Generalized-barrier parameters for single-barrier structures and resonance width in asymmetrical double-barrier structures

Superlattices and Microstructures, Vol. 23, No. 6, 1998 Article No. sm960159

Generalized-barrier parameters for single-barrier structures and resonance width in asymmetrical double-barrier structures X. D. Z HAO , H. YAMAMOTO , K. TANIGUCHI Department of Information Science, Fukui University, 3-9-1 Bunkyo, Fukui 910, Japan (Received 1 May 1996)

Generalized-barrier parameters are studied theoretically for rectangular and trapezoidal single-barriers as well as parallel-plane ones, which are basic units for double-barrier and multiple-barrier structures. Analytical expressions for these parameters are derived by taking into account the position-dependent effective-mass effect. Furthermore, the expressions for the generalized-barrier parameters of optical single-layer thin film structures are considered and permit direct application of previous results to thin film interference filters also. The heights of resonant peaks and the resonance widths in asymmetrical double-barrier structures are studied as functions of the generalized-barrier parameters. c 1998 Academic Press Limited

Key words: resonance condition, generalized-barrier parameters, resonance widths.

1. Introduction The resonant tunnelling phenomena in quantum-well and superlattice structures have been extensively studied experimentally and theoretically [1–18] due to their possible applications to electronic and optoelectronic devices. The generalized-barrier parameters method was proposed for the first time [10, 11] and used to obtain simple expressions of the transmission coefficients and the resonance conditions in double-, triple- and multiple-barrier structures with arbitrary potential profiles [11–13]. Furthermore, both the generalized-barrier parameters method and the analytical expressions such as the transmission coefficients and the resonance conditions obtained in [11–13] are valid for studying optical multilayer thin-film structures, in which no Coulomb interaction exists. Quantum-mechanical electron waves in semiconductors and electromagnetic optical waves in dielectrics exhibit transmission, reflection, interference and diffraction characteristics which are analogous to each other [16, 17]. In this work, we will derive analytical expressions of the generalized-barrier parameters for some complicated single-barrier structures, for example the trapezoidal and parallel-plane single-barriers. These expressions are important when using the generalized-barrier parameters method. In Section 2, the analytical expressions of the generalized-barrier parameters for single-barrier structures are derived by four kinds of methods: the Airy function method, the rectangular single-barrier approximation, the WKB approximation [4] and a new WKB approximation. The analytical expressions of the generalized-barrier parameters are also given for the optical single-layer thin film structures with an arbitrary refractive index profile. As a typical example, the generalized-barrier parameters for a trapezoidal single-barrier are examined by the above four kinds of methods in Section 3. In addition, we also derive the analytical expressions of both resonance 0749–6036/98/061309 + 14

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c 1998 Academic Press Limited

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Region 1

2 9i

E

3 9t

9r

Vf 1

mf*1

mb* X1 = 0

mf*1 X2 = Lb

x

Vf 2 Fig. 1. Potential energy diagram of single-barrier structure.

width and height of resonant peak for the unity resonance and the under-unity one in asymmetrical doublebarrier structures in Section 4, which are important for analysing the peak-to-valley current ratios through a double-barrier structure under an applied electric field [14]. By using the analytical expressions for the generalized-barrier parameters, the shape of the transmission spectrum for an asymmetrical double-barrier structure is investigated in Section 5. It is confirmed that the shape of the transmission spectrum for the structure is a Lorentzian near resonance energies.

