Phase time in Kane type barrier tunneling

Physics Letters A 374 (2009) 139–143

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Physics Letters A www.elsevier.com/locate/pla

Phase time in Kane type barrier tunneling S. ¸ Çakmaktepe ∗ , M. Boztas¸ Physics Department, Kilis 7 Aralik University, Kilis, Turkey

a r t i c l e

i n f o

Article history: Received 26 May 2009 Received in revised form 2 September 2009 Accepted 21 October 2009 Available online 24 October 2009 Communicated by P.R. Holland

a b s t r a c t The behavior of Wigner phase delay time in the reflection mode is studied taking into account the real band structure of Kane type semiconductor quantum ring. It’s calculated the analytical expression for the saturated delay time. It’s shown that the saturated delay time is independent of the width of the opaque barrier. © 2009 Elsevier B.V. All rights reserved.

PACS: 03.65.-w 73.40.Gk 84.40.Az 73.23.-b Keywords: Tunneling Electronic transport

1. Introduction Electronic and optical properties of semiconductor nanostructures such as quantum wires and rings have become a very active field of research with respect to both theory and experiment [1]. Quantum tunneling is one of the most important phenomenons with wide range of applications in modern technology. The tunneling time is essential for designing the frequency response of many types of optoelectronic devices such as lasers, photo detectors, modulators, etc. In spite of its remarkable success in different applications, there still remains the basic question, calling, how much time a particle takes to transverse the barrier (so-called tunneling time problem). The immerse potential of this concept has lead to the study of various timescales to understand the time a particle takes to tunnel through a barrier [2]. Some of them include dwell time, Larmor time, complex times, sojourn times [3], timescale based on Bohm’s view [4], phase delay time, etc. In the present study, it’s concentrated mainly on the concepts of Wigner phase delay time. Phase delay time is usually taken as the difference between the times at which the peak of the incident Gaussian wave packet arrives at the barrier. The phase delay time (τ ) is expressed in terms of the derivative of phase shift of the scattering matrix with respect to energy. Since it’s in-

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ception, Wigner phase delay time has been a quantity of interest from a fundamental as well as technological point of view [5]. In the case of a quantum tunneling it has been shown that in the opaque barrier limit the phase delay time does not depend on the barrier width. This phenomenon is called as Hartman effect [6]. This implies that for sufficiently large barriers the effective velocity of the particle can become arbitrarily large, even higher than the speed of the light in the vacuum. It’s demonstrated by experiments [7–9] that tunneling photons travel with superluminal group velocities. Theoretically Hartman effect has been generalized to different cases including double barriers, various geometric structures and in the presence of Aharonov–Bohm flux. The model of a ring coupled to a wire which in turn is connected to an electron reservoir was originally put forth by M. Buttiker [10] to investigate the effect of dephasing on the persistent current. In subsequent work Akkermans et al. [11] related the reflection phase directly to thermodynamic quantities. Mystery surrounding of this effect has also been addressed recently by Winful [12–14]. He thinks that the observed short time delay is due to energy storage and release and has nothing to do with propagation. He argued that the tunneling particle or wave packet is not really traveling with superluminal velocity but actually a standing wave. Büttiker and Landauer [15] raised objection that the peak is not a reliable characteristic of packets distorted during the tunneling process. The ‘phase time’ statistics is intimately connected with dynamic admittance of microstructures [16]. This ‘phase time’ is also directly related to the density of states. A simple semi classical derivation

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S. ¸ Çakmaktepe, M. Boztas¸ / Physics Letters A 374 (2009) 139–143

Fig. 1. Schematic diagram of an InSb quantum ring in the presence of Aharonov– Bohm flux.

