Ballistic Transmission of the Relativistic Quasielectrons …
161
ψ I =
1
√
2
⎛
⎝
cos φe
iϕ
1
sin φe
−iϕ
⎞
⎠ e
ik x x e
ik y y
+
r
√
2
⎛
⎝
− cos φe
iϕ
1
− sin φe
iϕ
⎞
⎠ e
−ik x x e
ik y y
ψ I I =
a
√
2
⎛
⎝
cos φe
iθ
−1
sin φe
−iθ
⎞
⎠ e
iq x x e
ik y y
−
b
√
2
⎛
⎝
cos φe
−iθ
1
sin φe
iθ
⎞
⎠ e
−iq x x e
ik y y
ψ I I I =
t
√
2
⎛
⎝
cos ϕe
−iφ
1
sin ϕe
−iφ
⎞
⎠ e
ik x x e
ik y y
,
(4)
where φ and θ are the angle of incidence and the refraction angle, respectively; the
quasimomentums in the out-of-barrier region k x and in the barrier region q x are equal
to
k x =
E 2 − k 2
y ; q x =
(U − E)
2
β
− k 2
y ;
(5)
tgθ =
q y
q x
; q y = k y ; the units with v F1 = 1; h = 1 are adopted.
Using the appropriate matching conditions [21, 22]
√
v F1 [cos ϕψ I (−ε) + sin ϕψ I I I (−ε)] =
√ v F2 [cos ϕψ I (ε) + sin ϕψ I I I (ε)], ε → 0,
(6)
we can deduce the expression for the transmission coefficient T
T = 16 cos
2
θ cos
2
φ
f
2
p + f
2
m − 2 f p f m cos(2q x D)
f p = 2 − 2 cos(θ + φ) − sin
2
(2ϕ)(sin θ + sin φ)
2
;
f m = 2 + 2 cos(θ − φ) − sin
2
(2ϕ)(sin θ + sin φ)
2
.
(7)
3 Results and Discussion
Figures 1, 2, and 3 show the dependence of the transmission coefficient T on the
angle of incidence of the quasielectrons on the structure considered for such values
of parameters:
for Fig. 1: E = 2; U = 3; β = 0.5;
for Fig. 2: E = 3; U = 4; β = 1;
for Fig. 3: E = 3; U = 4; β = 5; other parameters as shown in Fig. 1.