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Table of Contents

Table of ContentsTransport in graphene nanoribbons

IntroductionBand structure of 2D graphene

Constructing the grapheneBand structure: k-point samplingAnalysis of the resultsCalculate the band structure using DFT

Band structure of an armchair ribbonBuilding the archair ribbon structureBand structureAnalyzing the results

Transport properties of a zigzag nanoribbonPristine ribbonAnalyzing the resultsCreating a distortion in the ribbonTransport properties of the distorted ribbonNon-self-consistent calculationSelf-consistent calculationComparing the transmission spectra

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Transport in graphene nanoribbonsTransport in graphene nanoribbonsVersion:Version: 2016.0

Research on fundamental aspects and device applicat ions of graphene is a very act ive f ield. This tutorialillust rates how QuantumATK can be ut ilized to study various applicat ions related to graphene nanoribbons,ranging from a simple, inf inite sheet of graphene to more complex junct ions.

The main purpose of the tutorial is to demonstrate the basic work-f low in QuantumATK, how to buildsystems with nanoribbons and compute fundamental quant it ies such as band st ructures and t ransmissionspectra using QuantumATK.

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This tutorial is based on operat ions in the graphical user interface QuantumATK and involves almost noscript ing at all. It is assumed that you are familiar with the basic operat ions in QuantumATK, asint roduced in the Int roduct ion to QuantumATK tutorial.

In t roduct ionIn t roduct ion

The geometry of graphene is simple and regular, and the infinite, planar st ructure can easily be created,either by hand or by taking a single layer from the crystal st ructure of graphite.

Fig. 57 A graphene sheet.

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However, to create a device- like st ructure, the infinite sheet must be cut into a suitable shape. A commonshape for elect ronics applicat ions is a so-called graphene nanoribbon (GNR). These can be quitecumbersome to set up in an effect ive manner, not least considering the hydrogen passivat ion of theedges, which is needed in f inite st ructures in order to sat isfy all carbon valence elect ron bonds.

QuantumATK has a specif ic graphene nanoribbon Builder tool, and you will in this tutorial learn how to use it ,and how to customize the st ructures it produces to make more advanced shapes of graphene. This willallow you to

1. calculate the band st ructure of ideal, inf inite graphene nanoribbons, of either armchair or zigzag type,as a funct ion of the ribbon width;

2. int roduce doping or defects, on the edge or elsewhere, and study the influence on the t ransmissionspectrum;

3. use basic ribbons as building blocks for your own device ideas, such as e.g. Z-shaped nanoribbonjunct ions.

This tutorial will show examples of each type of system.

Band st ruct ure of 2D grapheneBand st ruct ure of 2D graphene

Before start ing with the nanoribbons, you will do a quick int roductory calculat ion of the band st ructure ofan infinite 2D sheet of graphene. This will also serve as a way to refresh the basic concepts and work f lowin QuantumATK.

Co nst ruc t i ng t he graphe neCo nst ruc t i ng t he graphe ne

Start QuantumATK, create a new project and give it a name, then select it and click OpenOpen. Launch thebuilder by clicking the BuilderBuilder icon on the tool bar.

Graphene, and many other crystals, molecules and fullerenes, are available in the built - in database inQuantumATK. In the Builder, click Add ‣ From Database and locate Graphene (while you are typing, you willsee that graphite is included in the database too).

Double-click the line to add the st ructure to the Stash, or click the in the lower right -hand corner.

Now send the st ructure to the Script Generat orScript Generat or by clicking the “Send To” icon in the lower right -hand corner of the window.

B and st ruc t ure : k- po i nt sampl i ngB and st ruc t ure : k- po i nt sampl i ng

You will now set up the band st ructure calculat ion. It is inst ruct ional for this part icular system to comparethe results of a DFT calculat ion with the semi-empirical method; as you will see, they are very similar.

The f irst step is to use the Extended Hückel method in the ATK-SE calculator to determine the required k-point sampling. Using this faster method saves t ime compared to doing the same invest igat ion with DFT.The results should, for this aspect , be very similar.

In the Script Generat orScript Generat or, change the default output f ile to graphene_band.hdf5 and add the followingblocks to the script :

New Calculat orNew Calculat or Bandst ruct ureBandst ruct ure

You should the default parameters, except the for the Hückel basis set parameters: Open the NewNewCalculat orCalculat or by double-clicking it .

Select the “ATK-SE: Extended Hückel” calculator.Under Huckel basis setHuckel basis set , select the “Cerda.Carbon [graphite]” basis set .

Make sure to use this basis set whenever studying graphene (or carbon nanotubes) with the extendedHückel method. Close the dialogue by clicking OKOK in the lower right -hand corner.

Next , open the Bandst ruct ureBandst ruct ure block. A pre-defined route in the Brillouin zone is set up, correspondingto a three-dimensional hexagonal Bravais lat t ice. Leave it like that for now; this point will be discussedbelow.

