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Resonance condition for asymmetrical triple-barrier structures
Superlattices and Microstructures, Vol. 23, No. 6, 1998 Article No. sm960163
Resonance condition for asymmetrical triple-barrier structures X. D. Z HAO , H. YAMAMOTO , Y. N AKANO , K. TANIGUCHI Department of Information Science, Fukui University, 3-9-1 Bunkyo, Fukui 910, Japan
(Received 9 February 1996) The resonance condition for triple-barrier structures with arbitrary potential is studied systematically. The quasisymmetrical triple-barrier (QST) resonance mode and the quasiasymmetrical double-barrier (QAD) resonance mode may both exist in asymmetric triplebarrier structures. The QST consists of two submodes: a normal mode (doublet) and a degenerate mode (singlet). The critical condition for distinguishing the two modes is examined. It is confirmed that there are both unity resonant transmission and below-unit resonant transmission in the asymmetrical triple-barrier structure. Furthermore, the wavefunctions of an electron at resonance level are calculated and the confining phenomenon is studied. c 1998 Academic Press Limited
Key words: resonance condition, asymmetrical triple-barrier structures, resonance modes.
1. Introduction Since the pioneering work by Esaki et al. [1, 2], the resonant tunnelling in semiconductor heterostructures has been studied extensively [3–20]. The triple-barrier structure is the simplest multiple-barrier structure which allows us to understand the interaction between two wells. Experiments about resonant tunnelling in the triple-barrier and the double-well structures have been reported [6, 7], and some interesting features have been found: the current peak-to-valley ratio was enhanced relative to the double-barrier diode by using these structures. Theoretical studies have been also carried out on the triple-barrier structures with different potential profiles: rectangular [8–11], trapezoidal [12], slowly changed [13, 14], and δ-doping potential profiles [15]. Usually, the Airy function method is used for the triple-barrier structure with a trapezoidal potential profile [12] and the WKB approximation is used for the triple-barrier structure where the potential changes slowly (i.e. the change of potential energy within a de Broglie wavelength is small compared with the kinetic energy) [13, 14]. It is noteworthy that the analytical expressions for the transmission coefficient and the resonance condition in the previous theories [8–15] are modified as a result of the different potential profiles of triple-barrier structures. In these theoretical studies, eight different expressions for the transmission coefficient in different triple-barrier structures were given. The resonance condition in the triple-barrier structure with trapezoidal potential profile was not obtained [12]. In other words, the common points for the triple-barrier structures with different potential profiles have not been thoroughly considered: analytical expressions for the transmission coefficient and the resonance condition for transmission through triple-barrier structures were obtained only for some special cases [8–15]. Therefore, it is necessary to study systematically the transmission coefficient and the resonance condition for the triple-barrier structure with an arbitrary potential. In this paper, ‘the generalized single-barrier parameters method’ [16, 17] is used to study in detail the transmission coefficient, the resonance condition and the resonant transmission modes for asymmetrical 0749–6036/98/061273 + 11
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c 1998 Academic Press Limited
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A
V 1
Region 2
4
3
5
6
7
Vmax3 Vmax2 Vmax1 E Vw3 Vw2 Vw1 B
X
Lwf2 X1
Lwf1 X2 X3
V 1 Region 2
X4 4
3
Vmax3 Vmax2 Vmax1
X5
X6 5
E X
Vw3 Vw2 Vw1 line 1
line 2
Fig. 1. Potential energy diagram of the triple-barrier structure with arbitrary potential.
triple-barrier structures with arbitrary potential profiles. The confining phenomena at QST and QAD modes are also investigated for two typical triple-barrier structures. The analytical expressions for transmission coefficients and resonance conditions obtained here do not rely upon barrier potential profiles and many kinds of single-barrier structures with complicated potential profiles may be used simultaneously to investigate the tunnelling transmission characteristics in the triple-barrier structure. The generalized single-barrier parameters have been calculated numerically for arbitrary barrier structures [16, 17] and analytically for rectangular and trapezoidal structures as well as parallel-plane ones [18], and, in this paper those for δ-doping barrier structures are also included.
2. Transmission coefficient in asymmetrical triple-barrier structures In this section we shall derive first an analytical expression for the transmission coefficient in an asymmetrical triple-barrier structure with arbitrary potential. The schematic energy diagram of the asymmetrical triple-barrier structure, formed by three generalized single barriers (regions 2, 4, 6) and two flat well regions (regions 3, 5), is shown in Fig. 1A, where Vwj is the potential height, m ∗wj the electron effective mass, and L w f j the width for the jth flat well region ( j = 1, 2). If L w f 1 and L w f 2 are reduced to zero, the schematic energy diagram shown in Fig. 1A will become that shown in Fig. 1B, i.e. regions 3 and 5 become line 1 and line 2, respectively. Here, line 1 and line 2 are called the dividing barrier lines (DBL), by which we can distinguish the left and right generalized single-barrier from the middle one. With the aid of the transfer matrix N j (the transfer matrix for the jth generalized single-barrier) whose elements are functions of generalized single-barrier parameters [17], the transmission coefficient for an
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asymmetrical triple-barrier structure is expressed as T3g = |R + R1 exp(i82,3 ) + R2 exp[i(81,2 + 82,3 )] + R3 exp(i81,2 )|−2 = [R 2 + R12 + R22 + R32 + 2R1 R3 cos(81,2 − 82,3 ) + 2R R2 cos(81,2 + 82,3 ) +2(R R3 + R1 R2 ) cos(81,2 ) + 2(R R1 + R2 R3 ) cos(82,3 )]−1 ,
(1)
where we have 81,2 = θ1,R + θ2,L + π + 2κw1 L w f 1 , 82,3 = θ2,R + θ3,L + π + 2κw2 L w f 2 ,
(2a) (2b)
R = (T1 T2 T3 )− 2 , 1
R1 = R2 = R3 =
−1 T1 2 [(T2−1 −1 T2 2 [(T1−1 −1 T3 2 [(T1−1
(3a)
− 1)(T3−1 − 1)] ,
(3b)
− 1)(T3−1 − 1)] ,
(3c)
− 1)(T2−1 − 1)] ,
(3d)
1 2 1 2 1 2
and κwj = [2m ∗wj (E + Vwj )] 2 /h¯ 1
( j = 1, 2 and h¯ (= h/2π ) is the reduced Planck constant).