of the Hartman effect valid for the potential of a barrier of general shape in one dimension is presented in [17]. Bandopadhyay et al. [18] verified the Hartman effect beyond one dimension and in the presence of Aharonov–Bohm flux. In another study Bandopadhyay and Jayannavar [19] studied the phase delay time for tunneling in the reflection mode. Their system consists of a circular loop connected to a single wire of semi-infinite length in the presence of Aharonov–Bohm flux. Çakmaktepe studied the tunneling delay time in the transmission mode taking into account the real band structure of InSb type semiconductor quantum ring and compared with that of a parabolic band structure [20]. However, the experimental advantages of using narrow-gap semiconductors for the reduced dimensionality systems make it necessary to account for the real band structure of these materials. In the current Letter, we study analytically the above mentioned effect in a simple semiconductor quantum ring consisting of a circular loop connected to a single wire of semi-infinite length as shown in Fig. 1 taking into account both parabolic and the real band structure of InSb type (Kane model; narrow band gap, strong spin–orbit interaction) semiconductors. The Kane problem determines the energy spectrum taking into consideration the interaction of three bands; conduction band, valence band and spin–orbit splitting band. Valence and conduction band interaction is characterized via the only matrix element P (so-called Kane’s parameter). The systems of Kane equations including the potentials have the form [21,22]:

P k−



−( E − V )ψ1 − √ ψ3 + 2

P kz

P k+

3

3

P k−



P k+ P k z ψ4 + √ ψ5 3 6

−( E − V )ψ2 − √ ψ4 + 6

P k−

P kz

3

3

3

P k z ψ5 + √ ψ6 2

P k+

− √ ψ1 − ( E − V + E g )ψ3 = 0,  

2

2 3 2 3

P k+

P k z ψ1 − √ ψ2 − ( E − V + E g )ψ4 = 0, 6 P k−

P k z ψ2 + √ ψ1 − ( E − V + E g )ψ5 = 0, 6

P k−

√ ψ2 − ( E − V + E g )ψ6 = 0, 2

P kz

P k+

3

3

√ ψ1 + √ ψ2 − ( + E − V + E g )ψ7 = 0,

P k−

√

3

P kz

· ψ1 − √ ψ2 − ( + E − V + E g )ψ8 = 0. 3

 −

h¯ 2 2m0

where

∇2 −

εp =

3E ( E + E g )( E + E g + )

ε p (3E + 3E g + 2)

2m0 h¯ 2

 ψ1 = 0,

(9)

P 2 = 23.42 eV for InSb and P is the Kane parame-

ter [25]. In the stationary case the incoming particles are represented by a plane wave e ikx of unit amplitude. The wave function in different regions (which are solutions of Kane equations) in the absence of magnetic flux are as follows:

ψ0 (x0 ) = e ikx0 + Re −ikx0

(in region I),

ψ3 (x3 ) = A 3 e

iq3 x3

+ B 3e

−iq3 x3

(in region II), (in region III), (in region IV).

(10) (11) (12) (13)

Here R is the reflection coefficient and k is the wave vector of electrons in the lead and in the intermediate free space between two barriers inside the ring

P k+

+ √ ψ 7 − √ ψ 8 = 0,

In the present paper, the scattering problem is approached using the quantum waveguide theory for the mesoscopic structures of waveguide type [23,24]. Substituting expressions (3)–(8) into formulas (1)–(2) and assuming that k x = k y = 0, we obtain the following equation in the z-direction in one-dimensional form;

ψ2 (x2 ) = A 2 e ikx2 + B 2 e −ikx2 (1)

2

2. Theory

ψ1 (x1 ) = A 1 e iq1 x1 + B 1 e −iq1 x1

2

+ √ ψ 7 + √ ψ 8 = 0,

Here, ψi are envelope functions, P is the Kane parameter, E g the band gap energy,  the value of spin orbital splitting, and k± = . −i ∇ The system considered in the present study is schematized in Fig. 1. As it’s seen in Fig. 1, there is a potential barrier (or barriers) inside the loop, while the potential in the connecting lead is set to be zero. In addition, we impose Aharonov–Bohm flux through the loop. The incident energy of the free propagating electron E in the semi-infinite wire is less than the barrier potential height V . The impinging electrons in sub-barrier regime travels as an evanescent mode thought the circumference of the ring and the reflection or the conductance involve contributions from both Aharonov–Bohm effect as well as quantum tunneling. We analyze the phase time of the reflected wave. It’s shown for an InSb quantum ring that this phase time in the opaque barrier regime becomes independent of the length of the circumference of the ring and the magnitude of the Aharonov–Bohm flux so generalized the Hartman effect. Analysis of the phase delay time in the present study, are studied following the method described by Bandopadhyay et al. [19].

