NEGF Simulation of Electron Transport in Carbon Nano-tubes and Graphene Nanoribbons

Transcription

1 2009 Summer School: Electronics from the Bottom Up NEGF Simulation of Electron Transport in Carbon Nano-tubes and Graphene Nanoribbons Samiran Ganguly and Seokmin Hong Purpose of the Experiment and Simulation: To convey a basic understanding of how electron transport through any given device may be solved using Non-equilibrium Green s Function approach and specifically through Carbon Nanotubes and Graphene Nano-ribbons in both Armchair and Zigzag configurations. Expected outcomes: After the lab session it is expected that the participant will have a basic understanding and appreciation of the following: NEGF formalism of quantum transport How a Hamiltonian of a given device may be written using the tight-binding method How Self-energy matrices may be calculated using iterative solution of surface Green s Functions How to include phase breaking processes (example to be covered is the electron-electron interaction) How to plot the band structure of the given device Calculate transmission through the device This lab session is meant to provide some hands-on practice to the lecture G5 of 2009 NCN@Purdue Summer School titled NEGF Simulation of Graphene Nanodevices by Prof. Supriyo Datta. 1

2 2009 Summer School: Electronics from the Bottom Up Introduction Graphene is essentially one-atom thick sheet of carbon atoms arranged in a hexagonal structure. Each carbon atom is thus sp 2 -hybridized leaving a free p orbital (conventionally p z orbital) with a single electron which then form delocalized π bonds. This can give rise to high conductivity in a graphene sheet. There are two associated carbon nano-structures which are also of great interest as material for future devices: Graphene Nano-ribbon, which is essentially a thin strip of graphene (which exhibits quantization of bands due to finite width, the box boundary condition) and Carbon Nano-tube which is essentially a nano-ribbon rolled up along its length (and hence has a periodic boundary condition) Deping upon how the nano-ribbon is cut or nano-tube grown, the structure of the device can vary, thereby giving different electrical properties. The structure of the device is often specified in terms of chirality of the device. For a visual definition of the chirality see the figure 1 attached at the of this handout. In this experiment we simulate 2 special chiralities: Armchair and Zigzag. However, note that the nano-tube configuration is as per the chirality of the transverse direction (i.e. transverse to the electron transport) and in nano-ribbon it is as per the longitudinal direction. See figure 2 for a better understanding. Description of the Hamiltonian For a better understanding of the following description, please refer to figure 3. We define a 4-atom unit cell (outlined by blue) which we will use to construct our Hamiltonian. The tight binding energy t is approximated to 3eV (the strength of the π bond). Using this information and the geometry of the unit cell we construct the matrix: 0 t 0 0 t 0 t 0 u 0 t 0 t 0 0 t 0 for each individual unit cell. We further see that the connection between the unit cells is defined by the matrices: t t t Therefore, deping upon the structure of the device, we can decide which of the connection are longitudinal and which ones are the transverse l w 2

3 2009 Summer School: Electronics from the Bottom Up Now to convert the problem into a 1-D problem, we can define another one single column of each individual unit cell as one single super -unit cell using the following matrix: u w 0 w' u. 0.. And the connection between these two super -unit cells is given by: l l. 0.. The complete Hamiltonian may, then, be written as: H 0 0 ' 0 0 ' Band structure Using the same tight binding method to create a 1-D array of unit cells, we may then proceed to calculate the band-structure of the device by using: ika ika h k e e Which can be derived from time-indepent formulation of Schrodinger s Equation (see chapter 5, Quantum Transport: Atom to Transistor, Supriyo Datta) The sub-bands are simply given by the Eigen values of hk of the previous expression. 3

4 2009 Summer School: Electronics from the Bottom Up MATLAB Code The provided MATLAB code calculates the transmission and sub-bands of a GNR and CNT device with Armchair and Zigzag edges. The following is a description of the MATLAB code: The code has been divided into 4 different sections: Device Setup This section defines some fundamental constants, the tight binding parameter t and accepts some inputs from the user regarding the type of device, structure and the width of the device (in terms of unit cells of 4-atoms). Since we are working in ballistic limit (we have considered only phase breaking processes) the transport through the device is indepent of the length, therefore to reduce computational time, we have taken the length of the device as 1. NEGF Setup This section is used to setup the Hamiltonian. This section uses some MATLAB commands to construct the matrices, described in the section Description of the Hamiltonian, in a concise way. To understand how they have been constructed, it would be useful to keep the MATLAB code and the above mentioned section side-by-side for comparison. Short descriptions of the commands are as below: diag: used to either extract a diagonal matrix or to construct a diagonal matrix. Diag(a,b) uses the array a as the diagonal elements, and b as the offset from the primary diagonal, +1 means one above the primary diagonal and -1 means one below the primary diagonal. This form can be used to quickly place elements in diagonal and off diagonal elements. ones and zeros: these are used to construct matrices fully composed of 1 and 0. eye: used to construct an identity matrix. kron: used for Kronecker tensor product. kron(x,y) constructs a big matrix composed of all the possible element-by element products of the matrices x and y. This can be used to construct the Hamiltonian in a concise way, when it is desired to put complete matrices in certain positions in the resulting matrix, which is not possible with the diag command. linspace: used to construct a vector with linearly spaced elements. linspace(p,q,r) creates an array whose elements range from p to q with r equally spaced elements. Use the command help or doc followed by command name to get more help on the commands 4