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The different components of the script inherit the global output f ile name. It is possible to specify aunique f ile name for each analysis block, but usually it makes most sense to collect all results in thesame file.

You are now ready to run the calculat ion: Send the script to the Job ManagerJob Manager by clicking the “SendTo” but ton and by select ing “Job Manager” in the menu that shows up.

In the Job Manager, click to run the job. The calculat ion takes only seconds to complete.

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

Once the calculat ion is f inished, the graphene_band.hdf5 f ile should be visible in the Project Files list inQuantumATK. On the LabFloorLabFloor, select “Item Type” from the drop-down menu as shown in the belowfigure. Select the Bandstructure item, and click the Bandst ruct ure Analyz erBandst ruct ure Analyz er but ton in the right -handplugins panel.

The calculated band st ructure is visualized. However, on close inspect ion, you can see that is does notappear to be correct , since the Brillouin zone K point is supposed to coincide with the Fermi level.

The cause of the poor band st ructure is insufficient k-point sampling; the default is to have just one pointin each direct ion, and this fails to properly describe the graphene elect ronic st ructure.

Therefore, open the minimized Script Generat orScript Generat or window again, open the Calculator block, and enter9x9x1 k-points under “Basic” set t ings.

You should also adjust the symmetry points used for the band st ructure calculat ion to avoid the f latsegments, which correspond to paths out of the planar graphene Brillouin zone. The only relevant k-pointsin a 2D hexagonal lat t ice are G=(0,0,0), K=(1/3,1/3,0), and M=(0,1/2,0), in units of the reciprocal lat t icevectors.

This can easily be done in the Scripter (by double-clicking the Bandstructure block and edit ing the path),but to demonstrate how easy it is to use a script produced by the Script Generator as a template, you willnow learn to do that by hand instead!

Send the script to the Edit orEdit or by clicking the icon in the Scripter. Locate the line with the symmetrypoints, and edit it as shown below.

bulk_configuration.setCalculator(calculator)nlprint(bulk_configuration)bulk_configuration.update()nlsave('graphene_band.nc', bulk_configuration)

# -------------------------------------------------------------# Bandstructure# -------------------------------------------------------------bandstructure = Bandstructure( configuration=bulk_configuration, route=['G', 'M', 'K', 'G'], points_per_segment=20, bands_above_fermi_level=All )nlsave('graphene_band.nc', bandstructure)

Run the calculat ion by sending the script from the Edit orEdit or to the Job ManagerJob Manager. When thecalculat ion f inishes, a new Bandstructure object will be located on the LabFloorLabFloor. Select it and plot it asbefore. The Fermi level is now exact ly at the K point , as illust rated below.

Cal c ul at e t he band st ruc t ure usi ng DF TCal c ul at e t he band st ruc t ure usi ng DF T

You now know the required k-point sampling, and you can use this also for the DFT calculat ion. To switchmethod, return to the Script Generat orScript Generat or, open Calculator block, and select the AT K- DFTAT K- DFT calculator.Set the k-point sampling to 9x9x1, but leave all other parameters at their default values.

T ipT ip

In case you closed the Script Generator window, you can reuse the geometry from the saved HDF5 f ileby dropping one of the bulk configurat ion inside it on the Scripter icon on the toolbar.

Keep again the same name for the output HDF5 f ile, and run the calculat ion. This will add yet anotherBandstructure item to the LabFloorLabFloor, which can be plot ted like before. On inspect ion, you will f ind that theresults are very similar to the Hückel band st ructure. The Hückel method therefore has good accuracy forthis system and will be used for the further examples, mainly to save t ime; the interested reader isencouraged to verify that the results obtained with DFT are very similar.

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As it turns out , get t ing an accurate alignment of the K point to the Fermi level in graphene is not just asimple matter of having a lot of k-points. The results for 12x12x1 or 13x13x1 k-points are, for instance,considerably worse than those for 10x10x1. There is a general t rend that more k-points are bet ter, buton the other hand, 3x3x1 points are bet ter than almost all other values up to 20x20x1. This behaviorcan be understood by studying how the k-points in the Monkhorst–Pack grids are dist ributed around K=(1/3,1/3,0); the ideal situat ion is to have several points close to or at the Fermi level, although the lat teris less favorable than having two points close by.

Band st ruct ure of an arm chair r ibbonBand st ruct ure of an arm chair r ibbon

Having invest igated the band st ructure of pure graphene, the focus is now on graphene nanoribbons (GNR),which are f inite, narrow st rips of graphene cut out from the infinite 2D sheet . There are two primary waysto cut out such a ribbon – these two st ructures are known as armchair or zigzag ribbons, a nomenclaturethat refers to the shape of the edges. Notably, an armchair ribbon is an unrolled zigzag nanotube! Armchairribbons are predicted to be semiconduct ing, and you will now compute the band st ructure of such a ribbonto confirm this.