 (2) (3)

(4)

k=

2m0 3E ( E + E g )( E + E g + )

 q1 = and

h¯ 2

,

2m0 3( V 1 − E )( V 1 − E − E g )( V 1 − E − E g − ) h¯ 2



ε p (3E + 3E g + 2)

ε p (3( V 1 − E ) + 3E g + 2)

2m0 3( V 2 − E )( V 2 − E − E g )( V 2 − E − E g − )

(14)

(15)

(5)

q2 =

(6)

are the wave vectors in the InSb semiconductor quantum ring for the non-parabolic band model where V 1,2 = 0. It’s accepted that E  E g in Eqs. (14) and (15) to obtain the wave vector in the lead and the wave vector in InSb semiconductor quantum ring for the parabolic band model. The origin of the coordinates of x0 and x1 is assumed to be at J and that for x2 and x3 are at P 1 and P 2 , respectively. At P 1 , x1 = lb1 , at P 2 , x2 = w and

(7) (8)

h¯ 2

ε p (3( V 2 − E ) + 3E g + 2)

(16)

S. ¸ Çakmaktepe, M. Boztas¸ / Physics Letters A 374 (2009) 139–143

at J , x3 = lb3 , where lb1 and lb3 are the length of the two barriers separated by a well region of length w inside the ring. The total circumference of the ring is L = lb1 + lb3 + w. Griffith’s boundary conditions are used to solve this problem [26];

ψ0 (0) = ψ1 (0) = ψ3 (0)

(17)

and

   ∂ψ1 (x1 )  ∂ψ3 (0)  ∂ψ0 (x0 )  + + =0 ∂ x0  J ∂ x1  J ∂ x3  J

1

P1

2

(19)

= 0,

(20)

P1

at P 2 :

ψ1 ( w ) = ψ3 (0),   ∂ψ3 (x3 )  ∂ψ2 (x2 )  +   ∂x ∂x 2

P2

3

(23)

A 3 exp(−κ3lb3 ) exp(i α3 ) + B 3 exp(κ3lb3 ) − 1 − R = 0,

(24)



 ik(1 − R ) + κ1 A 1 − B 1 exp(−i α1 )

− κ3 A 3 exp(−κ3lb3 ) exp(i α3 ) − κ3 B 3 exp(κ3lb3 ) = 0,

(25)

A 1 exp(−κ1lb1 ) exp(i α1 ) + B 1 exp(−κ1lb1 ) − A 2 (26)

κ1 A 1 exp(−κ1lb1 ) exp(i α1 ) + κ1 B 1 exp(κ1lb1 ) − ik A 2 − ikB 2 exp(−i α2 ) = 0,

at the junction J . All the derivatives are taken either outward or inward from the junction [23]. Similar boundary conditions are held at P 1 and P 2 i.e. at P 1 :

ψ1 (lb1 ) = ψ2 (0),   ∂ψ2 (x2 )  ∂ψ1 (x1 )  +   ∂x ∂x

1 + R − A 1 − B 1 exp(−i α1 ) = 0,

− B 2 exp(−i α2 ) = 0, (18)

141

(27)

A 2 exp(ikw ) exp(i α2 ) + B 2 exp(−ikw ) − A 3

− B 3 exp(−i α3 ) = 0,

(28)

ik A 2 exp(ikw ) − ikB 2 exp(−ikw ) exp(−i α2 ) − κ3 A 3

− κ3 B 3 exp(−i α3 ) = 0

(29)

with κ1 and κ3 being the imaginary wave vectors in the presence of rectangular barriers of strength V 1 and V 3 respectively inside the ring. 3. Results and discussion

(21)

= 0.

(22)

P3

A gauge for the vector potential is chosen in which the magnetic field appears only in the boundary conditions rather than explicitly in the Hamiltonian [23]. The electrons propagating clockwise and anticlockwise will pick up opposite phases. The electrons propagating in the clockwise direction from J will pick up phases i α1 at P 1 , i (α1 + α2 ) at P 2 and i (α1 + α2 + α3 ) at J after traversing once along the loop. The total phase around the loop is α1 + α2 + α3 = 2πϕ /ϕ0 , where ϕ and ϕ0 are the magnetic flux and flux quantum, respectively. Hence, by the above boundary conditions we get propagating waves [24,27,28];

