# Young's modulus graphene nanoribbons

• A

## Main Question or Discussion Point

Young’s modulus is given as the second derivative of the total energy with respect to the strain divided
by the equilibrium volume.
Can help me for the calculate equilibrium volume and young's modulus?

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Do you have the expression for energy as a function of strain figured out already?

Matlab code for calculate band structure graphene nanoribbons under strain is below:

clear;
clc;
close all;

NU=12; % Number of atoms in unit cell
Nbnd=4*NU; % number of bands
q=0.03;
w=0.41;
aa=2.232*(1+q);
a=3.866*(1+q);

X(1)=1.9330*(1+q);
Y(1)=0;

xswitch = 0;

for ixy=2:NU
if mod(ixy,2)==1
Y(ixy)=Y(ixy-1)+aa;
else
Y(ixy)=Y(ixy-1)+aa*sind(30);

X(ixy) = xswitch;

if (ixy+1)<=NU
X(ixy+1)=xswitch;
end

if xswitch == 0
xswitch =aa*cosd(30);
else
xswitch = 0;
end
end
end

for iz=1:NU
if mod(iz,2)==1
Z(iz)=0.46152;
else
Z(iz)=0;
end

end

sho=0;
for is=[0,-1,1]
for ks=1:NU
sho=sho+1;
XT(sho)=X(ks)+is*a;
YT(sho)=Y(ks);
ZT(sho)=Z(ks);
Ax(sho)=is*a;
No(sho)=ks;
end
end
figure(1)
plot(XT,YT,'*')
Ax=Ax/a;

for ik=1:101
K(ik)=(-pi+(ik-1)*((2*pi)/100))*(1-q);

H=H0(Nbnd);

for is=1:NU
for js=1:sho
dis=sqrt(((XT(is)-XT(js))^2)+((YT(is)-YT(js))^2));
if abs(dis-2.232)<0.1 & abs(No(is)-No(js))>0
l=(XT(is)-XT(js))/dis;
m=(YT(is)-YT(js))/dis;
n=(ZT(is)-ZT(js))/dis;
h1=hamiltonian1(l,m,n);
h2=hamiltonian2(l,m,n);

H((No(is)-1)*4+1:No(is)*4,(No(js)-1)*4+1:No(js)*4)=H((No(is)-1)*4+1:No(is)*4,(No(js)-1)*4+1:No(js)*4)+h1*exp(i*K(ik)*Ax(js))+h2*exp(i*K(ik)*Ax(js));

end
end

end

E(ik,1:Nbnd)=sort(real(eig(H)));

pl(ik)=(ik-1)/100;

end

figure(2)
plot(E)

[V,D]=eig(H);
f=diag(D);
g=f<0;
g1=f(g);
r=sum(g1)

So is this yes or no? jim mcnamara
Mentor
A suggestion: learn about formatting code on PF. What you posted is HARD to read, IMO. And hopefully the real code does not look like what you posted.