Well you have given us quite a puzzler; little background info and no context.
I'm assuming you are looking for the value Ee since you didn't give us a value for it. But are you sure your equation is correct? I couldn't find a (real) solution, only complex, so I guessed you made a sign error.
If the equation really is K*cot(K*R)= +k (note sign change), then we have a solution as shown below.
Vc = 0.5;
c = 3e8;
m0 = 0.511e6/c^2; me1 = 0.067*m0; me2=0.039*m0;
hr = 6.58e-16;
k= @(Ee) sqrt(2*me1*Ee/hr^2);
K= @(Ee) sqrt(2*me2*(Vc-Ee)/hr^2);
f = @(Ee) K(Ee).*cot(K(Ee)*R)-k(Ee);
Ee0 = 0.01;
Ee = fzero(@(Ee) f(Ee), Ee0)
Ee = linspace(0,0.02)';
plot(Ee, f(Ee)); yline(0);