This literature may be helpful for you Phys. Rev. Lett. 98, 089901 (2007)

Thanks Vipin,but it s very conceptual question that when Fermi energy comes into picture,there will be flow of current if electrons lies in the vicinity of chemical potential only,but in some papers people have written fermi energy and some where only energy,so i am confused actually.

Thanks

Hi!

For studying the charge and spin transport, you should find Fermi energy of ribbon, at first. For doing that, you should find the energy dispersion curve. You can write the Hamiltonian of a super cell including some unit cells and called it H00. Then you can write the interaction Hamiltonian between this super cell with its left and right  neighbor super cells and called them H10 and H01, respectively. Finally you can find the eigenvalues of H=H00+H01exp(ika)H01+exp(-ika)H10, for different k-values. Here, H01 and H10 should be dagger of each other, otherwise H will not be a Hermitian matrix.

Here, a is the lattice constant of your one dimensional lattice of ribbon.

After drawing E(k), you can find Fermi energy which is at the middle of maximum of valance and minimum of conduction bands.

If you want to inject the electrons from left (right) side to right(left) side, the electron energies should be higher than the Fermi energy which you have found it before. Therefore, you should draw for example the conductance curve versus  (E-EF) where E is electron energy and EF is Fermi energy.

If you apply a voltage along the length of ribbon the Fermi energy is changed and you should solve the non-equilibrium problem. Of course if you assume that you deal only with the energies which is very close to EF you can assume that EF does not change and you solve equilibrium problem. Many published articles use the assumptions. Otherwise you should solve Schrodinger or Dirac equation with Poisson equation simultaneously on self consistent method.

Now if you apply a magnetic field perpendicular to the plane of the ribbon it does not work on electrons and you should only change hopping constant, t, to t . exp(i. integral(A .dl ) ) which A is vector potential related to magnetic field. The integral should be calculated along the hopping trajectory between two atoms.

But if you apply an electric field perpendicular to the plane of the ribbon or the flow direction of electrons (i.e., current direction), it can  split the spins. When it is perpendicular to the plane, you should add  the Rashba  spin orbit  interaction terms to Hamiltonian. If it is perpendicular to the flow direction of electrons (i.e. current direction) you should assume at one edge of ribbon the potential is zero (reference of potential) and calculate the potential of each atom by using the formula E . Ym where E is the electric field and Ym is the y-coordinate of each atom. The term should be added to the onsite energy of each atom.

By using the method you can study the charge and spin transport in ribbon .

Best Regards

Thanks  Simchi for your kind reply,yes i am using Schrodinger equation with NEGF with self consistent approach only.there are hopping between the leads also as you have said,between supercells ,As you know that band gap also depends upon the width of the ribbon alo,i am getting LDOS ,EK diagram as well as Conductance curve,with this approach.Only changing the hopping from  t to texp as you have written will add the magnetic field term or i need to change the hoppings between leads as well,Here third nearest  neighbor interaction also has been taken into account.BUT what else i can work here thats what i am thinking here now.

Regards

Hi!

Generally, it is assumed that, the magnetic field is only applied in transport channel region. Therefore it is not necessary you consider it in H00, H01, and H10 matrices when you are going to calculate the surface Green's functions gL and gR. 

Best Regards

Thanks but why there are needs of block matrices like in case of self energies which were coming non zeros from both sides,which means that leads have been connected to channel .why we are converting block to sparse matrices then and vice-versa .

regards

Dear Simchi,Can you please let me know why magnetic field terms will not come in contacts ,since self energies will be non zeros there?

regards

Hi!

It is only an assumption. Of course peoples want to study the effect of magnetic field on transport channel. Also, since you should find coupling matrix between left and right leads with transport channels i.e., HCL , and HCR for for going that you should add at least one super cell of left and right  lead ribbon to transport channel, it means that you can consider the effect of magnetic field in self energy. If gL and gR be surface Green's functions then the self energies will be

Self- energy of left lead=HCL . gL . HCL'

 Self- energy of right lead=HCR . gR . HCR'

Best Regards

H.Simchi

Thanks Simchi,need to calculate more terms since i have calculated LDOS,Conductance and EK relationI am not yet cleared about the effect of magnetic field effect upon contacts,although you have said that it is assumption.

regards

Dear Simchi,is it necessary to define or to calculate self energies at each Fermi level or not? and my self energies are coming in complex forms,real as well as imaginary components,i am not able to understand it physically interpretation,Can you please if you have any idea about this.

best wishes

Hi!