B ui l di ng t he arc hai r r i bbo n st ruc t ureB ui l di ng t he arc hai r r i bbo n st ruc t ure

Generat ing nanoribbons with QuantumATK could not be easier – there is a BuilderBuilder plugin specif icallydesigned for this task. Launch the Builder and click Add ‣ From Plugin ‣ Nanoribbon. This launches theGraphene Ribbon tool.

In the widget that opens, you can design many different types of nanoribbons, not just graphene but alsoB-N, etc. By select ing a linear combinat ion nA+mB of the two unit cell vectors A and B in the grapheneplane, ribbons of any chirality can be constructed (the “chirality” label comes from the fact that thesest ructure correspond to an unrolled (n,m) nanotube). For simplicty, you can also direct ly choose to buildarmchair or zigzag ribbons, and specify their width (which must be even for zigzag ribbons).

Select an armchair ribbon of width 8 (carbon atoms), and click BuildBuild to add it to the Stash.

Send the st ructure to the Script Generat orScript Generat or.

B and st ruc t ureB and st ruc t ure

Like before we will use the Extended Hückel method for the calculat ions. Change the default outputf ilename to armchair.hdf5 , and add the following two blocks to the script :

New Calculat orNew Calculat or Bandst ruct ureBandst ruct ure

Open the Calculator block to edit it :

Select the “ATK-SE: Extended Hückel” calculator.Select the “Cerda.Carbon [graphite]” basis set for carbon. The default “Hoffmann” basis set should

work well for the hydrogen.

In this case you need only 1 k-point in the direct ions perpendicular to the direct ion of the ribbon, since thesystem is f inite in these direct ion, while the system is periodic in the C direct ion. A large number of k-points is needed in this direct ion, since the unit cell is quite short and graphene has a rather peculiar point -like Fermi surface as we saw above; 100 points should be suff icient :

The default suggested route in the Brillouin zone from G=(0,0,0) to Z=(0,0,1/2) for the band st ructure isindeed the appropriate one, so you just need to increase the number of points to 200 to obtain nicelysmooth curves:

Run the calculat ion using the Job ManagerJob Manager.

Anal yz i ng t he re sul t sAnal yz i ng t he re sul t s

Return to the main QuantumATK window and use the Bandst ruct ure Analyz erBandst ruct ure Analyz er to plot the Bandstructureitem from the f ile armchair.hdf5 . The computed band st ructure is shown in the f igure below. By zoomingin a bit you should see a small band gap band gap at the Γ point . The armchair nanoribbon is thereforesemiconduct ing.

As an extension of this exercise, you are recommended to redo the calculat ions for different types ofribbons (armchair/zigzag) and for different widths. The band gap is st rongly dependent on both factors!

In the next sect ion, you will invest igate a zigzag ribbon, which is metallic rather than semiconduct ing.

Transpor t proper t ies of a zigzag nanor ibbonTranspor t proper t ies of a zigzag nanor ibbon

In this sect ion you will do some t ransport calculat ions. One of the most basic t ransport system that canbe made with graphene is a perfect , inf inite zigzag nanoribbon. Such ribbons are metallic and display noelast ic scat tering, i.e. perfect conduct ivity. In the second part you will invest igate what happens whenscat tering is int roduced by distort ing the edges of the ribbon.

P r i st i ne r i bbo nP r i st i ne r i bbo n

In a prist ine device system there is no scat tering from the cent ral region and, as illust rated in the following,

you can obtain the t ransmission spect rum from the propert ies of the elect rode.

Return to the BuilderBuilder and open the Nanoribbon widget from Add ‣ From Plugin ‣ Nanoribbon. Build azigzag nanoribbon with a width of 8 carbon atoms.

Transfer the st ructure to the Script Generat orScript Generat or and add the following blocks to the script .

New Calculat orNew Calculat or Bandst ruct ureBandst ruct ure T ransmissionSpect rumT ransmissionSpect rum

Change the default output f ile to zigzag_transport.hdf5 .

Edit the Calculator set t ings:

Select the Ext ended HückelExt ended Hückel calculatorSet the k-point sampling to 1x1x100.Change the Hückel basis set to “Cerda.Carbon [graphite]” for C.

Next , open the the Bandstructure block and set points per segment to 200.

Finally, open the TransmissionSpectrum block and set the energy range from −4 eV to +4 eV.

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The odd number of energy points ensures that there will be an energy point at the Fermi level.

Run the script using the Job ManagerJob Manager. It should take only a few minutes to f inish.

