Resonance-enhanced group delay times in an asymmetric single quantum barrier

Chun-Fang Li1 Department of Physics, Shanghai University, 99 Shangda Road, Shanghai 200444, People’s Republic of China State Key Laboratory of Transient Optics Technology, Xi’an Institute of Optics and Precision

arXiv:quant-ph/0404080v2 5 Jan 2005

Mechanics, Academia Sinica, 322 West Youyi Road, Xi’an 710068, People’s Republic of China

Abstract

It is shown that the transmission and re?ection group delay times in an asymmetric single quantum barrier are greatly enhanced by the transmission resonance when the energy of incident particles is larger than the height of the barrier. The resonant transmission group delay is of the order of the quasibound state lifetime in the barrier region. The re?ection group delay can be either positive or negative, depending on the relative height of the potential energies on the two sides of the barrier. Its magnitude is much larger than the quasibound state lifetime. These predictions have been observed in a microwave experiment by H. Spieker of Braunschweig University. PACS numbers: 03.65.Xp, 73.23.Ad

1 Mailing

address: Department of Physics, Shanghai University, 99 Shangda Road, Shanghai 200444, People’s

Republic of China. Email: c?i@sta?.shu.edu.cn

1

The tunnelling time of particles through single or multiple quantum barriers has drawn much attention [1, 2, 3, 4, 5] with the advent of techniques for the fabrication of semiconductor tunnelling devices, such as single-electron tunnelling transistors [6], resonant tunnelling diodes [7], quantum cascade lasers [8], and resonant photodetectors [9]. Theoretical [10, 11, 12, 13, 14, 15, 16, 17, 18, 19] as well as experimental [2, 20, 21, 22, 23, 24, 25, 26, 27] investigations have been extensively made on this problem. It was found that the group delay (also referred to as the phase time in the literature [1]) for particles tunnelling through a potential barrier saturates to a constant value in the opaque limit [4, 28, 29, 30]. This is the so-called Hartman e?ect [31]. The re?ection group delay is the same as the transmission one in a symmetric con?guration [15]. It will be shown in this Letter that the re?ection group delay from a single barrier of asymmetric con?guration can be negative and is greatly enhanced by the transmission resonance. The negative group delays in both re?ection and transmission were previously discovered. But they all occur in quantum-well structures, such as single quantum wells [32, 33, 34], double-barrier quantum wells [35, 36], and their optical analogues [5, 37]. What we consider here is such a case in which particles are scattered by an asymmetric single barrier, the height of which is less than the energy of incident particles. Quasibound states were predicted and observed in such a situation [38]. This is a classically allowed motion. The particle in the barrier region has a real classical moving velocity which speci?es a classical traversal time τc . It is found that the transmission and re?ection group delays are both enhanced by transmission resonance. The re?ection group delay can be either positive or negative, depending on the relative height of the potential energies on the two sides of the barrier. The negative resonant re?ection group delay corresponds to a transmission probability that is larger than 1. The resonant transmission group delay is of the order of the quasibound state lifetime in the barrier region and is larger than the classical time τc . And the magnitude of resonant re?ection group delay is much larger than the lifetime of the quasibound state in the barrier region. The re?ected wave packet is considered without taking into account the interference between the incident and the re?ected waves [1, 35]. The height of the potential barrier, extending from 0 to a, is V0 . The values of the potential energies on the left and right handed sides of the barrier are V1 and V2 , respectively. It is assumed that V0 > V1 and V0 > V2 . Let a beam of particles be incident from the left, and let be ψin (x) = A exp(ik1 x) the Fourier component of the incident wave packet, where k1 = [2?(E ? V1 )]1/2 /? , h and ? is the mass of incident particles. In the following, we will assume that the energy, E, of incident particles is larger than the height, V0 , of the potential barrier. Denoting, respectively, by B exp(?ik1 x) and F exp[ik2 (x ? a)] the corresponding Fourier components of the re?ected and transmitted wave packets, then the Schr¨dinger equation and boundary conditions at x = 0 and o x = a give r ≡