2. Calculation of generalized-barrier parameters 2.1. Generalized-barrier parameters for some single-barriers First, we consider the generalized-barrier parameters for the single-barrier with arbitrary potential. In the schematic energy diagram of the single-barrier structure in Fig. 1, we define the generalized barrier as the region in which a reflected wave occurs for an incident wave. The single-barrier structure is composed of one generalized barrier (region 2) and two flat bottom regions (regions 1 and 3), where L b is the generalized barrier width, m ∗b is the effective mass for the generalized barrier and V f j and m ∗f j are the barrier height and the effective mass for the jth flat bottom region ( j = 1, 2), respectively. The three regions are specified by the coordinates (x1 , x2 ). It is assumed that an electron with energy E is incident from the left and transmits to the right along the x-direction, we may express the wavefunctions in the three regions as follows 91 (x) = A1 exp(iκ1 x) + B1 exp(−iκ1 x), 92 (x) = A2 f A (x) + B2 f B (x),

(1) (2)

93 (x) = A3 exp(iκ2 x) + B3 exp(−iκ2 x),

(3)

and where κ j = [2m ∗f j (E + V f j )] 2 /h¯ , 1

( j = 1, 2).

(4)

Here, h¯ (= h/2π) is the reduced Planck constant and f A (x) and f B (x) are the forward and backward waves in region 2, respectively. Coefficients A j and B j are constants to be determined from the boundary conditions by matching the

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wavefunctions and their first derivatives at each discontinuity point x j ( j = 1, 2): 9 j (x j ) = 9 j+1 (x j )

(1/m j )[d9 j (x j )/d x] = (1/m j+1 )[d9 j+1 (x j )/d x],

and

where m 1 = m ∗f 1 ,

m 2 = m ∗b ,

m 3 = m ∗f 2 ,

x1 = 0

and

x2 = L b .

Using the transfer matrix method we obtain the following relation [11],     1  κ´ 2 2 A3 A1 =N = B1 B3 κ´ 1 T − 2 exp[i(−φ + κ2 L b )] 1 (T −1 − 1) 2 exp[i(−δ − π/2 + κ2 L b )] 1

×

1

(T −1 − 1) 2 exp[i(δ + π/2 − κ2 L b )] 1 T − 2 exp[i(φ − κ2 L b )]

!

A3 B3

 , (5)

where κ´ j = κ j /m ∗f j ( j = 1, 2), T is the transmission coefficient of the generalized-barrier and N is the transfer matrix for the generalized-barrier. If we assume the transmitted wavefunction with unity amplitude at x = L b is ψt = exp(ik2 L b ),

(6a)

then the incident wavefunction ψi and the reflected wavefunction ψr at x = 0 may be derived from (5) as follows  1 κ´ 2 2 − 12 T j exp(−iφ)exp(ik2 L b ), (6b) ψi = κ´ 1 and  1 1 κ´ 2 2 −1 (T − 1) 2 exp[−i(δ + π/2)]exp(ik2 L b ). (6c) ψr = κ´ 2 It is evident that φ is the wave phase difference between transmitted wave and incident wave and δ + π/2 is that between transmitted wave and reflected wave. T , φ and δ are derived as    s s 2 !2 −1 1 p 1 κ´ 1 κ´ 2  1   −d + (7a) b κ´ 1 κ´ 2 + c p T = 1 + a  , 4 κ´ 2 κ´ 1 4 κ´ 1 κ´ 2   p c √ 1 − b κ´ 1 κ´ 2 κ´ 1 κ´ 2   q (7b) φ = arctan  q , κ´ 1 κ´ 2 a κ´ + d κ´ 2

and

1



q q κ´ 1 κ´ 2 − d a κ´ 2 κ´ 1  δ = arctan  p b κ´ 1 κ´ 2 + c √ 1

  ,

(7c)

κ´ 1 κ´ 2

where 1 a= K b=

"

# d f B (x) d f A (x) − f B (L b ) f A (0) , d x x=L b d x x=0

1 [ f B (0) f A (L b ) − f A (0) f B (L b )], K

(8a) (8b)

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"

# d f A (x) 1 d f A (x) d f B (x) d f B (x) , c= − K d x x=0 d x x=L b d x x=0 d x x=L b " # d f B (x) d f A (x) 1 f A (L b ) , − f B (0) d= K d x x=0 d x x=L b     d f B (x) d f A (x) − f B (L b ) , K = f A (L b ) dx dx x=L b x=L b

(8c) (8d)

and the relation ad − bc = 1

(9)

holds. Here a, b and c are called the independent coefficients. The reflection phase differences θ L (at x = 0) and θ R (at x = L b ) may be given by θ L = φ − δ − π/2,

(10a)

θ R = φ + δ − π/2.

