SCIENCE CHINA Physics, Mechanics & Astronomy • Research Paper •
March 2011 Vol.54 No.3: 388–392 doi: 10.1007/s11433-010-4237-1
Investigation on tunneling in optoelectronic devices with consideration of subwaves WANG XianPing1,2*, YIN Cheng2, SANG MingHuang1, DAI ManYuan1,2 & CAO ZhuangQi2 2
1 Department of Physics, Jiangxi Normal University, Nanchang 330027, China; Department of Physics, the State Key Laboratory on Fiber Optic Local Area Communication Networks and Advanced Optical Communication Systems, Shanghai Jiao Tong University, Shanghai 200240, China
Received August 21, 2010; accepted September 8, 2010; published online January 21, 2011
Since novel optoelectronic devices based on the peculiar behaviors of the tunneling probability, e.g., resonant tunneling devices (RTD) and band-pass filter, are steadily proposed, the analytic transfer matrix (ATM) method is extended to study these devices. For several examples, we explore the effect of the scattered subwaves on tunneling; it is shown that the resonant or band-pass structures in tunneling probability are determined by the phase shift results from the scattered subwaves. quantum tunneling, WKB approximation, the ATM method, subwaves PACS: 03.65.Xp, 73.40.Gk, 85.30.De
Quantum tunneling, where a particle has a probability to penetrate a classically forbidden region, is one of the most striking consequences of quantum mechanics. The Josephson Effect in solid state physics, fusion in nuclear physics, and instantons in high energy physics are all manifestations of this phenomenon. Since the probability of tunneling decreases exponentially with the barrier width and the square root of the particle mass, tunneling is very important for light particles such as electrons in nanostructure. To date novel optoelectronic devices based on the peculiar behaviors of the tunneling probability are constantly proposed. For example, placing two suitable potential wells beside a smooth barrier, one will obtain a fine electronic on/off switch which provides maximum or minimum tunneling probability via adjusting the external electric field [1]. Gaussian function modulated or randomly chosen “ ” shape modulated superlattice structures can be used to function as effective band-pass energy filters which allow the incident electrons to be nearly totally transmitted in the passband and completely reflected in the stopband [2–6]. *Corresponding author (email:
[email protected]) © Science China Press and Springer-Verlag Berlin Heidelberg 2011
By a judicious choice of the single-negative materials, a tunable omnidirectional double-channel filter was numerically demonstrated [7]. Numerous theoretical studies on calculating the tunneling probability of nanostructures have been reported in the literature [8–14]. The most widely used method among them is the semiclassical WKB approximation since its derivative procedure is analytical and the resulting tunneling formula is simple [14]. However, since the first-order WKB wave function ignores all the higher terms of O( n ) , the WKB approximation is only expected to work well near the semiclassical limit 0 . In the previous derivation, ATM quantization condition [15] has led to the concept of scattered subwaves which originate inherently from the inhomogeneity of the potential and correspond to the higher terms of O( n ) which are neglected in the WKB approximation, and then the eigenenergies can be evaluated to arbitrary precision [16–19]. As the phase contribution devoted by the scattered subwaves (denoted by s ) is independent of n and integrable for all known shape invariant potentials (SIPs), the reason why the SWKB approximation is exact for all SIPs has been phys.scichina.com
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made clear [20]. Subsequently, the subwave concept was extended to explore the quantum time issue, and a generalized reflection time formula was obtained [21]. It was found that the Hartman effect in quantum tunneling stems from the decrease of scattered subwaves [22]. Recently, the explicit expression for the reflection and transmission probabilities through an arbitrary potential barrier was proposed [23] and demonstrated that the quantum reflection is nothing but the reflection of subwaves [24]. Thus, a question naturally arises: what is the role of subwaves in the quantum tunneling probability? In this paper, we will show that the variation trend of tunneling probability is similar to that of the imaginary part of S, i.e., both of them display the same resonant or band structures. The examples illustrated in this paper confirm the generality of the proposed conclusion.
1 Theory 1.1
WKB tunneling expressions
The first-order WKB wave function can be written as [14] WKB
i exp p( x)dx , p ( x) 1
(1)
in which all the higher terms of ( n ) are ignored and the term p( x) denotes the local classical momentum. For the case of a particle with energy E incidents on a single barrier with two classical turning points xl , xr , by matching the WKB wave function at these two points, one will obtain the analytical tunneling expression TWKB ( E ) B , 2
(2)
1 xr p( x)dx. Obvi xl ously, eq. (2) approaches unity at the top of the barrier where the two turning points coalesce, but the exact result is always smaller than unity. Due to Kemble [25,26], an improved formula
where B exp Im m and m
TKemble ( E ) 1 1 B
2
2
ATM tunneling expression
In Figure 1, consider a particle of mass m with incident energy E impinging on an arbitrary continuous potential that has its support on the interval [0,L]. Based on the ATM method, the tunneling probability expression reads [23] T 1 rr ,
(5)
L r (0) r ( L) exp i2 K ( x)dx 0 . r L 1 r (0)r ( L) exp i2 K ( x)dx 0
(6)
The two terms r (0), r ( L) are given by r (0) r ( L)
0 (0) , 0 (0)
( L) 0 with 0 2 mE . Here we introduce ( L) 0
the wavenumber of the general waves K ( x) which consists of the main waves and the scattered subwaves [15,23]. The relation between the main wavenumber ( x) and the classical momentum p( x) is ( x) p( x) . As it has already been pointed out in our previous work [15] that the wavenumber of subwaves can be written as q (q 2 2 ), where q( x) ( x) ( x) is the minus logarithmic derivative of the wave function, we denote differentiations with respect to x by primes throughout this paper. Thus the explicit expression for S becomes
S
L
0
q dx. q2 2
(7)
It is desirable to elaborate the concept of scattered subwaves here. Let’s consider a simple ladder potential barrier as shown in Figure 2 and a particle approaching from the left. The wave function takes the form
(3)
is much better for all barriers which can be approximated by an inverted parabola function, V (r ) r 2 . With the connection parameters fixed, another alternative method leads to T ( E ) B 1 4 B ,
1.2
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Figure 1 An arbitrary continuous potential profile V(x) and the planewave components of the incoming, reflected and transmitted wave.