B A g1 g2 F A 1 g2

=

exp[i(φ2 ? φ1 )], and t ≡

=

exp(iφ2 ), where k2 = [2?(E ? V2 )]1/2 /? , h

2

k0 = [2?(E ? V0 )]1/2 /? , non-negative number g1 and real number φ1 are de?ned by a complex h number as follows, g1 exp(?iφ1 ) = i k2 k0 k2 1 (1 ? ) cos(k0 a) ? ( ? ) sin(k0 a), 2 k1 2 k0 k1 (1)

and non-negative number g2 and real number φ2 are de?ned similarly by another complex number as follows, g2 exp(?iφ2 ) = According to de?nition (1), we have tan φ1 = 1/k0 ? k0 /k1 k2 tan(k0 a), 1/k2 ? 1/k1 (3) i k2 k0 1 k2 (1 + ) cos(k0 a) ? ( + ) sin(k0 a). 2 k1 2 k0 k1 (2)

which shows that φ1 will change its sign by exchanging k1 and k2 . This property will have important e?ect on the group delay of re?ected particles. According to Eq. (2), we have tan φ2 = 1/k0 + k0 /k1 k2 tan(k0 a), 1/k2 + 1/k1 (4)

which shows that φ2 is symmetric between k1 and k2 . We can also see from Eqs. (3) and (4) that φ1 and φ2 can be exchanged from one to another by changing the sign of k1 . This symmetry between φ1 and φ2 will simplify our calculation of the group delay. First, let us look at the group delay τt of transmitted particles. It is de?ned as h(dφ2 /dE) [39, 15] ? and is given by τt =

2 τc k2 k2 k0 k0 k2 k0 sin 2k0 a 2 (1 + k )[ k + k ? (1 ? k 2 )( k ? k ) 2k a ], 4g2 1 0 1 0 2 0 1

where τc = a/vc is the time taken for classical particles to travel through the barrier region, vc =

1 h(dk0 /dE) ?

=

hk0 ? ?

is the classical velocity of the particles in the barrier region. It is clear that τt is in

general di?erent from the classical time τc . Furthermore, it is easy to show that τt can be larger as well as less than τc . In fact, when k0 a = mπ (m = 1, 2, 3 . . .), τt reduces to τtmax ≡ τt |k0 a=mπ =

2 k1 k2 + k0 τc . k0 (k1 + k2 )

(5)

If the energy of incident particles is so close to the height of the potential barrier that k0 is much less than k1 and k2 , τtmax will be larger than τc . On the other hand, when k0 a = (m + 1/2)π, τt becomes τt |k0 a=(m+1/2)π =

1+k2 /k1 k2 /k0 +k0 /k1 τc .

It is less than τc under the above-mentioned condition.

In Fig. 1 is shown the dependence of τt upon the thickness a of the barrier, where V0 /E = 0.95, V1 /E = 0, V2 /E = 0.3, and a is re-scaled to be k0 a. We see that τt is maximum at k0 a = mπ. It will be useful to introduce a quantity T , called transmission probability, which is de?ned as T ≡ |t|2 =

2 k0 (k1 2 (k1 2 2 4k0 k1 . 2 2 2 ? k0 )(k2 ? k0 ) sin2 k0 a

+ k2

)2

+

(6)

3

2.5 2 1.5 τt 1 0.5 0

0

2

4 k0a

6

8

10

Figure 1: The dependence of the transmission group delay τt upon the thickness a of the barrier, where V0 /E = 0.95, V1 /E = 0, V2 /E = 0.3, a is re-scaled to be k0 a, and τt is in units of τc . The transmission coe?cient, de?ned as the ratio of the transmitted probability current density to the incident probability current density [40], is then (k2 /k1 )T . When k0 a = mπ, T reaches its maximum, Tmax =

4 (1+k2 /k1 )2 .