(10b)

and T , θ L and θ R have been defined as a set of generalized-barrier parameters [11]. In fact, T , φ and δ are also the equivalent generalized-barrier parameters whose physical meaning is mentioned above. It is seen whether the independent coefficients or the generalized-barrier parameters consist always of three independent variables. Second, let us consider the generalized-barrier parameters for the trapezoidal single-barrier shown in Fig. 2A, which is an important single-barrier structure since the rectangular single-barrier will become the trapezoidal one when an electric field is applied. Using the Airy function approach [6], we can express the wavefunction in the region 2 as 9(x) = A2 × Ai[ρ(x)] + B2 × Bi[ρ(x)],

(11)

where ρ(x) = (2m b 1V /L b h¯ 2 )1/3 (x + η), η = (L b /1V )(Vb − E), Vb is the maximum barrier height and 1V /L b is the applied electric field. It is understood that the coefficients a, b, c and d can be obtained if f A (x) in eqns (8a)–(8d) is replaced by Ai[ρ(x)] and f B (x) by Bi[ρ(x)]. Third, for the rectangular barrier shown in Fig. 2B, the wavefunction in region 2 may be expressed as 9(x) = A2 cosh(βx) + B2 sinh(βx),

(12)

where β ≡ [2m ∗b (Vb − E)] 2 /h¯ . 1

For the parallel-plane single-barrier shown in Fig. 2C, the wavefunction in region 2 may be written as the following form by the WKB approximation Z x  −1/2 cosh β(x) d x 9(x) = A2 β(x)  0Z x  −1/2 sinh β(x) d x , (E < Vmin and E > Vmax ), (13) +B2 β(x) 0

where β(x) ≡ [2m ∗b (Vb (x) − E)] 2 /h¯ , 1

Vb (x) is the potential profile in region 2, Vmin is the minimum barrier height in region 2 and Vmax is the

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Region 1

2

3

V Vb

1Vb

Vf 1 0 Vf 2

x A

Vb Vf 1 0 Vf 2

x B

Vbmax Vbmin Vf 1 0

x

Vf 2

C

Lb n1

n

n2

n (x) Lb

D

Fig. 2. Potential energy diagram of some typical single-barrier structures: A, trapezoidal single-barrier; B, rectangular one; C, parallelplane one.

maximum. The coefficients a, b, c and d for the rectangular and parallel-plane single-barrier are given by eqns (A1)–(A8). It is seen that the rectangular barrier is a special case of the parallel-plane barriers. Equation (7a), i.e. the transmission coefficient of the parallel-plane single-barrier structure, can be rewritten as ! #−1  Z Lb ´ β(L ´ b) β(0) κ´ 2 κ´ 1 κ´ 2 κ´ 1 + + + β(x) d x . sinh2 T = ´ β(L ´ b) κ´ 2 κ´ 1 κ´ 1 κ´ 2 β(0) 0 (14) 1 1 If we assume that β(x) = β = [2m ∗ (Vb − E)] 2 /h¯ , (x ∈ [0, L b ]) and κ1 = κ2 = (2m ∗ E) 2 /h¯ for the "

1 1 + 2 4



κ´ 2 κ´ 1 + κ´ 2 κ´ 1



1 + 4

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B1

B2

B1

Lw

w

B2

0 Lw

A

B

Fig. 3. Potential energy diagram of A, a complicated single-barrier structure; B, that of a double-barrier structure.