You can Calculate self energy using

sigma-left= HCL . gL .HCL'

sigma-right=HCR . gR . HCR'

where gL and gR are surface Green's function and Hs are coupling matrices between channel and leads.

If you want to consider quasi-particles as a free particle in many body theory, you should pay its expense. the expense is adding self energy in the Green's function of free particle as below:

Green=1/(E-Ei-self energy (-/+) i . epsilon)

where Ei is eigenvalues and epsilon is very small positive number.

In the other word, now you can assume that electrons in channel behave as  free particle. Of course as above formula shows, self energy change the position of eigenvalues in energy axis and it can be shown that it broad them.

Best Regards

Thanks Simchi for your kind reply,it is happening like broadening will occur ,like Lorentzian or Gaussian kind of shape ,what you said in your last words,but i am not able to understand why my self energies left as well as right terms are coming in complex forms,as well as gamma's which are equal to =i(sigma-siagma^dagger)

all the terms are coming in complex terms ,in the form of matrices,why these are coming in complex forms ,like one element of gamma_r you can say 1.666872313932557e-06 + 0.000000000000000e+00i,this kind of valu it is coming ,although i have taken hopping to be t,no magnetic field or electric field terms ,still i have not introduced.Please throw some light here.

Regards

Dear Simchi sir

Can you please let me know why these self energies are coming in complex forms what it means physics wise.

regards

Hi!

sigma is complex. it appears in Green function as:

Green=1/(E-Ei-real of self energy- i . imaginary of self energy)

it can be converted to:

Green =(E-Ei-real of self energy)/((E-Ei-real of self energy)^2+( imaginary of self energy)^2) + (i.imaginary of self energy)/((E-Ei-real of self energy)^2+( imaginary of self energy)^2)

It can be shown that, first term is seen in experiments as real data of sample and second part is seen as background which can be deleted in data refining.

You can see the book of Inkson in many body physics for more explanations.

But, Gama which is  

Gama=-i . ( (x+iy) - (x-iy))= - i. ( 2iy)= 2y, where (x+iy) is self energy.

is a real number. Other wise, we can not use it in formula of transport.

Best Regards 

Thanks Simchi,if we'll go beyond third nearest neighbour then what will be the expected outcome.Broadening and selfenergies should be affected as i understand ,what do you think over it sir,I am following Ferry and Goodnick for this.

regards

Hi!

As I know, if we add second neighbors effects, we will see a shift in position of Fermi energy (see below reference) but I do not think that the third neighbors effects have serious effect on previous results.

A. H. C. Neto et al., REVIEWS OF MODERN PHYSICS, VOLUME 81, JANUARY–MARCH 2009

Best Regards

H.Simchi

Thanks Simchi,yes i have downloaded this review article but in my calculation i am keeping length=3,and width =4,t=1,nearest neighbour hopping parameters ,while hoppings inside the leads is t,whether onsite potential is 0 only,but i am getting characterstics of armchair graphene nano ribbon ,if i want to do it for zigzag case ,what i will have to do,and i want to do for Landau levels,as i understand it should come after introduction of magnetic field sir,what is your opinion about this,please throw some light here.

regards

Hi!

Landau levels should be calculated by calculation SigmaXX when we assume SigmaXX=0 i.e., Fermi energy is placed within localized states. If you calculate it , it will be quantized and proportional to Radical(absolute value of n) , n= ....-3,-2,-1,0,1,2,3,..... or Radical(absolute value of B), where B is magnetic field.

Based on previous explanation, you can calculate SigmaXX . Please see the previous reference.

Best Regards

H.Simchi

Thanks Simchi sir for your kind reply,as i have posted my question also that i am doing this calculation for armchair and so my results of EK relation is coming as shown in review article also,but sir eg i want to do it for zigzag case what parameter i should change sir and i am getting my retarded Green function as a matrices,depending upon length and width of ribbon ,but sir in each element ,one small term has been subtracted ,as i understand sir,its the same term appearing which i have added in energy like [(E+ieta)I-H] this eta must be coming in these elements sir,am i right in this case sir or not?like one element i am showing here 0.0175-0.0000i .