The resonance condition k0 a = mπ for transmission through a single

barrier is the same as that for the quasilocalization of the states in the barrier region [38]. It can be shown that the resonant transmission time (5) is of the order of the quasibound state lifetime in the barrier region [12]. The symmetry of φ2 between k1 and k2 means that the transmission group delay is the same whether the incident particles come to the barrier from left-handed or right-handed side [15]. Now let us pay our attention to the re?ection group delay, which is τr = τt + τ1 as is seen from the expression of r, where τ1 = ?? h and

2 g1 =

dφ1 k2 k2 k0 k 2 k2 k0 sin 2k0 a τc ? (1 ? 0 )( ? ) = ? 2 (1 ? )[ ? ], 2 dE 4g1 k1 k0 k1 k1 k0 k2 2k0 a 1 k2 k0 k2 1 (1 ? )2 cos2 (k0 a) + ( ? )2 sin2 (k0 a). 4 k1 4 k0 k1

(7)

(8)

Note: (1) When the energy of incident particles is so close to the height of the potential barrier that k1 ? k2 and k0 ? k2 , the second term on the right-handed side of Eq. (8) is usually much larger

2 than the ?rst term unless k0 a = mπ, at which g1 is minimal. As a result, near its maximum, τ1 can

be approximated as τ1 ≈ ? k2 k2 k0 τc 2 (1 ? k )( k ? k ). 4g1 1 0 1

k k ?k2

(9)

And its maximum has a value of τ1max = ? k01 2 ?k0 ) τc . Comparing with Eq. (5), we see that the (k1 2 magnitude of τ1max is much larger than the resonant transmission group delay, which means that

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the re?ection group delay, τr , is dominated by τ1 near resonances, and its sign is determined by the sign of τ1 . This shows that the magnitude of the resonant re?ection group delay is much larger than the quasibound state lifetime in the barrier region. It is clear from Eq. (9) that τ1 can be negative as well as positive. When k1 > k2 (V1 < V2 ), τ1 (and hence τr ) is negative. On the other hand, when k1 < k2 (V1 > V2 ), τ1 (and hence τr ) is positive. These properties of τ1 can also be inferred from Eq. (3). As a result, when the re?ection group delay for incidence from the left is positive, it is negative for incidence from the right for the same con?guration. (2) When k1 > k2 , Tmax > 1. This shows that the negative peaks of the re?ection group delay correspond to a transmission probability that is larger than 1. The fact that the transmission probability can be larger than 1 is not at odds with the law of probability conservation. It is the probability current density, rather than the probability itself, that is in direct connection with probability conservation in quantum scattering. In fact, the transmission coe?cient, (k2 /k1 )T , is always less than 1. (3) The re?ection coe?cient |r|2 does not vanish at the transmission resonance, so that the re?ected wave packet is well de?ned under the condition that follows (Eq. (10)). (4) For the case of far from resonances, k0 a = (m + 1/2)π, τ1 becomes τ1 |k0 a=(m+1/2)π =

1?k2 /k ? k2 /k0 ?k0 1 1 τc . Under the above mentioned conditions (k1 ? k2 and k0 ? k2 ), its magnitude is /k

much smaller than the corresponding transmission group delay, τt |k0 a=(m+1/2)π . This shows that the re?ection group delay is almost the same as the transmission one when the energy of incident particles is far from resonances. In Fig. 2 is shown the dependence of τr upon the thickness a of the barrier, where a is rescaled to be k0 a. The solid curve corresponds to negative-peak group delay, where all the physical parameters are the same as in Fig. 1. The dashed curve corresponds to positive-peak group delay, where V0 /E = 0.95, V1 /E = 0.3, and V2 /E = 0. It is seen that the peaks of the group delay occur at k0 a = mπ and are much larger than the peaks of the transmission group delay whether they are negative or positive. The symmetries of φ1 and φ2 between k1 and k2 discussed before mean that the transmission group delay and re?ection group delay satisfy the average principle [15], τr1 + τr2 = 2τt , where τr1 and τr2 are re?ection group delays for the incident particles coming from left-handed side and right-handed side, respectively. Next, we discuss the validity of the above theoretical results. To this end, let us look at the dependence of the re?ection group delay on the energy of incident particles, which is shown in Fig. 3, where V1 = 0, V2 = 0.3V0 , E ∈ [V0 , 1.15V0 ], a = 10/(0.3?V0 )1/2 , and the incidence energy E is re-scaled to be k0 a. The half width of the peak of the re?ection group delay, which can be