parallel-plane single-barrier (E < Vmin ), then (14) may be simplified to " #−1 Vb2 0 2 , sinh (β L b ) T = 1+ 4E(Vb − E) which is the well-known expression of the transmission coefficient in the rectangular single-barrier structure. To distinguish (14) from the old expression of transmission coefficient by WKB approximation (i.e. T = RL exp[−2 0 b β(x) d x]) we call eqn (14) the new expression of transmission coefficient by WKB approximation (the new WKB expression). 2.2. Generalized-barrier parameters for optical single-layer thin film structure As an expansion, let us consider the generalized-barrier parameters for the optical single-layer thin film structure as shown in Fig. 2D, where n is the refractive index in the single-layer thin film (i.e. in region 2), L b the thickness of the thin film, n 1 and n 2 are the refractive indexes of the left-side and the right-side optical material, respectively. For the thin film structures with an arbitrary refractive index profile, we may use a refractive index profile function n(x). If we replace the parameters κ´ j ( j = 1, 2) in eqns (7a)–(7c) by 2π n j /λ (λ: the free-space wavelength) and replace the parameters β and β´ in eqns (A1)–(A4) by i2π n/λ (or ´ replace β(x) and β(x) in eqns (A5)–(A8) by i2πn(x)/λ), then eqns (7a)–(7c) give the generalized-barrier parameters for the thin film structure n 1 /n/n 2 (or n 1 /n(x)/n 2 ), which are basic parameters for studying multilayer thin-film optical filters [18]. 2.3. Generalized-barrier parameters for a compound single-barrier The analytical expressions of generalized-barrier parameters in some single-barrier are given in Section 2.1. Here, we consider the generalized-barrier parameters for the compound single-barrier shown in Fig. 3A, which may be regarded as a special double-barrier structure with zero well width. We have derived the generalizedbarrier parameters (i.e. T(1+2) , θ(1+2),L and θ(1+2),R ) for the compound single-barrier in [13].

3. Calculating generalized-barrier parameters by different approximations As a typical example, we will investigate the energy variation of the transmission coefficient and the energy variation of reflection phase difference for a GaAs/Ga0.7 Al0.3 As/GaAs type single-barrier structure under

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1.0

Transmission coefficient

0.8

0.6

0.4 1 0.2 2 0.0

3

4 0.0

0.1

0.2

0.3

0.4

Energy (eV) Fig. 4. Transmission coefficient versus electron energy in a GaAs/Ga0.7 Al0.3 As/GaAs single-barrier structure (L b = 4 nm and 1Vb = 0.1 eV) by different methods (1, the Airy function method; 2, the rectangular single-barrier approximation; 3, the new WKB expression given here; 4, the old WKB expression).

an applied electric field by different approximations mentioned in Section 2. The schematic energy diagram of the single-barrier structure is the same as that shown in Fig. 2A. Here, the conduction-band offset at the heterojunction interfaces between Ga0.7 Al0.3 As and GaAs is 0.3 eV, the effective mass in GaAs is assumed to be 0.067m 0 (m 0 is the free electron mass) and that in Ga0.7 Al0.3 As is 0.092m 0 . First, the transmission coefficient versus energy is shown in Fig. 4 for the GaAs/Ga0.7 Al0.3 As/GaAs type single-barrier structure: L b1 = 4 nm and 1Vb = 0.1 eV, where curve 1 is calculated by the Airy function method, curve 2 is calculated by the rectangular single-barrier approximation (using the average barrier RL height, Vb = (1/L b ) 0 b Vb (x) d x and the expressions of generalized-barrier parameters for the rectangular single-barrier structure to calculate approximately the generalized-barrier parameters for the trapezoidal or parallel-plane single-barrier), curve 3 is calculated by the new WKB expression proposed above and curve 4 is calculated by the old WKB expression. It is seen that the result calculated by rectangular single-barrier approximation agrees with the Airy function method very well and the new WKB expression is valid when the electron energy is smaller than the minimum barrier height or greater than the maximum. Second, the right reflection phase difference versus energy for the same single-barrier structure is shown by curve 1 in Fig. 5 by the Airy function method, curve 2 by the rectangular single-barrier approximation and curve 3 by the new WKB expression for E < Vmin (0.2 eV) and E > Vmax (0.3 eV). It is seen that the results calculated by the rectangular single-barrier approximation and the new WKB expression are close to that by the Airy function method. Finally, the right reflection phase difference versus energy for the thin single-barrier structure L b1 = 1 nm and 1Vb = 0.025 eV is shown by curve 1 in Fig. 6 by the Airy function method, curve 2 by the rectangular single-barrier approximation and curve 3 by the new WKB expression for E < 0.275 eV and E > 0.3 eV. It is understood that the new WKB approximation and the rectangular single-barrier approximation are valid for calculating the generalized-barrier parameters of the parallel-plane single-barrier structures (e.g. trapezoidal single-barrier structure) when the barrier width is thin.