Anal yz i ng t he re sul t sAnal yz i ng t he re sul t s

Once the calculat ion f inishes, locate the T ransmissionSpect rumT ransmissionSpect rum item on the LabFloor, and open theT ransmission Analyz erT ransmission Analyz er plugin. As expected for a perfect 1D system, the t ransmission exhibits asequence of steps with integer t ransmission.

The most prominent feature of the plot is the enhanced t ransmission around the Fermi level. This is due toa peculiarity in the band st ructure of the zigzag ribbon (see the band st ructure below), and interest inglyenough, the enhancement is retained even if the ribbon is distorted, at least in the way that will beinvest igated in the next example.

Cre at i ng a di st o r t i o n i n t he r i bbo nCre at i ng a di st o r t i o n i n t he r i bbo n

So far you have basically just confirmed that armchair ribbons are semiconduct ing and zigzag ribbons aremetallic (at least as long as elect ron spin is not considered). It is now t ime to do something a lit t le bit moreinterest ing. The next system you will study is a distorted ribbon; more specif ically you will make aconstrict ion in the perfect zigzag ribbon, and see how this inf luences the t ransport propert ies.

Return to the BuilderBuilder and open the “Repeat” tool (available from “Bulk Tools” in the plugin panel). SetC=12 and click ApplyApply . Then press Ctrl+R to reset the 3D view.

Select the atoms indicated in the f igure below by drawing a rectangle with the mouse, while holding downCtrl .

Delete the atoms by pressing the Delete key. The dangling bonds that have now been created should besaturated by adding hydrogen atoms. To do this, click the “Passivate” but ton on the top toolbar.

Finally, convert the periodic st ructure to a device geometry by using the Device Tools ‣ Device From Bulktool.

The builder will detect any pat tern of periodicity close to the edges of the st ructure, and suggest theseas elect rodes. You may see these suggest ions in the dropdown combo boxes, as indicated in the f igureabove.

For the current st ructure, several periodicity pat terns can be found, corresponding to a certain number ofrepet it ions of the zigzag unit cell. The suggested value (7.383 Å), corresponds to 3 repet it ions, and this isa suitable length. Thus, no further adjustments are needed. Click OKOK, to build the device geometry for thedistorted zigzag nanoribbon.

T ranspo rt pro pe r t i e s o f t he di st o r t e d r i bbo nT ranspo rt pro pe r t i e s o f t he di st o r t e d r i bbo n

In the following you will learn how to perform the calculat ion of the zero-bias conductance of thedistorted ribbon using open boundary condit ions, f irst non-self consistent ly and then self-consistent ly.

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Calculat ions at f inite bias must always be performed fully self-consistent ly, since the non-self-consistent model do not produce a correct voltage drop across the device.

N o n- se l f - c o nsi st e nt c al c ul at i o nN o n- se l f - c o nsi st e nt c al c ul at i o n

Send the device geometry to the Script Generat orScript Generat or, and set the default output f ile name tozigzag_transport.hdf5 (i.e. the same file name as used for the bulk t ransmission calculat ion above, so that

the new calculat ions are saved in that same file).

Then add the New Calculat orNew Calculat or and T ransmissionSpect rumT ransmissionSpect rum script blocks.

Set up the Calculator block to use the Ext ended Hückel (Device)Ext ended Hückel (Device) calculator with the“Cerda.Carbon[graphite]” Hückel basis set for carbon. Set also the energy range in theTransmissionSpectrum block to −4 eV to +4 eV.

Run the script using the Job ManagerJob Manager. It takes a minute or so, but while it is running you can alreadyproceed and prepare the next step.

Se l f - c o nsi st e nt c al c ul at i o nSe l f - c o nsi st e nt c al c ul at i o n

You should also perform a self-consistent calculat ion with open boundary condit ions. This requires only a

Next

single change to the Calculator block:

Remove the t ick mark from No SCF it erat ionNo SCF it erat ion in the “Iterat ion Control Parameters” calculator set t ings.Send the script to the Job ManagerJob Manager and run the job.

Run the script using the Job ManagerJob Manager. This t ime the job will take somewhat longer (around 10minutes).

Co mpar i ng t he t ransmi ssi o n spe c t raCo mpar i ng t he t ransmi ssi o n spe c t ra

Let us now compare the computed t ransmission spect ra of the perfect ribbon and the distorted ones. Onthe LabFloorLabFloor, select It em T ypeIt em T ype in the “Group by” menu. Then select all three the t ransmissionspectrumobjects and click the Compare Dat aCompare Dat a plugin.

Fig. 58 Comparison of the transmission spectra of a perfect ribbon (blue) and a distorted ribbon calculated non-selfconsistently (green) and selfconsistently (red).

Comparing the results you see that the st rong t ransmission at the Fermi level in the perfect ribbon is alsopresent in the distorted ribbon. Apart from this, the t ransmission spect ra for the distorted GRN arest rongly suppressed due to scat tering by the distort ion.

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