5

30 25 20 15 10 5 0 ?5 ?10 ?15 ?20 ?25

τr

0

1

2

3

4

5 k0a

6

7

8

9 10

Figure 2: The dependence of the re?ection group delay τr upon the thickness a of the barrier, where a is re-scaled to be k0 a, and τr is in units of τc . The solid curve corresponds to the negative-peak group delay, where all the physical parameters are the same as in Fig. 1. The dashed curve corresponds to the positive-peak group delay, where V0 /E = 0.95, V1 /E = 0.3, and V2 /E = 0. approximately obtained from its dominant part τ1 , is ?E = k0 |k1 ? k2 | h ? sin?1 . 2 2 2 2 τc [(k1 ? k0 )(k2 ? k0 )]1/2

For a Gaussian-shaped wave packet, its energy half-width δE is related to its time spreading w by δE · w = h/2. For the above theoretical calculation to be valid, that is, for the distortion of the ? re?ected wave packet to be negligible, it is required that δE ≤ ?E. This results in a restriction on the thickness of the barrier, a ≤ 2vc w sin?1

2 [(k1

k0 |k1 ? k2 | . 2 2 2 ? k0 )(k2 ? k0 )]1/2

(10)

Because of the analogy between Schr¨dinger’s equation in quantum mechanics and Helmholtz’s o equation in electromagnetism [4, 5, 33], the predictions presented here have been observed in a socalled G-band waveguide of width 47.5mm by H. Spieker of Braunschweig University in Germany [41], where the asymmetric barrier structure was obtained by reducing the inside width of the waveguide, leading to e?ective widths of 40.5mm and 30.5mm. The resonance-enhancement of the times is clearly shown, and both the positive and negative resonant peaks of the re?ection time is much larger than the resonant peak of the transmission time. In a word, the re?ection and transmission group delays in an asymmetric single quantum barrier are greatly enhanced by the transmission resonance when the energy of incident particles is larger than the height of the barrier. The re?ection group delay can be negative as well as positive, depending on the relative height of the potential energies on the two sides of the barrier. The negative resonant re?ection group delay corresponds to a transmission probability that is larger than 1. The resonant transmission group delay is of the order of the quasibound state lifetime in the 6

10 5 0 ?5 ?10 ?15 ?20 ?25 ?30 ?35 ?40

τr

0

1

2

3

4

5 k0a

6

7

8

9 10

Figure 3: The dependence of the re?ection group delay τr on the energy E of incident particles, where V1 = 0, V2 = 0.3V0 , E ∈ [V0 , 1.15V0 ], a = 10/(0.3?V0)1/2 , E is re-scaled to be k0 a, and τr is in units of τc . barrier region and is larger than the classical traversal time. The magnitude of resonant re?ection group delay is much larger than the lifetime. These phenomena may have potential applications in electronic devices, such as novel quantum-mechanical delay lines. It should be pointed out that the negative re?ection group delay does not imply a negative propagation velocity. As a matter of fact, the negative group delay results from the reshaping [17, 42] of the re?ected wave packet due to the di?erent phase shifts φ2 ? φ1 for its di?erent Fourier components.

Acknowledgments

The author thanks G. Nimtz for his helpful discussions and suggestions. This work was supported in part by the National Natural Science Foundation of China (Grants 60377025 and 60407007), Shanghai Municipal Education Commission (Grants 01SG46 and 04AC99), Science and Technology Commission of Shanghai Municipal (Grants 03QMH1405 and 04JC14036), and the Shanghai Leading Academic Discipline Program.

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