For the evanescent regime in the problem, the wave vector q in the loop is replaced by i κ . Once R is known the reflection phase time τ can be calculated from the energy derivative of the phase of the reflection amplitude [5] as

τ = h¯

∂ Arg[ R ] . ∂E

(30)

The concept of phase time was first introduced by Wigner [5] to estimate how long a quantum mechanical wave packet is delayed by the scattering obstacle. To obtain an analytical expression for the reflection amplitude for a semiconductor quantum ring system as shown in Fig. 1, it’s solved Eqs. (23)–(29) using C = 0, D = 0, A 2 = A 1 , B 2 = B 1 ,

Fig. 2. For an InSb ring with a barrier of strength of V 1 through its circumference, the phase time τ is plotted as function of ring’s circumference L in the absence of magnetic flux. Incident energy is set to be E = 1. (For interpretation of the references to color in this figure, the reader is referred to the web version of this Letter.)

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Fig. 3. Plot of τ versus L for parabolic and Kane models in an InSb quantum ring, with E = 1, referred to the web version of this Letter.)

ϕ = 0. (For interpretation of the references to color in this figure, the reader is

Fig. 4. Plot of τ versus ϕ in InSb quantum ring, for different L in the Kane model. (For interpretation of the references to color in this figure, the reader is referred to the web version of this Letter.)

κ1 = κ2 in the wave functions (12) and (13) and x1 = L at J 1 after a complete transversal along the circumference. The reflection amplitude is

R=

−κ1 (2 Cos α − exp(kL )) + i 2k exp(kL )

κ1 (2 Cos α − exp(kL )) + i 2k exp(kL )

(31)

where α = α1 + α2 + α3 . Let’s set the units of h¯ and 2m unity. All the physical quantities are taken in dimensionless units, i.e. all the barrier strengths V n in units of band gap energy E g (V n ≡ V n / E g ), all the barrier widths lbn in units of inverse wave vector k−1 (lbn ≡ klbn ), where k = E g ,

and the reflection phase time τ in units of inverse band gap energy E (τ ≡ E g τ ). We can now proceed to analyze the behavior of τ as a function of physical parameters. In Fig. 2, phase time τ for the reflection mode is shown as a function of L of an InSb semiconductor quantum ring by using the non-parabolic model for different values of incident energies in the absence of magnetic flux ϕ . In Fig. 2, phase time τ is shown as a function of the length L of an InSb semiconductor quantum ring using the Kane model for different values of incident energies. We see that in the absence of magnetic flux τ evolves as a function of L and asymptotically saturates to a value τs which are independent of L confirming the

S. ¸ Çakmaktepe, M. Boztas¸ / Physics Letters A 374 (2009) 139–143

Hartman effect. The saturation value τs increases with increasing incident energy and the corresponding τs values for V = 4, 5, 5.25, 5.5 are 0.99, 0.52, 0.46, 0.42, respectively. In Fig. 3, the phase time τ is plotted as a function of the length of the InSb semiconductor quantum ring by using both parabolic and Kane models, again in the absence of magnetic flux. It can be seen easily that two models have different τs saturation values where τs = 0.52 for Kane model and τs = 0.37 for parabolic band model. Fig. 4 is plotted to see the effect of magnetic flux on Hartman effect. The same system is considered but in the presence of Aharonov–Bohm flux. It’s seen from Fig. 4 that τ is periodic with periodicity ϕ0 . It can be also seen from Fig. 4 that, in the large length limit the visibility of these oscillations vanishes. Fig. 4 indicates that the reflection time in the presence of opaque barrier is independent of Aharonov–Bohm flux. Since the mean value of reflection phase time isn’t changed for different values of L. 4. Conclusion The Hartman effect is generalized in the presence of Aharonov– Bohm flux taking into account the real band structure of InSb type semiconductors. It’s done by studying the phase delay time in a circular geometric ring in the reflection mode. It’s shown that the phase delay time for a given incident energy does not depend on the barrier thickness or on the magnetic flux. It’s also shown that parabolic and non-parabolic band models have different saturated delay times. Our reported results may be beneficial in the further understanding of quantum mechanical tunneling time and the designing new high-speed quantum electronic devices and the frequency response of many types of electronic devices such as lasers, photo detectors and modulators.

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