--------------------------------------------------------------------------------

Loading Modules + MPI : 1.34 s 1.34 s 19.27% |=============| -------------------------------------------------------------------------------- Total : 6.94 s

One can see that the calculated values for the current are essent ially the same as those calculated in theSect ion Calculat ion of the current with and without elect ron-phonon interact ions, despite the fact thatthe present calculat ions are much faster.

Usi ng t he bul k dynami c al mat r i x and H ami l t o ni an de r i vat i ve sUsi ng t he bul k dynami c al mat r i x and H ami l t o ni an de r i vat i ve s

Another opt ion to speed up the calculat ions is to use the dynamical matrix and Hamiltonian derivat ives ofthe bulk elect rodes, instead of those of the device. However, this is possible only for devices with ast ructure which is t ranslat ionally invariant along the

-direct ion, apart from doping, elect rostat ic regions and applied bias voltage.

The scripts needed to calculate the dynamical matrix and the Hamiltonian derivat ives of the bulkelect rodes can be downloaded from here: dynmat_bulk.py, dHdR_bulk.py. Running the twocalculat ions on 4 CPUs will take less than 2 minutes.

In order to calculate the current inelast ic t ransmission spect rum using the bulk dynamical matrix andHamiltonian derivat ives, repeat the steps followed in Sect ion Calculat ing the inelast ic t ransmissionspectrum, with the changes described in the following.

First , click the Script generat orScript generat or, add the following blocks, and modify their parameters as follows:

Analysis from File

Load the DeviceConfigurat ion object from the tranmission_V-0.4.hdf5 f ile.

Analysis ‣ Inelast icTransmissionSpectrum

Set parameters as described in the Inelast ic t ransmission spect rum sect ion.

Remove the DynamicalMatrix and HamiltonianDerivat ives blocks, add two more Analysis from Fileblocks right after the Analysis from File already present , and set their propert ies as follows:

Click on the second Analysis from File block from the top, select the

dynamic_bulk.hdf5 f ile and load the DynamicalMatrix object contained in the f ile.

C

Click on the second Analysis from File block from the top, select the

dHdR_bulk.hdf5 f ile and load the HamiltonianDerivat ives object contained in the f ile.

In the Global IOGlobal IO opt ions, set the name of the Def ault out put f ileDef ault out put f ile to xloe_V-0.4_bulk.hdf5 . In this case,you need to modify the Inelast ic T ransmission Spect rumInelast ic T ransmission Spect rum block in the Edit orEdit or as follows:

1 2 3 4 5 6 7 8 91011121314

inelastic_transmission_spectrum = InelasticTransmissionSpectrum( configuration=device_configuration, bulk_dynamical_matrix=dynamical_matrix, bulk_hamiltonian_derivatives=hamiltonian_derivatives, energies=numpy.linspace(-0.5, 0.5, 25)*eV, kpoints=kpoints, qpoints=qpoints, self_energy_calculator=RecursionSelfEnergy(), energy_zero_parameter=AverageFermiLevel, infinitesimal=1e-06*eV, phonon_modes=All, method=XLOE, spectral_representation=True, electrode_extensions=[0, 0],

Send the script to the Job managerJob manager using the but ton, save it as xloe_V-0.4_bulk.py and press the but ton to run the calculat ion. Also in this case, the calculat ion will take only 20 minutes on 24 CPUs. You

can also download the full script from here: xloe_V-0.4_bulk.py.

Once the calculat ion is f inished, run the script current .py as:

atkpython current.py transmission_V-0.4.hdf5 xloe_V-0.4_bulk.hdf5 .

If you use Windows, look at the how to run the script .

The result ing output will be:

+------------------------------------------------------------------------------+ | | | QuantumATK 2017.2 [Build 997d195] | | | +------------------------------------------------------------------------------+ -------------------------------------------------------------------------------- TransmissionSpectrum read from file : transmission_V-0.4.hdf5 InelasticTransmissionSpectrum read from file : xloe_V-0.4_bulk.hdf5 -------------------------------------------------------------------------------- Current (w/o interactions) = -3.71e-09 nA Current (w interactions) = -6.42e-06 nA --------------------------------------------------------------------------------

Timing: Total Per Step %

--------------------------------------------------------------------------------

Loading Modules + MPI : 1.33 s 1.33 s 2.17% |=============| -------------------------------------------------------------------------------- Total : 61.15 s (1m01.15s)

It can be seen that the calculated values for the current is similar to that calculated using the dynamicalmatrix and Hamiltonian derivat ives of the device configurat ion, although there are differences due to the

Next

fact that in the present case, the loss t ranslat ional invariance of the Hamiltonian derivat ives due to theapplied bias is not taken into account .

ReferencesReferences

[FPBJ07] T . Frederiksen, M. Paulsson, M. Brandbyge, and A.-P. Jauho. Inelast ic t ransport theory from firstprinciples: Methodology and applicat ion to nanoscale devices. Phys. Rev. B, 75:205413, May 2007.doi:10.1103/PhysRevB.75.205413.

[LCF+14] (1, 2) J.-T . Lü, R. B. Christensen, G. Fot i, T . Frederiksen, T . Gunst , and M. Brandbyge. Efficient calculat ionof inelast ic vibrat ion signals in electron t ransport : Beyond the wide-band approximat ion. Phys. Rev. B,89:081405, Feb 2014. doi:10.1103/PhysRevB.89.081405.

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