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Reflection phase difference θR (degree)

60

30

0 3

1

–30

2

–60

–90 0.0

0.1

0.2

0.3

0.4

Energy (eV) Fig. 5. Right-hand reflection phase difference versus electron energy in the GaAs/Ga0.7 Al0.3 As/GaAs single-barrier structure (L b = 4 nm and 1Vb = 0.1 eV) by three different methods (1, the Airy function method; 2, the rectangular single-barrier approximation; 3, the new WKB approximation expression).

Reflection phase difference θR (degree)

3 –60

1

2 –80

–100

–120 0.0

0.1

0.2

0.3

0.4

Energy (eV) Fig. 6. Transmission coefficient versus electron energy in a GaAs/Ga0.7 Al0.3 As/GaAs single-barrier structure (L b = 1 nm and 1Vb = 0.025 eV) by three different methods (1, the Airy function method; 2, the rectangular single-barrier approximation; 3, the new WKB approximation expression).

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4. Resonance width and height of resonant peak In the above sections, the analytical expressions of the generalized-barrier parameters for some singlebarrier are derived and examined. These are the basic expressions for calculating the transmission coefficients of double- and multiple-barrier structures. It has been proven that the transmission coefficient of asymmetrical double-barriers may be written as [11] − 12

T2a = [T1−1 T2−1 + (T1−1 − 1)(T2−1 − 1) − 2T1

−1

T2 2 (T1−1 − 1) 2 (T2−1 − 1) 2 cos(θc )]−1 , 1

1

(15)

where θc = θ1,R + θ2,L + 2κw L w f , (κw =

[2m ∗w (E

(16)

1 2

− Vw )] /h¯ )

L w f is the width of the flat well region in the double-barrier structure, m ∗w is the effective mass, Vw is the potential height, T1 is the transmission coefficient of the left-hand barrier in the double-barrier structure, T2 is the transmission coefficient of the right-hand barrier, θ1,R is the reflection phase difference at the right edge of the left barrier, θ2,L that at the left edge of the right barrier, θc is called the circular phase difference in the flat well region [12]. The condition θc = 2nπ,

(n : 0, 1, 2 . . .),

(17a)

gives the jth resonance energy level (Er, j ) ( j = n + 1, j = 1, 2, . . .), at which a resonant peak occurs and the resonant peak becomes unity for T1 (Er, j ) = T2 (Er, j ).

(17b)

To obtain the resonance width (half width at half maximum) and the height of the resonant peak, we rewrite (15) as follows TM , (18) T2a = 1 + [Ds sin(θc /2)]2 where T1 T2 , (19a) TM = [1 − (1 − T1 )1/2 (1 − T2 )1/2 ]2 and 2(1 − T1 )1/4 (1 − T2 )1/4 . (19b) Ds = 1 − (1 − T1 )1/2 (1 − T2 )1/2 It is seen that eqn (18) becomes 0 = TM (Er, j ), T2a

(20)

when condition (19a) holds. Because the variations of T1 (E) and T2 (E) are much less than the variation of T2a near each resonant peak, i.e. T1 (E) ≈ T1 (Er, j ) and T2 (E) ≈ T2 (Er, j ) near the resonant peak, TM (Er, j ) may be regarded as the height of the jth resonant peak. If the function Ds sin(θc /2) near a point E = Er, j is expressed approximately as # " d sin(θc /2) Ds sin(θc /2) ≈ Ds (Er, j ) sin(nπ ) + (E − Er, j )Ds (Er, j ) dE E=Er, j Ds (Er, j ) dθc , (21) = (E − Er, j ) 2 d E E=Er, j