5 2009 Summer School: Electronics from the Bottom Up In line 81, we have implemented the periodic boundary condition for the CNT device by connecting the 1 st and the n th element of the super -unit cell by adding the two additional and elements. NEGF Calculation In this section we calculate the transmission and the band structure of the device. The transmission is calculated for energies -1 ev to 1 ev. Increasing the number of energy points make the calculation more time consuming. We then proceed to calculate the self energy matrices 1 and 2 using the iterative method as outlined in chapter 8, Quantum Transport: Atom to Transistor, Supriyo Datta. Since 1 is for the left contact, as per the structure of the Hamiltonian 1 g 1, because the left lead is connected by and right by ( g 1 is the surface Green s function for left contact, denoted as 1). Similarly, for contact 2 or the right contact 2 g2. See figure 3. Since acts as connection to the contacts, they have to be placed at proper locations along the longitudinal direction, which are done using the lines 119 and 120 (at the left most and right most points). We then calculate the Green s function of the device self consistently, including the electronelectron interaction, which is described by the phase-breaking potential D. See the lecture for the details. We then calculate the transmission of the device at the given energy. This process is repeated for all the energy points, to give us transmission through the device at energy range selected previously. We also calculate the sub bands using the formulation described previously in the section Band structure. Plot figures This section is used to plot the transmission and sub bands. A point to note: Increasing the width of the devices increases the computation time quite significantly, so for the purpose of the lab, use the values of 7, 8 and 9 which should run in a reasonable amount of time. 5

6 2009 Summer School: Electronics from the Bottom Up Further exploration Explain the steps observed in the Energy v Transmission plots in terms of sub-bands at the given energy. Explain which bands are degenerate for the GNR and CNT. If needed, change the energy range of the simulation. The code given to you is limited in the sense of chiralities it can simulate. Think about how you extract the unit cell for any given chirality and calculate sub-bands and transmission through it. The unit cell we have defined cannot simulate a monolayer Graphene nano-ribbon. Think about how you can define the unit cell to simulate a monolayer GNR. A part of the code which can calculate the Transmission for the coherent transport, compare the plots of coherent and incoherent transports and note the effect if increasing the value of D, the phase breaking potential. (defined in line 40) Another part of the code that is commented out calculates the Correlation function. Create a surface plot the diagonal elements of the correlation function with the energy. (diagonal elements of the correlation function corresponds to the electron density at that site) 6

7 Fig. 1: Chirality in a Graphene sheet a 2 a 1 Zigzag n,0 na ma n, m 1 2 Armchair nn, Image idea courtesy User:Kebes from Wikipedia

8 Fig2: Configurations of a GNR and CNT Armchair GNR Zigzag GNR Armchair CNT Zigzag CNT

9 Fig. 3: Definition of Unit cell and construction of the Hamiltonian u u 2 u 1 u t t u t t 0 0 t 0 t 0 u w 0 '. 0 t 0 t w u t 0 l l. 0.. In GNR Armchair case: 1 l, 2 win Zigzag case: 2 l, 1 w In CNT Armchair and Zigzag configurations are reversed. H 0 0 ' 0 0 '

10 %************************************************************************ % MATLAB code to generate Transmission and Sub-bands plots % for GNR and CNT devices % Developed by Samiran Ganguly % For NCN Summer School Purdue %************************************************************************ %% Device Setup clear all; clc; %define constants q = 1.6e-19; hbar = 1.06e-34;h = 2*pi*hbar;m = 9.1e-31;zplus=i*1e-3; %t has been given as a parameter t = -3; f1 =1; f2 = 0; fprintf(1,'enter the type of device you want to simulate GNR/CNT\n') fprintf(1,'1 for GNR\n'); fprintf(1,'2 for CNT\n'); device = input('valid inputs are 1 or 2 : '); if (~((device == 1) (device == 2))) device = 1; fprintf(1,'enter the configuration of device you want to simulate Armchair/Zigzag\n') fprintf(1,'1 for Armchair\n'); fprintf(1,'2 for Zigzag\n'); type = input('valid inputs are 1 or 2 : '); if (~((type == 1) (type == 2))) type = 1; %Points in GNR/CNT width NW = input('enter number of Unit Cells in the width : '); if isempty(nw) NW = 12; %Points in GNR/CNT length NL = 1; %Points in GNR/CNT unit cell NU = 4; %Define any scatterer as a delta potential barrier %No scatterer %Define phase breaking potential D = 1e-12; %Convergence criteria errmax = 1e-3; %Total H will be of LxWxU size NT = NL*NW*NU; fprintf(1,'simulating...\n'); if (device == 1) fprintf(1,'device: Graphene Nano-Ribbon\n'); else fprintf(1,'device: Carbon Nanotube\n'); if (type == 1) fprintf(1,'type: Armchair\n'); else fprintf(1,'type: Zigzag\n'); %% NEGF Setup

