H1[kx_,ky_]:=t1*{{0, Exp[I*kx*b], 0, Exp[I*ky*b], 0, 0},
                 {Exp[-I*kx*b], 0, Exp[I*ky*b], 0, 0, 0},
                 {0, Exp[-I*ky*b], 0, Exp[-I*kx*b], 0, 0},
                 {Exp[-I*ky*b], 0, Exp[I*kx*b], 0, 0, 0},
                 {0, 0, 0, 0, 0, 0},
                 {0, 0, 0, 0, 0, 0}}

H2[kx_, ky_] := t2*{{0, 0, 0, 0, Exp[I*(-kx*Sqrt[3]/2 + ky/2)*b], Exp[I*(kx/2 - ky*Sqrt[3]/2)*b]},
                    {0, 0, 0, 0, Exp[I*(kx*Sqrt[3]/2 + ky/2)*b], Exp[I*(-kx/2 - ky*Sqrt[3]/2)*b]},
                    {0, 0, 0, 0, Exp[I*(kx*Sqrt[3]/2 - ky/2)*b], Exp[I*(-kx/2 + ky*Sqrt[3]/2)*b]},
                    {0, 0, 0, 0, Exp[I*(-kx*Sqrt[3]/2 - ky/2)*b], Exp[I*(kx/2 + ky*Sqrt[3]/2)*b]}, {Exp[-I*(-kx*Sqrt[3]/2 + ky/2)*b], Exp[-I*(kx*Sqrt[3]/2 + ky/2)*b], Exp[-I*(kx*Sqrt[3]/2 - ky/2)*b], Exp[-I*(-kx*Sqrt[3]/2 - ky/2)*b], 0, 0},
                    {Exp[-I*(kx/2 - ky*Sqrt[3]/2)*b], Exp[-I*(-kx/2 - ky*Sqrt[3]/2)*b], Exp[-I*(-kx/2 + ky*Sqrt[3]/2)*b], Exp[-I*(kx/2 + ky*Sqrt[3]/2)*b], 0, 0}}

H[kx_, ky_] := H1[kx,ky] + H2[kx,ky]

ev[kx_, ky_] := Eigenvalues[H[kx, ky]]

Plot3D[{ev[kx, ky][[4]]}, {kx, -2*Pi, 2*Pi}, {ky, -2*Pi, 2*Pi}]

ev[kx, ky][[4]] // Re

ClearAll[vals, H1, H2];
vals = {t1 -> -1, t2 -> -1, b -> 1/(1 + Sqrt[3])};
H1[kx_, ky_] :=
  t1*{{0, Exp[I*kx*b], 0, Exp[I*ky*b], 0, 0}, {Exp[-I*kx*b], 0,
      Exp[I*ky*b], 0, 0, 0}, {0, Exp[-I*ky*b], 0, Exp[-I*kx*b], 0,
      0}, {Exp[-I*ky*b], 0, Exp[I*kx*b], 0, 0, 0}, {0, 0, 0, 0, 0,
      0}, {0, 0, 0, 0, 0, 0}} /. vals;

H2[kx_, ky_] :=
  t2*{{0, 0, 0, 0, Exp[I*(-kx*Sqrt[3]/2 + ky/2)*b],
      Exp[I*(kx/2 - ky*Sqrt[3]/2)*b]}, {0, 0, 0, 0,
      Exp[I*(kx*Sqrt[3]/2 + ky/2)*b],
      Exp[I*(-kx/2 - ky*Sqrt[3]/2)*b]}, {0, 0, 0, 0,
      Exp[I*(kx*Sqrt[3]/2 - ky/2)*b],
      Exp[I*(-kx/2 + ky*Sqrt[3]/2)*b]}, {0, 0, 0, 0,
      Exp[I*(-kx*Sqrt[3]/2 - ky/2)*b],
      Exp[I*(kx/2 + ky*Sqrt[3]/2)*b]}, {Exp[-I*(-kx*Sqrt[3]/2 + ky/2)*
        b], Exp[-I*(kx*Sqrt[3]/2 + ky/2)*b],
      Exp[-I*(kx*Sqrt[3]/2 - ky/2)*b],
      Exp[-I*(-kx*Sqrt[3]/2 - ky/2)*b], 0,
      0}, {Exp[-I*(kx/2 - ky*Sqrt[3]/2)*b],
      Exp[-I*(-kx/2 - ky*Sqrt[3]/2)*b],
      Exp[-I*(-kx/2 + ky*Sqrt[3]/2)*b],
      Exp[-I*(kx/2 + ky*Sqrt[3]/2)*b], 0, 0}} /. vals;

H[kx_, ky_] := H1[kx, ky] + H2[kx, ky];
ev[kx_, ky_] := Eigenvalues[H[kx, ky]];

Plot3D[ev[kx, ky], {kx, -2*Pi, 2*Pi}, {ky, -2*Pi, 2*Pi},
 BoxRatios -> {1, 1, 2}]

a = Table[
   Sort@ev[kx, ky], {kx, -2*Pi, 2*Pi, Pi/10.}, {ky, -2*Pi, 2*Pi,
    Pi/10.}];

Table[ListPlot3D[a[[All, All, n]]], {n, 6}]