(4)
Here, we call 8 j, j+1 the characteristic phase difference in the well region between the jth barrier and the ( j + 1)th barrier ( j = 1, 2), T j is the transmission coefficient of the jth generalized single-barrier structure ( j = 1–3), θ j,L and θ j,R are the reflection phase difference at the left and right edge of that, respectively. T j , θ j,L and θ j,R are called the generalized single-barrier parameters for the jth generalized single-barrier structure [17]. The analytical expressions of the generalized single-barrier parameters have been obtained for the rectangular and the trapezoidal as well as the parallel-plane potential structures [17, 18], which are shown in Figs 2A–C. Recently, the effects of a δ-doped impurity sheet have been considered [15, 21]. Here, we will give the generalized single-barrier parameters for the δ-doped barrier shown in Fig. 2D. Assuming the potential of the δ-doped barrier has the form δ, where is δ-potential strength and δ is δ-function, we have !−1 2 , (5a) Tj = 1 + 0 0 κ j κ j+1 h¯ 4 " # # " 0 (κ j+1 − κ 0j )h¯ 2 2|| − π/2, (5b) − arctan θ j,L = arctan − 0 2|| (κ j + κ 0j+1 )h¯ 2 and
"
θ j,R
2|| = arctan − 0 (κ j + κ 0j+1 )h¯ 2
#
" + arctan
(κ 0j+1 − κ 0j )h¯ 2 2||
# − π/2,
(5c)
1
where κ j = [2m ∗j (E + V j )] 2 /h¯ , κ 0j = κ j /m ∗j , m ∗j is the electron effective mass of the jth region in Fig. 2D, and V j is the potential height of that.
3. Resonance condition for asymmetrical triple-barrier structures In an asymmetrical double-barrier structure, b1 w1 b2 , only a single well w1 exists. On the other hand, in an asymmetrical triple-barrier structure, b1 w1 b2 w2 b3 , double wells, w1 and w2 , exist, which are coupled
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A
B Region j
j+1
m*j
m*j + 1
Vj
Vj + 1 C
D
Fig. 2. Four typical single-barrier structures.
QST mode b1 w1 b2
w2 A
Left barrier b1
b1 w1
b3 Energy level
Right barrier b(2 + 3)
b2 B1
w2
Left barrier b(1 + 2)
b3
b1 w1 QAD mode
Right barrier b3
b2
w2
b3
B2
Fig. 3. A, Illustration of QST mode; B, Illustration of QAD mode.
through a central barrier b2 as seen in Fig. 3. We may understand that there are two kinds of resonance modes in the asymmetrical triple-barrier structure. One is the quasisymmetrical triple-barrier resonance mode (QST mode), at which the electron energy coincides with an eigenenergy in both wells, and the other is the quasiasymmetrical double-barrier resonance mode (QAD mode), at which the electron energy coincides with an eigenenergy in either well. Usually, only the QAD mode exists in the asymmetrical triple-barrier structure. In this section, we will investigate the resonance conditions for the above two modes.
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3.1. Resonance condition for QST mode Here, let us investigate the resonance condition for the QST mode in asymmetrical triple-barrier structures. It is seen that eqn (1) may be simplified to T3s = [(R − R2 )2 + (R1 + R3 )2 + 4R R2 cos2 8 + 2(R + R2 )(R1 + R3 ) cos 8]−1 ,
(6)
for 81,2 = 8 + 2n 1 π,
(n 1 : 0, 1, 2, . . .),
(7a)
82,3 = 8 + 2n 2 π,
(n 2 : 0, 1, 2, . . .),
(7b)
and i.e. for 81,2 = 82,3 ± 2nπ,
(n = 0, 1, 2, . . .).
(8)
Here, eqn (8) is called the QST mode condition, since this condition is always satisfied in symmetrical triple-barrier structures and may be satisfied in an asymmetrical triple-barrier structure for a given value of energy. First, let us assume that T1 , T2 , and T3 are constants or almost constants near a resonant peak. This assumption may be satisfied since T3s varies with energy E abruptly near each resonant peak and T1 , T2 , and d 2 T3s 3s T3 varies with energy E smoothly. By the local maximum conditions, dT d8 = 0 and d82 < 0, we obtain the following resonance conditions q 2 1 − C3s , (C3s < 1), (9a) 8 = π ± arctan C3s (9b) 8 = π, (C3s = 1), and 8 = π,
(C3s > 1),
(9c)
where
√ p p 1 1 − T2 1 + 1 − T1 + √ + 1 − T3 . (10) C3s ≡ √ 4 1 − T1 1 − T3 It is seen that (9a) corresponds to two resonant peaks (normal code), (9c) one resonant peak (degenerate mode), and (9b) is the critical relation between (9a) and (9c). When the critical relation holds, the resonant transmission of the doublet will reduce to that of the singlet. Here, we call (9a) the normal mode condition, (9b) the critical condition, (9c) the degenerate mode condition, and call these three conditions the phase difference conditions for the resonance in the QST mode (PDCRQST). The resonant tunnelling will occur in an asymmetrical triple-barrier structure when one of the above three conditions is satisfied. From the relation C3s = 1, the critical transmission coefficient of the middle barrier in the asymmetrical triple-barrier structure is given by 2 p p 1 1 + 1 − T1 + √ + 1 − T3 . (11) T2−critical = 1 − 16 √ 1 − T1 1 − T3 It is understood that there is only a degenerate mode in an asymmetrical triple-barrier structure when T2 < T2−critical , and there is only the normal mode when T2 > T2−critical . Next, if we assume further that the condition T1 = T3 = T
(12)
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is satisfied at a given value of energy E, then eqn (6) is simplified to 0 2 −1 ) ] , T3s0 = [1 + K 2 (cos 8 + C3s
where we have
s 2 K = T
and 0 C3s
1−T , T2
√ √ 1 − T2 1 + 1−T . = √ 2 1−T
(13)
(14a)
(14b)
It is evident that T3s0 = 1,
(15)
0 cos 8 + C3s = 0.