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Region 1

2

3

4

5

6

7

E 0

A 1Vb3 1Vb2 E 0

0.3 eV

1Vb1

Lw

Lb2

Lb1

B2

B1

0.6 eV

1Vb4

Lb3

B

Lb4 B4

B3 BR

BL

Fig. 7. Potential energy diagram of some complicated double-barrier structures.

then the transmission coefficient of a double-barrier structure near the jth resonant peak can be expressed as T2a, j = where 1Er, j =

TM (Er, j )  2 , E−E 1 + 1Er,r,j j

!−1 Ds (Er, j ) dθc , 2 d E E=Er, j

(22)

(23)

1Er, j is the resonance width of the jth resonant peak, by which the resonant state lifetime may also be estimated [4].

5. Discussion on the transmission spectrum In this section, we will calculate the transmission coefficients and investigate the shape of the transmission spectrum for the double-barrier structure GaAs/Ga0.7 Al0.3 As/GaAs/Ga0.4 Al0.6 As/GaAs in which the GaAs well is sandwiched between the Ga0.7 Al0.3 As and Ga0.4 Al0.6 As 0-point potential barriers. It is assumed that the conduction-band offset at the heterojunction interfaces between Ga0.4 Al0.6 As and GaAs is 0.6 eV, the

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1

Transmission coefficient

10–1

10–2 2 10–3

10–4

1

10–5 0.00

0.05

0.10

0.15

0.20

0.25

Energy (eV) Fig. 8. Transmission coefficient versus electron energy in some complicated double-barrier structures (1, L b1 = 1 nm, 1Vb1 = 0.05 eV, L b2 = 5 nm, 1Vb2 = 0.25 eV, L b3 = 5 nm, 1Vb3 = 0.25 eV, L b4 = 1 nm, 1Vb4 = 0.05 eV and L w = 1.564 nm; 2, L b3 = 2.975 nm, 1Vb3 = 0.1488 eV, L w = 1.765 nm) and the others are the same as the above parameters.

effective mass in Ga0.4 Al0.6 As 0.117m 0 which is obtained from m ∗ (Ga0.4 Al0.6 As) = m ∗ (GaAs) + 0.6 × [m ∗ (AlAs) − m ∗ (GaAs)] (m ∗ (AlAs) = 0.15m 0 ) and the dielectric constants of Ga0.7 Al0.3 As equals that of Ga0.4 Al0.6 As. The schematic energy diagram of the double-barrier structure is shown in Fig. 7A, where an electric field is applied and the conduction-band of the substrates (regions 2 and 6), are bent due to the accumulation effect of electrons [15]. Considering the confined effect in the well (region 4), we know that the density of the electron in the well is far greater than that in the other regions when the resonant transmission occurs in the double-barrier structure. This means the applied electric field in the well is much less than that in other regions in this case. Thus, the schematic energy diagram shown in Fig. 7A may be simplified to that shown in Fig. 7B, in which the left-side barrier in the double-barrier structure is formed of two trapezoidal single-barriers (B1 and B2 ), i.e. a compound single-barrier, the right-side barrier is formed of B3 and B4 . The simplified energy diagram will be used to calculate the transmission coefficient for the double-barrier structure under an applied electric field 1Vb (= 1Vb1 + 1Vb2 + 1Vb3 + 1Vb4 ). First, the transmission coefficients versus electron energy are shown respectively by curve 1 in Fig. 8 for the double-barrier structure (Vb1 = Vb4 = 0.05 eV, L b1 = L b4 = 1 nm, L b2 = L b3 = 5 nm and L w = 1.564 nm) in which the under-unity resonance occurs at E = 0.05 eV (i.e. θc (0.05 eV) = 0 is satisfied) and by curve 2 in Fig. 8 for the double-barrier structure (Vb1 = Vb4 = 0.05 eV, L b1 = L b4 = 1 nm, L b2 = 5 nm, L b3 = 2.975 nm and L w = 1.765 nm) in which the unity resonance occurs at E = 0.05 eV (i.e. both θc (0.05 eV) = 0 and T1 (0.05 eV) = T2 (0.05 eV) are satisfied simultaneously). Next, the transmission coefficient versus electron energy calculated by eqn (18) (solid line) and by eqn (22) (square dots) near the resonant peak of curve 1 and curve 2 in Fig. 8 is shown in Figs 9 and 10 respectively. It is seen that the energy variation of the transmission coefficient shown by the square dots agrees very well with that of the solid curve. Therefore, it is considered that the transmission spectrum is Lorentzian in shape and that (22) is a good approximation to (18) in the neighbourhood of resonance energies. These mean that