11 %Define Hamiltonians and alpha, beta matrices fprintf(1,'hamiltonian Setup...please wait\n'); alphau = t*diag(ones(1,nu-1),1)+t*diag(ones(1,nu-1),-1); betaw = zeros(nu); betal = zeros(nu); %Armchair Structure for GNR if ((type == 1)&&(device == 1)) betaw(2,1) = t; betaw(3,4) = t; betal(4,1) = t; elseif ((type == 2)&&(device == 1)) %Zigzag Structure for GNR betaw(4,1) = t; betal(2,1) = t; betal(3,4) = t; elseif ((type == 1)&&(device == 2)) %Armchair Structure for CNT betaw(4,1) = t; betal(2,1) = t; betal(3,4) = t; else %Zigzag Structure for CNT betaw(2,1) = t; betaw(3,4) = t; betal(4,1) = t; alpha = kron(diag(ones(1,nw)),alphau)+kron(diag(ones(1,nw- 1),1),betaw)+kron(diag(ones(1,NW-1),-1),betaw'); if (device == 2) alpha = alpha+kron(diag(ones(1,1),nw-1),betaw')+kron(diag(ones(1,1),1- NW),betaw); beta = kron(diag(ones(1,nw)),betal); H = kron(diag(ones(1,nl)),alpha)+kron(diag(ones(1,nl- 1),1),beta)+kron(diag(ones(1,NL-1),-1),beta'); %Define Energy grid for calculation of Transmission E = linspace(-1,1,51); %Define the matrices for NEGF T = zeros(1,length(e)); g1 = inv(e(1)*eye(nw*nu)-alpha); g2 = inv(e(1)*eye(nw*nu)-alpha); Es = zeros(nt); Esin = zeros(nt); fprintf(1,'done\n'); %% NEGF Calculations fprintf(1,'running NEGF...please wait\n') fprintf(1,'energy range = %d ev to %d ev\n',min(e),max(e)); fprintf(1,'%s\n',['[',blanks(length(e)),']']); fprintf(1,' '); %Run for each energy for k = 1:length(E) %Calculate surface GFs self consistently %Calculate for beta' err = 100; while(err>errmax) g1new = inv((e(k)+zplus)*eye(nw*nu)-alpha-beta'*g1*beta); err = (sum(sum(abs(g1new-g1))))/(sum(sum(abs(g1new+g1)))); g1 = g1new; sigma1 = beta'*g1*beta; %Calculate for beta

12 err = 100; while(err>errmax) g2new = inv((e(k)+zplus)*eye(nw*nu)-alpha-beta*g2*beta'); err = (sum(sum(abs(g2new-g2))))/(sum(sum(abs(g2new+g2)))); g2 = g2new; sigma2 = beta*g2*beta'; %Calculate self energy matrices E1 = kron(diag([1 zeros(1,nl-1)]),sigma1); E2 = kron(diag([zeros(1,nl-1) 1]),sigma2); %Calculate broadening G1 = 1i*(E1-E1'); G2 = 1i*(E2-E2'); %Calculate G, Gn, A, T for coherent transport % G = inv((e(k)+1i*etaplus)*eye(nt)-h-e1-e2); % T(k) = real(trace(g1*g*g2*g')); % Gn = f1*(g*g1*g')+f2*(g*g2*g'); % A = i*(g-g'); %Calculate G, Gn, A, T self consistently including the phase breaking processes err = 100; while(err>errmax) G = inv((e(k)+zplus)*eye(nt)-h-e1-e2-es); Esnew = D*G; err = sum(sum(abs(esnew-es)))/sum(sum(abs(esnew+es))); Es = Esnew; % err = 100; % while(err>errmax) % Gn = f1*(g*g1*g')+f2*(g*g2*g')+(g*esin*g'); % Esinnew = D*Gn; % err = sum(sum(abs(esinnew-esin)))/sum(sum(abs(esinnew+esin))); % Esin = Esinnew; % % A = i*(g-g'); T(k) = real(trace(g1*g*g2*g')); fprintf(1,' '); fprintf(1,'\n'); fprintf(1,'done\n'); %Calculate Subbands ka = linspace(-pi,pi,501); eige = zeros(length(ka),nw*nu); fprintf(1,'calculating Sub-bands...please wait\n'); for k = 1:length(ka) [V,D] = eig(alpha + beta*exp(i*ka(k))+ beta'*exp(-i*ka(k))); eige(k,:) = diag(d); fprintf(1,'done\n'); %% Plot Figures %Figure 1: Transmission and sub-band comparison figure(1); subplot(1,2,1); plot(t,e); xlabel('t(e)'); ylabel('e[ev]'); title('energy v Transmission'); xlim([0 max(t)+0.5]);

13 ylim([min(e) max(e)]); subplot(1,2,2); plot(ka/pi,eige); xlabel('ka/\pi'); ylabel('e[ev]'); title('sub-bands'); ylim([min(e) max(e)]); %Figure 2: Sub-bands figure(2); plot(ka/pi,eige,'b'); xlabel('ka/\pi'); ylabel('e[ev]'); title('sub-bands');