(16)
for Furthermore, (16) is reduced to the following equations q 02 1 − C3s 0 , (C3s < 1) (i.e. the normal mode condition), 8 = π ± arctan 0 C3s
(17a)
and 8 = π,
0 = 1) (i.e. the critical condition). (C3s
(17b)
Here, eqn (12) is called the maximum condition for the resonant peak value in QST mode (MCPVQST), which is always satisfied in the symmetrical triple-barrier structures. It is understood that the transmission coefficient becomes unity when both MCPVQST and the normal mode condition or both MCPVQST and the critical condition are satisfied simultaneously. Equation (13) is a simple relation for the transmission coefficient in symmetrical triple-barrier structures. 0 > 0, then we have On the other hand, if cos 8 + C3s T3s0 < 1,
(18)
and the resonance condition is 8=π
0 > 1) (C3s
(i.e. the degenerate mode condition).
(19)
It means that only below-unity resonant tunnelling occurs in any triple-barrier structure when the degenerate mode condition is satisfied. 3.2. Resonance condition in QAD mode For the case of 81,2 6 = 82,3 ± 2nπ,
(n : 0, 1, 2, . . .),
(20)
which is different from (8), we may assume that there are two different quasidouble-barrier structures in the asymmetrical triple-barrier structure. One is the structure b(1+2) w2 b3 , and the other is the structure b1 w1 b(2+3) , as shown in Fig. 3B1 and B2. Here, b(1+2) is the left equivalent-barrier which is composed of b1 w1 b2 , and b(2+3) , the right equivalent-barrier, composed of b2 w2 b3 . Equation (20) is called the QAD mode condition, which is necessary for the QAD mode to exist.
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To investigate the resonance condition in the quasidouble-barrier structures, b1 w1 b(2+3) and b(1+2) w2 b3 , we may rewrite eqn (1) as −1 − 1)] 2 exp(i81,(2+3) )|−2 , T3a1 ≡ T3g = |(T1 T(2+3) )− 2 + [(T1−1 − 1)(T(2+3)
(21a)
−1 − 1)(T3−1 − 1)] 2 exp(i8(1+2),3 )|−2 , T3a2 ≡ T3g = |(T(1+2) T3 )− 2 + [(T(1+2)
(21b)
1
1
or 1
1
where T(1+2) is the transmission coefficient for the left equivalent single-barrier b1 w1 b2 , T(2+3) the transmission coefficient for the right equivalent single-barrier b2 w2 b3 , 81,(2+3) the characteristic phase difference in the well region between the barrier b1 and the equivalent single-barrier b2 w2 b3 , and 8(1+2),3 the characteristic phase difference in the well region between b1 w1 b2 and b3 . The expressions for T(1+2) , T(2+3) , 81,(2+3) , and 8(1+2) , 3 are given in the Appendix. It is evident that the transmission coefficient in the asymmetrical triple-barrier structure becomes unity for 81,(2+3) = π + 2nπ,
(n = 0, 1, 2, . . .),
(22a)
and T1 = T(2+3) ,
(22b)
or for 8(1+2),3 = π + 2nπ,
(n = 0, 1, 2, . . .),
(23a)
and T(1+2) = T3 .
(23b)
The below-unity resonant transmission will occur if either the condition (22a) or (23a) is satisfied. Here, we call eqns (22a) and (23a) the phase difference conditions for the resonance in QAD mode (PDCRQAD), and eqns (22b) and (23b) the maximum condition for the resonant peak value in QAD mode (MCPVQAD). It is seen that the resonance conditions in QAD mode are similar to that in the double-barrier structure [16, 17].
4. Tunnelling transmission characteristics and confining phenomenon In the above sections, the transmission coefficient and resonance condition for the asymmetrical triplebarrier structure were derived and discussed. Here, let us investigate the energy variation of the transmission coefficient and the probability density of an electron wavefunction. As typical examples, let us consider the energy variation of the transmission coefficient for the asymmetrical triple-barrier structure Ga0.7 Al0.3 As/GaAs/Ga0.7 Al0.3 As/GaAs/AlAs, where the effective mass in GaAs is assumed to be 0.067 m 0 , the effective mass in Ga0.7 Al0.3 As is 0.092 m 0 [22] and that in AlAs is 0.15 m 0 [23]. It is assumed that the conduction-band offset at the heterojunction interfaces between Ga0.7 Al0.3 As and GaAs is 0.3 eV, and the conduction-band offset between GaAs and AlAs 1.0 eV [24]. First, the energy variation of the transmission coefficient is shown for L b1 = L b2 = 4 nm, L b3 = 1.368 nm, L w1 = 4.043 nm, and L w2 = 4.585 nm by the real linepin Fig. 4, where both MCPVQST (T1 = T3 ) and the 2 ) are satisfied simultaneously at E = 0.1 eV normal mode condition (81,2 = 82,3 = π − arctan C13s 1 − C3s by selecting suitable values of L w1 , L w2 and L b3 . It is seen that the resonant transmission with the doublet occurs and the peak value at E = 0.1 eV (the first peak in the doublet) is unity. The second peak of the doublet is almost unity because the energy difference between the two peaks in the doublet is small and both (T1 ≈ T3 ) and 81,2 ≈ 82,3 is satisfied simultaneously at this peak. Second, the energy variation of the transmission coefficient is shown for L b1 = L b2 = 4 nm, L b3 = 1.886 nm, L w1 = 3.8 nm, and L w2 = 4.682 nm by the dashed line in Fig. 4, where both MCPVQAD (T(1+2) = T3 ) and PDCRQAD 8(1+2),3 = π are satisfied simultaneously by selecting suitable values of L w2
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1 0.9 Transmision coefficient
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.095
0.1
0.105
0.11 0.115 Energy (eV)
0.12
0.125
Fig. 4. Transmission coefficient vs. electron energy in two asymmetrical triple-barrier structures: A, L b1 = L b2 = 4 nm, L b3 = 1.368 nm, L w1 = 4.043 nm, and L w2 = 4.585 nm (real line); B, L b1 = L b2 = 4 nm, L b3 = 1.886 nm, L w1 = 3.8 nm, and L w2 = 4.682 nm (dashed line).
