# Fill the region above two curves

I have need to fill the region above two curves as shown below. How can I fill only the darker region without filling the two intersections as well?

Clear[t, a0, \[Alpha], \[Delta], \[CapitalTheta], lmin, lmin2, ebar, \
nmin, nmin2]
a0 = 1.059*10^-9;
\[Alpha] = 10;
\[Delta] = 1/100;
\[CapitalTheta] = 435;
ebar[t_] := ((t/\[CapitalTheta])^2)*
NIntegrate[x/(Exp[x] - 1), {x, 0, \[CapitalTheta]/t}];
nmin[t_] := (\[CapitalTheta]*\[Alpha])/(4*t*
ebar[t])*((4*ebar[t]/\[Alpha]) + 1)^2;
nmin2[t_] := (2 \[Alpha]/\[Delta])*(\[CapitalTheta]/t)*ebar[t];
lmin[t_] := a0*nmin[t];
lmin2[t_] := a0*nmin2[t];
p1 = LogLogPlot[{lmin2[t], lmin[t]}, {t, 10^-1, 10^6},
PlotRange -> All, PlotTheme -> "Monochrome", PlotStyle -> {Black},
Frame -> True, FrameTicksStyle -> Directive[16, "TR"],
FrameLabel -> {{Style[Row[{Subscript["l", "min"], " (m)"}], 18,
"TR"], None}, {Style[Row[{"Temperature", " (K)"}], 18, "TR"],
None}}, ImageSize -> 500,
FrameTicks -> {{ Table[{10^k, Superscript[10, k]}, {k, -9, 1, 2}],
None}, {Table[{10^k, Superscript[10, k]}, {k, -1, 6, 1}], None}},
Filling -> Top]


Clear[t, a0, \[Alpha], \[Delta], \[CapitalTheta], lmin, lmin2, ebar, \
nmin, nmin2]
a0 = 1.059*10^-9;
\[Alpha] = 10;
\[Delta] = 1/100;
\[CapitalTheta] = 435;
ebar[t_] := ((t/\[CapitalTheta])^2)*
NIntegrate[x/(Exp[x] - 1), {x, 0, \[CapitalTheta]/t}];
nmin[t_] := (\[CapitalTheta]*\[Alpha])/(4*t*
ebar[t])*((4*ebar[t]/\[Alpha]) + 1)^2;
nmin2[t_] := (2 \[Alpha]/\[Delta])*(\[CapitalTheta]/t)*ebar[t];
lmin[t_] := a0*nmin[t];
lmin2[t_] := a0*nmin2[t];
p1 = LogLogPlot[{lmin2[t], lmin[t], Max[lmin2[t], lmin[t]]}, {t,
10^-1, 10^6}, PlotRange -> All, PlotTheme -> "Monochrome",
PlotStyle -> {Black}, Frame -> True,
FrameTicksStyle -> Directive[16, "TR"],
FrameLabel -> {{Style[Row[{Subscript["l", "min"], " (m)"}], 18,
"TR"], None}, {Style[Row[{"Temperature", " (K)"}], 18, "TR"],
None}}, ImageSize -> 500,
FrameTicks -> {{Table[{10^k, Superscript[10, k]}, {k, -9, 1, 2}],
None}, {Table[{10^k, Superscript[10, k]}, {k, -1, 6, 1}], None}},
Filling -> 3 -> Top]