Thanks sir

regards

Hi1

I can not understand what is your question exactly.

Generally eta=5E-3 is suitable for tight binding method.

The method is same for both armchair and zigzag. But since the structure of armchair and zigzag is not same the elements 0f H00, H01 nad H10 matrices are not same in armchair and zigzag. For example the matrix of a super cell including two unit cells, i.e., H00 is 

0    t    0    0

t     0    t    0

0    t     0   t

0   0     t   0

and H01 is:

0   0   0   0

t    0    0   0

0    0   0    t

0    0    0   0

Also, H01=Dager of H10

Best Regards

H.Simchi

Thanks Simchi sir,what i mean i that i want to add magnetic field as well Landau level in my system,earlier B was taken to be zero,so my Hamiltonian is becoming block matrices,as symmetric way,now i want to add electron-electron interaction in my system which earlier Schrodinger -Poisson solver was doing self consistently sir,but here how i can add,i am stuck off at this point sir.

with best wishes

Thanks Simchi sir,but i want to add electron-electron interaction like earlier i have done by adding Schrodinger -Poisson solver self consistently solving,as well as electron-phonon interaction.If only one field is applied  ,either of Electric and magnetic field time reversal symmetry of the energy bands  for electrons and holes is preserved ,but WHY this sir However combined effects on the energy dispersion is to break the time reversal symmetry for both electrons and holes and mix the energy bands.

regards

Dear Simchi sir,when i am adding the hopping term from t to texp(i.int(A.dl)),it is asking me for value of vector potential ,so sir can you tell me what value i should give of vector potential ,since as you have suggested that it will be equal to B=curl of A,but since i dont have magnetic field,so how i can come to know A(vector potential sir.Please share your views

regards

Hi!

WE assume graphene is placed in x-y plane and B is in z-direction such that B=(0,0,B0). Therefore A=(-B0y, 0, 0) or A=(0, B0x, 0).

Now we should calculate Int(A . dl) on joining line between two nearest neighbor atoms.

Best regards

H.Simchi

Ok Simchi sir thanks,what about the limits on integeral,and dl=dx^i+dy^J+dzk^,where i j and k are unit vectors,am i right at this point sir.

regards

Dear Simchi,i have added t=texp(i.int(B_0x times dy)),but what will be the value assighned to B_0X ,i'll have to define it ,otherwise i'll get error in my prog,when i am adding magnetic field term in the system.

regards

Hi!

Please study the attached references.

Best regards

Pardon me see below

Transport properties of two finite armchair

graphene nanoribbons

Luis Rosales and Jhon W Gonz´alez

Rosales and Gonz´alez Nanoscale Research Letters 2013, 8:1

http://www.nanoscalereslett.com/content/8/1/1

Thanks sir ,i have taken total flux is equal to -a*B*yn=if magnetic field is 4T,a=1.0,yn=4t means total is equal to -32.2624,total flux.

Hope this one is fine sir

regards

Thnaks Simchi,if i want to add temp in my calculation at very different temperature,then which quantity will be affected by temp ,please throw some light .

regards

Hi!

I have no experiences in this field. But I think at first you should study many-body theory and its related Green's function and write true self energies and so on.

Best Regards

ok sir thanks for  your praise able advise,

regards

Welcome

Following GD Mahan ,is it fine sir?

regards

Dear Simchi ,how  i can add electron-electron interaction in my system?

regards

Hi!

You should study many lectures. You can see the review article in below:

Valeri N. Kotov et al., arXiv:1012.3484v2 [cond-mat.str-el] 26 Feb 2012

Best Regards

Thank you very much sir,but i want to add e-e or e- phonon interaction in my system also sir,is it possible to add these interaction in the GNR system or not sir?

regards

Simchi sir,i have  calculated Conductance as well as LDOS from surface green's function,i have used the formula to determine current also I=q/h int(F_l-F_r)*T(E) ,but i dont knopw how to calculate volatge,since i want to plot IV curve,i cant use Ohm's law here since its in atomistic scale.Please let me know sir ,if you have any idea about this.

warm regards

Hello simchi sir,I am refering review article told by you,those people are taking hopping parameter -2.7eV,and they are saying to add for e-e interaction to add other self energy term due to e-e in Hamiltonian,but i dont understand why negative hopping energy ?Can you tell me if you understand this thing.

regards