Probability density |ψ|2
250
200
150
100
50
0 0
5
10 Distance X (nm)
15
Fig. 5. Probability density at the first peak of the real line in Fig. 4.
and L b3 at E = 0.1 eV. It is seen that the resonant transmission occurs and the peak value at E = 0.1 eV is unity and the second peak value is much smaller than unity. Finally, we shall calculate the probability density of an electron wavefunction to investigate the confining phenomenon by assuming that the amplitude of the transmission wave is unity. The wavefunction is calculated for the triple-barrier structure at the unity resonance state of the QST mode, which corresponds to the first
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Probability density |ψ|2
1600
1200
800
400
0 0
5
10 Distance X (nm)
15
Fig. 6. Probability density at the first peak of the dashed line in Fig. 4.
1
2
3
4
δ-potential E
Middle barrier Left barrier
Right barrier
Fig. 7. Potential energy diagram of the triple-barrier structure with a middle δ-doping barrier under an applied electric field.
peak of the real line in Fig. 4. The probability density at the unity resonance state is shown in Fig. 5. It is seen that the confining effect is strong in both wells at the resonant state of the QST mode. The wavefunction is also calculated for the triple-barrier structure at unity resonance for the QAD mode, which corresponds to the first peak of the dashed line in Fig. 4. The probability density at the unity resonance state is shown in Fig. 6. It is evident that the confining effect is strong in either well at resonance of the QAD mode. We can easily understand these results from Fig. 3. Here, we have considered a simplified asymmetrical triple-barrier structure. It can be confirmed, however, that the tunnelling transmission characteristics and confining phenomena for complicated triple-barrier structures are quite similar to those in the above cases. For example, we can easily investigate the tunnelling transmission characteristics for the asymmetrical triple-barrier structure shown in Fig. 7 by the analytical expressions of the generalized single-barrier parameters given in (5a)–(5c) and [18].
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5. Conclusion In this work, resonant tunnelling has been studied systematically in detail for asymmetrical triple-barrier structures with arbitrary potential profiles. The transmission coefficient for these structures is derived in the analytical form (1). It is found that the QST mode and the QAD mode may exist in asymmetric triplebarrier structures. The resonance conditions are given by eqns (9a)–(9c) and (12) for the QST mode, and by (22a)–(23b) for the QAD mode. The energy variation of the transmission coefficient, and the probability density for both the QST mode and the QAD mode are examined and shown for two typical asymmetrical rectangular triple-barrier structures in Figs 4–6. It is concluded that the unity resonant tunnelling with the doublet occurs if both the normal mode condition and MCPVQST hold simultaneously, the unity resonant tunnelling with singlet occurs if either both the critical condition and MCPVQST or both PDCRQAD and MCPVQAD hold simultaneously, and the below-unity resonant tunnelling will occur if either PDCRQST or PDCRQAD is satisfied in any triple-barrier structure. These resonance conditions enable us to select one desired energy level at which the transmission coefficient of the triple-barrier structure is unity. We believe that the generalized analytical criteria given here could help in classifying numerical results or experimental data and assist experimental efforts in designing optimum triple-barrier resonant-tunnelling systems.
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Appendix: The expressions of T(1+2) , T(2+3) , 81,(2+3) and 8(1+2),3 1
1
1 |(T2 T3 )− 2
1 + [(T2−1 − 1)(T3−1 − 1)] 2
T(1+2) = |(T1 T2 )− 2 + [(T1−1 − 1)(T2−1 − 1)] 2 exp(i81,2 )|−2 ,
(A1)
T(2+3) = exp(i82,3 )|−2 , 81,(2+3) = φ1 + φ(2+3) + δ1 − δ(2+3) + 2κw1 L w f 1 ,
(A2) (A3)
8(1+2),3 = φ(1+2) + φ3 + δ(1+2) − δ3 + 2κw2 L w f 2 ,
(A4)
and where
1
−1
−1 −1 (Tm Tn ) 2 sin(φm + φn + κwm L w f m )+[Tm − 1)(Tn − 1)] 2 sin(−δm + δn − κwm L w f m ) φ(m+n) = arctan 1 1 (Tm Tn ) 2 cos(φm + φn + κwm L w f m ) + [(Tm−1 − 1)] 2 cos(−δm + δn − κwm L w f m )
(A5)
and
1 1 − 21 − 21 −1 −1 2 2 (T −1) sin(−φ + δ −κ L )+T (T − 1) sin(φ + δ + κ L ) T m n wm w f m n m wm w f m n n m m δ(m+n) = arctan 1 , 1
1 − − Tm 2 (Tn−1 −1) 2 cos(−φm + δn −κwm L w f m )+Tn 2 (Tm−1 − 1) 2 cos(φn + δm + κwm L w f m )
1
(A6)
for the subscript m = 1, 2 and n = m + 1. Here, −π/2 < φ(m+n) < π/2, and −π/2 < δ(m+n) < π/2. Equations (A5) and (A6) correspond to eqns (16a) and (16b) in [20], respectively, when the energy E is outside the sudden change region for the reflection phase differences in the double barrier structure bm wm bn [20].