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Transmission coefficient

0.25

0.2

0.15

0.1

0.05

0 0.046 0.047 0.048 0.049 0.050 0.051 0.052 0.053 0.054 Energy (eV) Fig. 9. Transmission coefficient versus electron energy calculated by (18) (solid line) and by (22) (square dots) near the under-unity resonant peak of Fig. 8.

1

Transmission coefficient

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.046 0.047 0.048 0.049 0.050 0.051 0.052 0.053 0.054 Energy (eV) Fig. 10. Transmission coefficient versus electron energy calculated by (18) (solid line) and by (22) (square dots) near the unity resonant peak of Fig. 8.

eqn (19a) is valid for calculating approximately the height of the resonant peak and eqn (23) is valid for the resonance width in the asymmetrical double-barrier structure with arbitrary potential.

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6. Conclusion In this paper, generalized-barrier parameters are studied theoretically in detail for rectangular and trapezoidal single-barriers as well as parallel-plane ones. The expressions of the generalized-barrier parameters are derived in the analytical forms: (7a)–(7c). The energy variation of the transmission coefficient and that of the reflection phase difference are examined and shown for the trapezoidal single-barrier by the Airy function method, the rectangular single-barrier approximation and the new WKB approximation relations in Figs 4–6. It is understood that the rectangular single-barrier approximation may be a useful and convenient approximation for estimating the generalized-barrier parameters of the parallel-plane single-barrier structure. The new WKB approximation relations are also useful for estimating these parameters when the electron energy is smaller than the minimum of the barrier height or greater than the maximum of that. The analytical expressions of the generalized-barrier parameters are important for studying the tunnelling transmission characteristics in the double- and the multiple-barrier structures as well as optical multilayer thin-film ones. With the aid of these expressions, the resonance width and height of resonant peak as well as the transmission spectrum are also investigated for asymmetrical double-barrier structures. The expressions of the height of resonant peak and the resonance width are given in (19a) and (23). It is confirmed that the shape of the transmission spectrum for asymmetrical double-barrier structures is a Lorentzian near resonance energies as expressed by (22).

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]

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Appendix A: Independent coefficients for some single-barrier structures For the rectangular barrier shown in Fig. 2B a = cosh(β L b ), 1 b = − sinh(β L b ), β´ c = −β´ sinh(β L b ),

(A1) (A2) (A3)

1322

Superlattices and Microstructures, Vol. 23, No. 6, 1998

d = cosh(β L b ),

(A4)

where β´ = β/m ∗b . For the parallel-plane single-barrier shown in Fig. 2C at the cases E < Vmin and E > Vmax  Z Lb  a = cosh β(x) d x ,

(A5)

0

 Z Lb  1 sinh β(x) d x , b = −q 0 ´ β(L ´ b) β(0)  Z Lb  q ´ β(L ´ b ) sinh β(x) d x , c = − β(0) Z

Lb

d = cosh



(A7)

0

β(x) d x ,

0

where ´ β(x) = β(x)/m ∗b ,

(A6)

(x ∈ [0, L b ]).

(A8)