Resonance condition for asymmetrical triple-barrier structures X. D. Z HAO , H. YAMAMOTO , Y. N AKANO , K. TANIGUCHI Department of Information Science, Fukui University, 3-9-1 Bunkyo, Fukui 910, Japan
(Received 9 February 1996) The resonance condition for triple-barrier structures with arbitrary potential is studied systematically. The quasisymmetrical triple-barrier (QST) resonance mode and the quasiasymmetrical double-barrier (QAD) resonance mode may both exist in asymmetric triplebarrier structures. The QST consists of two submodes: a normal mode (doublet) and a degenerate mode (singlet). The critical condition for distinguishing the two modes is examined. It is confirmed that there are both unity resonant transmission and below-unit resonant transmission in the asymmetrical triple-barrier structure. Furthermore, the wavefunctions of an electron at resonance level are calculated and the confining phenomenon is studied. c 1998 Academic Press Limited
Key words: resonance condition, asymmetrical triple-barrier structures, resonance modes.
1. Introduction Since the pioneering work by Esaki et al. [1, 2], the resonant tunnelling in semiconductor heterostructures has been studied extensively [3–20]. The triple-barrier structure is the simplest multiple-barrier structure which allows us to understand the interaction between two wells. Experiments about resonant tunnelling in the triple-barrier and the double-well structures have been reported [6, 7], and some interesting features have been found: the current peak-to-valley ratio was enhanced relative to the double-barrier diode by using these structures. Theoretical studies have been also carried out on the triple-barrier structures with different potential profiles: rectangular [8–11], trapezoidal [12], slowly changed [13, 14], and δ-doping potential profiles [15]. Usually, the Airy function method is used for the triple-barrier structure with a trapezoidal potential profile [12] and the WKB approximation is used for the triple-barrier structure where the potential changes slowly (i.e. the change of potential energy within a de Broglie wavelength is small compared with the kinetic energy) [13, 14]. It is noteworthy that the analytical expressions for the transmission coefficient and the resonance condition in the previous theories [8–15] are modified as a result of the different potential profiles of triple-barrier structures. In these theoretical studies, eight different expressions for the transmission coefficient in different triple-barrier structures were given. The resonance condition in the triple-barrier structure with trapezoidal potential profile was not obtained [12]. In other words, the common points for the triple-barrier structures with different potential profiles have not been thoroughly considered: analytical expressions for the transmission coefficient and the resonance condition for transmission through triple-barrier structures were obtained only for some special cases [8–15]. Therefore, it is necessary to study systematically the transmission coefficient and the resonance condition for the triple-barrier structure with an arbitrary potential. In this paper, ‘the generalized single-barrier parameters method’ [16, 17] is used to study in detail the transmission coefficient, the resonance condition and the resonant transmission modes for asymmetrical 0749–6036/98/061273 + 11
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c 1998 Academic Press Limited
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A
V 1
Region 2
4
3
5
6
7
Vmax3 Vmax2 Vmax1 E Vw3 Vw2 Vw1 B
X
Lwf2 X1
Lwf1 X2 X3
V 1 Region 2
X4 4
3
Vmax3 Vmax2 Vmax1
X5
X6 5
E X
Vw3 Vw2 Vw1 line 1
line 2
Fig. 1. Potential energy diagram of the triple-barrier structure with arbitrary potential.
triple-barrier structures with arbitrary potential profiles. The confining phenomena at QST and QAD modes are also investigated for two typical triple-barrier structures. The analytical expressions for transmission coefficients and resonance conditions obtained here do not rely upon barrier potential profiles and many kinds of single-barrier structures with complicated potential profiles may be used simultaneously to investigate the tunnelling transmission characteristics in the triple-barrier structure. The generalized single-barrier parameters have been calculated numerically for arbitrary barrier structures [16, 17] and analytically for rectangular and trapezoidal structures as well as parallel-plane ones [18], and, in this paper those for δ-doping barrier structures are also included.
2. Transmission coefficient in asymmetrical triple-barrier structures In this section we shall derive first an analytical expression for the transmission coefficient in an asymmetrical triple-barrier structure with arbitrary potential. The schematic energy diagram of the asymmetrical triple-barrier structure, formed by three generalized single barriers (regions 2, 4, 6) and two flat well regions (regions 3, 5), is shown in Fig. 1A, where Vwj is the potential height, m ∗wj the electron effective mass, and L w f j the width for the jth flat well region ( j = 1, 2). If L w f 1 and L w f 2 are reduced to zero, the schematic energy diagram shown in Fig. 1A will become that shown in Fig. 1B, i.e. regions 3 and 5 become line 1 and line 2, respectively. Here, line 1 and line 2 are called the dividing barrier lines (DBL), by which we can distinguish the left and right generalized single-barrier from the middle one. With the aid of the transfer matrix N j (the transfer matrix for the jth generalized single-barrier) whose elements are functions of generalized single-barrier parameters [17], the transmission coefficient for an
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asymmetrical triple-barrier structure is expressed as T3g = |R + R1 exp(i82,3 ) + R2 exp[i(81,2 + 82,3 )] + R3 exp(i81,2 )|−2 = [R 2 + R12 + R22 + R32 + 2R1 R3 cos(81,2 − 82,3 ) + 2R R2 cos(81,2 + 82,3 ) +2(R R3 + R1 R2 ) cos(81,2 ) + 2(R R1 + R2 R3 ) cos(82,3 )]−1 ,
(1)
where we have 81,2 = θ1,R + θ2,L + π + 2κw1 L w f 1 , 82,3 = θ2,R + θ3,L + π + 2κw2 L w f 2 ,
(2a) (2b)
R = (T1 T2 T3 )− 2 , 1
R1 = R2 = R3 =
−1 T1 2 [(T2−1 −1 T2 2 [(T1−1 −1 T3 2 [(T1−1
(3a)
− 1)(T3−1 − 1)] ,
(3b)
− 1)(T3−1 − 1)] ,
(3c)
− 1)(T2−1 − 1)] ,
(3d)
1 2 1 2 1 2
and κwj = [2m ∗wj (E + Vwj )] 2 /h¯ 1
( j = 1, 2 and h¯ (= h/2π ) is the reduced Planck constant).
(4)
Here, we call 8 j, j+1 the characteristic phase difference in the well region between the jth barrier and the ( j + 1)th barrier ( j = 1, 2), T j is the transmission coefficient of the jth generalized single-barrier structure ( j = 1–3), θ j,L and θ j,R are the reflection phase difference at the left and right edge of that, respectively. T j , θ j,L and θ j,R are called the generalized single-barrier parameters for the jth generalized single-barrier structure [17]. The analytical expressions of the generalized single-barrier parameters have been obtained for the rectangular and the trapezoidal as well as the parallel-plane potential structures [17, 18], which are shown in Figs 2A–C. Recently, the effects of a δ-doped impurity sheet have been considered [15, 21]. Here, we will give the generalized single-barrier parameters for the δ-doped barrier shown in Fig. 2D. Assuming the potential of the δ-doped barrier has the form δ, where is δ-potential strength and δ is δ-function, we have !−1 2 , (5a) Tj = 1 + 0 0 κ j κ j+1 h¯ 4 " # # " 0 (κ j+1 − κ 0j )h¯ 2 2|| − π/2, (5b) − arctan θ j,L = arctan − 0 2|| (κ j + κ 0j+1 )h¯ 2 and
"
θ j,R
2|| = arctan − 0 (κ j + κ 0j+1 )h¯ 2
#
" + arctan
(κ 0j+1 − κ 0j )h¯ 2 2||
# − π/2,
(5c)
1
where κ j = [2m ∗j (E + V j )] 2 /h¯ , κ 0j = κ j /m ∗j , m ∗j is the electron effective mass of the jth region in Fig. 2D, and V j is the potential height of that.
3. Resonance condition for asymmetrical triple-barrier structures In an asymmetrical double-barrier structure, b1 w1 b2 , only a single well w1 exists. On the other hand, in an asymmetrical triple-barrier structure, b1 w1 b2 w2 b3 , double wells, w1 and w2 , exist, which are coupled
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A
B Region j
j+1
m*j
m*j + 1
Vj
Vj + 1 C
D
Fig. 2. Four typical single-barrier structures.
QST mode b1 w1 b2
w2 A
Left barrier b1
b1 w1
b3 Energy level
Right barrier b(2 + 3)
b2 B1
w2
Left barrier b(1 + 2)
b3
b1 w1 QAD mode
Right barrier b3
b2
w2
b3
B2
Fig. 3. A, Illustration of QST mode; B, Illustration of QAD mode.
through a central barrier b2 as seen in Fig. 3. We may understand that there are two kinds of resonance modes in the asymmetrical triple-barrier structure. One is the quasisymmetrical triple-barrier resonance mode (QST mode), at which the electron energy coincides with an eigenenergy in both wells, and the other is the quasiasymmetrical double-barrier resonance mode (QAD mode), at which the electron energy coincides with an eigenenergy in either well. Usually, only the QAD mode exists in the asymmetrical triple-barrier structure. In this section, we will investigate the resonance conditions for the above two modes.
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3.1. Resonance condition for QST mode Here, let us investigate the resonance condition for the QST mode in asymmetrical triple-barrier structures. It is seen that eqn (1) may be simplified to T3s = [(R − R2 )2 + (R1 + R3 )2 + 4R R2 cos2 8 + 2(R + R2 )(R1 + R3 ) cos 8]−1 ,
(6)
for 81,2 = 8 + 2n 1 π,
(n 1 : 0, 1, 2, . . .),
(7a)
82,3 = 8 + 2n 2 π,
(n 2 : 0, 1, 2, . . .),
(7b)
and i.e. for 81,2 = 82,3 ± 2nπ,
(n = 0, 1, 2, . . .).
(8)
Here, eqn (8) is called the QST mode condition, since this condition is always satisfied in symmetrical triple-barrier structures and may be satisfied in an asymmetrical triple-barrier structure for a given value of energy. First, let us assume that T1 , T2 , and T3 are constants or almost constants near a resonant peak. This assumption may be satisfied since T3s varies with energy E abruptly near each resonant peak and T1 , T2 , and d 2 T3s 3s T3 varies with energy E smoothly. By the local maximum conditions, dT d8 = 0 and d82 < 0, we obtain the following resonance conditions q 2 1 − C3s , (C3s < 1), (9a) 8 = π ± arctan C3s (9b) 8 = π, (C3s = 1), and 8 = π,
(C3s > 1),
(9c)
where
√ p p 1 1 − T2 1 + 1 − T1 + √ + 1 − T3 . (10) C3s ≡ √ 4 1 − T1 1 − T3 It is seen that (9a) corresponds to two resonant peaks (normal code), (9c) one resonant peak (degenerate mode), and (9b) is the critical relation between (9a) and (9c). When the critical relation holds, the resonant transmission of the doublet will reduce to that of the singlet. Here, we call (9a) the normal mode condition, (9b) the critical condition, (9c) the degenerate mode condition, and call these three conditions the phase difference conditions for the resonance in the QST mode (PDCRQST). The resonant tunnelling will occur in an asymmetrical triple-barrier structure when one of the above three conditions is satisfied. From the relation C3s = 1, the critical transmission coefficient of the middle barrier in the asymmetrical triple-barrier structure is given by 2 p p 1 1 + 1 − T1 + √ + 1 − T3 . (11) T2−critical = 1 − 16 √ 1 − T1 1 − T3 It is understood that there is only a degenerate mode in an asymmetrical triple-barrier structure when T2 < T2−critical , and there is only the normal mode when T2 > T2−critical . Next, if we assume further that the condition T1 = T3 = T
(12)
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is satisfied at a given value of energy E, then eqn (6) is simplified to 0 2 −1 ) ] , T3s0 = [1 + K 2 (cos 8 + C3s
where we have
s 2 K = T
and 0 C3s
1−T , T2
√ √ 1 − T2 1 + 1−T . = √ 2 1−T
(13)
(14a)
(14b)
It is evident that T3s0 = 1,
(15)
0 cos 8 + C3s = 0.
(16)
for Furthermore, (16) is reduced to the following equations q 02 1 − C3s 0 , (C3s < 1) (i.e. the normal mode condition), 8 = π ± arctan 0 C3s
(17a)
and 8 = π,
0 = 1) (i.e. the critical condition). (C3s
(17b)
Here, eqn (12) is called the maximum condition for the resonant peak value in QST mode (MCPVQST), which is always satisfied in the symmetrical triple-barrier structures. It is understood that the transmission coefficient becomes unity when both MCPVQST and the normal mode condition or both MCPVQST and the critical condition are satisfied simultaneously. Equation (13) is a simple relation for the transmission coefficient in symmetrical triple-barrier structures. 0 > 0, then we have On the other hand, if cos 8 + C3s T3s0 < 1,
(18)
and the resonance condition is 8=π
0 > 1) (C3s
(i.e. the degenerate mode condition).
(19)
It means that only below-unity resonant tunnelling occurs in any triple-barrier structure when the degenerate mode condition is satisfied. 3.2. Resonance condition in QAD mode For the case of 81,2 6 = 82,3 ± 2nπ,
(n : 0, 1, 2, . . .),
(20)
which is different from (8), we may assume that there are two different quasidouble-barrier structures in the asymmetrical triple-barrier structure. One is the structure b(1+2) w2 b3 , and the other is the structure b1 w1 b(2+3) , as shown in Fig. 3B1 and B2. Here, b(1+2) is the left equivalent-barrier which is composed of b1 w1 b2 , and b(2+3) , the right equivalent-barrier, composed of b2 w2 b3 . Equation (20) is called the QAD mode condition, which is necessary for the QAD mode to exist.
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To investigate the resonance condition in the quasidouble-barrier structures, b1 w1 b(2+3) and b(1+2) w2 b3 , we may rewrite eqn (1) as −1 − 1)] 2 exp(i81,(2+3) )|−2 , T3a1 ≡ T3g = |(T1 T(2+3) )− 2 + [(T1−1 − 1)(T(2+3)
(21a)
−1 − 1)(T3−1 − 1)] 2 exp(i8(1+2),3 )|−2 , T3a2 ≡ T3g = |(T(1+2) T3 )− 2 + [(T(1+2)
(21b)
1
1
or 1
1
where T(1+2) is the transmission coefficient for the left equivalent single-barrier b1 w1 b2 , T(2+3) the transmission coefficient for the right equivalent single-barrier b2 w2 b3 , 81,(2+3) the characteristic phase difference in the well region between the barrier b1 and the equivalent single-barrier b2 w2 b3 , and 8(1+2),3 the characteristic phase difference in the well region between b1 w1 b2 and b3 . The expressions for T(1+2) , T(2+3) , 81,(2+3) , and 8(1+2) , 3 are given in the Appendix. It is evident that the transmission coefficient in the asymmetrical triple-barrier structure becomes unity for 81,(2+3) = π + 2nπ,
(n = 0, 1, 2, . . .),
(22a)
and T1 = T(2+3) ,
(22b)
or for 8(1+2),3 = π + 2nπ,
(n = 0, 1, 2, . . .),
(23a)
and T(1+2) = T3 .
(23b)
The below-unity resonant transmission will occur if either the condition (22a) or (23a) is satisfied. Here, we call eqns (22a) and (23a) the phase difference conditions for the resonance in QAD mode (PDCRQAD), and eqns (22b) and (23b) the maximum condition for the resonant peak value in QAD mode (MCPVQAD). It is seen that the resonance conditions in QAD mode are similar to that in the double-barrier structure [16, 17].
4. Tunnelling transmission characteristics and confining phenomenon In the above sections, the transmission coefficient and resonance condition for the asymmetrical triplebarrier structure were derived and discussed. Here, let us investigate the energy variation of the transmission coefficient and the probability density of an electron wavefunction. As typical examples, let us consider the energy variation of the transmission coefficient for the asymmetrical triple-barrier structure Ga0.7 Al0.3 As/GaAs/Ga0.7 Al0.3 As/GaAs/AlAs, where the effective mass in GaAs is assumed to be 0.067 m 0 , the effective mass in Ga0.7 Al0.3 As is 0.092 m 0 [22] and that in AlAs is 0.15 m 0 [23]. It is assumed that the conduction-band offset at the heterojunction interfaces between Ga0.7 Al0.3 As and GaAs is 0.3 eV, and the conduction-band offset between GaAs and AlAs 1.0 eV [24]. First, the energy variation of the transmission coefficient is shown for L b1 = L b2 = 4 nm, L b3 = 1.368 nm, L w1 = 4.043 nm, and L w2 = 4.585 nm by the real linepin Fig. 4, where both MCPVQST (T1 = T3 ) and the 2 ) are satisfied simultaneously at E = 0.1 eV normal mode condition (81,2 = 82,3 = π − arctan C13s 1 − C3s by selecting suitable values of L w1 , L w2 and L b3 . It is seen that the resonant transmission with the doublet occurs and the peak value at E = 0.1 eV (the first peak in the doublet) is unity. The second peak of the doublet is almost unity because the energy difference between the two peaks in the doublet is small and both (T1 ≈ T3 ) and 81,2 ≈ 82,3 is satisfied simultaneously at this peak. Second, the energy variation of the transmission coefficient is shown for L b1 = L b2 = 4 nm, L b3 = 1.886 nm, L w1 = 3.8 nm, and L w2 = 4.682 nm by the dashed line in Fig. 4, where both MCPVQAD (T(1+2) = T3 ) and PDCRQAD 8(1+2),3 = π are satisfied simultaneously by selecting suitable values of L w2
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1 0.9 Transmision coefficient
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.095
0.1
0.105
0.11 0.115 Energy (eV)
0.12
0.125
Fig. 4. Transmission coefficient vs. electron energy in two asymmetrical triple-barrier structures: A, L b1 = L b2 = 4 nm, L b3 = 1.368 nm, L w1 = 4.043 nm, and L w2 = 4.585 nm (real line); B, L b1 = L b2 = 4 nm, L b3 = 1.886 nm, L w1 = 3.8 nm, and L w2 = 4.682 nm (dashed line).
Probability density |ψ|2
250
200
150
100
50
0 0
5
10 Distance X (nm)
15
Fig. 5. Probability density at the first peak of the real line in Fig. 4.
and L b3 at E = 0.1 eV. It is seen that the resonant transmission occurs and the peak value at E = 0.1 eV is unity and the second peak value is much smaller than unity. Finally, we shall calculate the probability density of an electron wavefunction to investigate the confining phenomenon by assuming that the amplitude of the transmission wave is unity. The wavefunction is calculated for the triple-barrier structure at the unity resonance state of the QST mode, which corresponds to the first
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Probability density |ψ|2
1600
1200
800
400
0 0
5
10 Distance X (nm)
15
Fig. 6. Probability density at the first peak of the dashed line in Fig. 4.
1
2
3
4
δ-potential E
Middle barrier Left barrier
Right barrier
Fig. 7. Potential energy diagram of the triple-barrier structure with a middle δ-doping barrier under an applied electric field.
peak of the real line in Fig. 4. The probability density at the unity resonance state is shown in Fig. 5. It is seen that the confining effect is strong in both wells at the resonant state of the QST mode. The wavefunction is also calculated for the triple-barrier structure at unity resonance for the QAD mode, which corresponds to the first peak of the dashed line in Fig. 4. The probability density at the unity resonance state is shown in Fig. 6. It is evident that the confining effect is strong in either well at resonance of the QAD mode. We can easily understand these results from Fig. 3. Here, we have considered a simplified asymmetrical triple-barrier structure. It can be confirmed, however, that the tunnelling transmission characteristics and confining phenomena for complicated triple-barrier structures are quite similar to those in the above cases. For example, we can easily investigate the tunnelling transmission characteristics for the asymmetrical triple-barrier structure shown in Fig. 7 by the analytical expressions of the generalized single-barrier parameters given in (5a)–(5c) and [18].
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5. Conclusion In this work, resonant tunnelling has been studied systematically in detail for asymmetrical triple-barrier structures with arbitrary potential profiles. The transmission coefficient for these structures is derived in the analytical form (1). It is found that the QST mode and the QAD mode may exist in asymmetric triplebarrier structures. The resonance conditions are given by eqns (9a)–(9c) and (12) for the QST mode, and by (22a)–(23b) for the QAD mode. The energy variation of the transmission coefficient, and the probability density for both the QST mode and the QAD mode are examined and shown for two typical asymmetrical rectangular triple-barrier structures in Figs 4–6. It is concluded that the unity resonant tunnelling with the doublet occurs if both the normal mode condition and MCPVQST hold simultaneously, the unity resonant tunnelling with singlet occurs if either both the critical condition and MCPVQST or both PDCRQAD and MCPVQAD hold simultaneously, and the below-unity resonant tunnelling will occur if either PDCRQST or PDCRQAD is satisfied in any triple-barrier structure. These resonance conditions enable us to select one desired energy level at which the transmission coefficient of the triple-barrier structure is unity. We believe that the generalized analytical criteria given here could help in classifying numerical results or experimental data and assist experimental efforts in designing optimum triple-barrier resonant-tunnelling systems.
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Appendix: The expressions of T(1+2) , T(2+3) , 81,(2+3) and 8(1+2),3 1
1
1 |(T2 T3 )− 2
1 + [(T2−1 − 1)(T3−1 − 1)] 2
T(1+2) = |(T1 T2 )− 2 + [(T1−1 − 1)(T2−1 − 1)] 2 exp(i81,2 )|−2 ,
(A1)
T(2+3) = exp(i82,3 )|−2 , 81,(2+3) = φ1 + φ(2+3) + δ1 − δ(2+3) + 2κw1 L w f 1 ,
(A2) (A3)
8(1+2),3 = φ(1+2) + φ3 + δ(1+2) − δ3 + 2κw2 L w f 2 ,
(A4)
and where
1
−1
−1 −1 (Tm Tn ) 2 sin(φm + φn + κwm L w f m )+[Tm − 1)(Tn − 1)] 2 sin(−δm + δn − κwm L w f m ) φ(m+n) = arctan 1 1 (Tm Tn ) 2 cos(φm + φn + κwm L w f m ) + [(Tm−1 − 1)] 2 cos(−δm + δn − κwm L w f m )
(A5)
and
1 1 − 21 − 21 −1 −1 2 2 (T −1) sin(−φ + δ −κ L )+T (T − 1) sin(φ + δ + κ L ) T m n wm w f m n m wm w f m n n m m δ(m+n) = arctan 1 , 1
1 − − Tm 2 (Tn−1 −1) 2 cos(−φm + δn −κwm L w f m )+Tn 2 (Tm−1 − 1) 2 cos(φn + δm + κwm L w f m )
1
(A6)
for the subscript m = 1, 2 and n = m + 1. Here, −π/2 < φ(m+n) < π/2, and −π/2 < δ(m+n) < π/2. Equations (A5) and (A6) correspond to eqns (16a) and (16b) in [20], respectively, when the energy E is outside the sudden change region for the reflection phase differences in the double barrier structure bm wm bn [20].