(4)
which was introduced in ref. [27]. Note that each WKB tunneling formula is approximate and has its own valid condition. Furthermore, they do not take into account the detailed structures below the incident particle energy and completely neglect the scattered subwaves.
Figure 2 A simple ladder potential barrier.
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A0 exp(i 0 x) B0 exp(i 0 x), A exp(i1 x) B1 exp( i1 x), ( x) 1 A2 exp(i 2 x) B2 exp(i 2 x), A exp(i x), 0 3 r
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x 0, 0 x d1 , d1 x d1 d 2 ,
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where i 2m( E Vi ) ( i 1, 2 ). Requiring the wave (8)
x d1 d 2 ,
function ( x) and its first derivative ( x) to be continuous yields the expression for refection coefficient, namely
r (0) r ( d1 d 2 ) exp i2 1 d1 2 d 2 tan 1 q 2 tan 1 q 1 B0 , A0 1 r (0)r (d d ) exp i2 d d tan 1 q tan 1 q 1 2 2 2 2 1 1 1
(9)
where r (0)
and
0 1 0 , r (d1 d 2 ) 2 0 1 2 0
q 2 tan tan 1 i 0 2 2 d 2 .
In this simplest tunneling case, the phase devoted by main waves (denoted by m ) can be decomposed as the sum of
1 d1 and 2 d 2 , and each term represents the plane wave tunneling through each part of the ladder potential barrier, respectively. It is interesting to note that the previous mentioned term s is reduced to tan 1 q 2 tan 1 q 1 . If we set V1 V2 , i.e. the potential becomes homogeneous, this term will be equal to zero. Clearly the term s represents the phase contribution picked up by the scattered subwaves that result from the interface between the potentials V1 and V2 . Since an arbitrary continuous potential profile can be divided into stack films that have constant potentials, one can easily see that the subwave concept refers to the scattered waves originating inherently from the inhomogeneity of the potential. When eqs. (5) and (6) are combined, it is easy to check that the real part of the phase in the exponential expression has no effect on the magnitude of the tunneling probability. It can only affect the phase of r . For simplicity, we concentrate on the image part of phase devoted by the main waves and scattered subwaves, hereinafter to be denoted by Im m and Im s , respectively. If the WKB tunneling expression is constructed in term of the first-order WKB wave eq. (1), there will generally be no reflection for the above-barrier energy, as in eq. (2). Its discrepancy with the exact result comes from the fact that Im m always decreases monotonically as impinging energy increases and remains zero at the top of the barrier. In contrast, the wavenumber of the subwaves takes the q( x) , i.e., the exact wavefunction, into account. Im s does not vanish for the above-barrier energy and depends sensitively on the potential in the whole range of x values, not just between the two turning points. In the next section, through concrete examples we will demonstrate that Im s is the dominant term in eq. (6); it is the scattered subwaves that determine the structure of the tunneling probability.
2
Examples
Quantum tunneling has a substantial impact on modern electronic devices, which are frequently realized with the advance of nanotechnology. Very recently a study [1] on a barrier with adjacent wells confirmed that the addition of the attractive potential well can substantially alter the tunneling probability T at energies below the barrier height. Such an electronic on/off switch can be obtained by adjusting an external electric field which supplies the energy for electrons to yield the maximum and minimum of T. The potential of this structure has the form V0 V1 , 1 exp x x0 a U ( x) V2 , 1 exp x x b 2
x 2 x0 ,
x 2 x0 ,
(10)
where V0 V1 0 , x2 2 x0 , and V2 is fixed such that U ( x) is continuous at the point x 2 x0 . The tunneling
probabilities as a function of incident energy E are plotted in Figure 3 for three cases: a smooth barrier with no well, with one adjacent well and with two adjacent wells. All the WKB expressions fail and each WKB expression provides the same tunneling probability for these three cases. While the same results by ATM tunneling expression and the numerical calculation are obtained [1] as expected, Figure 4(a) demonstrates that the tunneling probability can reach unity by adjusting the parameter V0 . The results provide a theoretical basis for the use of such properties to design an electronic on/off switch. In order to verify that the oscillation structure of tunneling probability follows the variation trend of the imaginary part of S , we plot Im m and Im s as functions of energy in Figure 5 for the two tunneling cases of Figure 3, and the corresponding plots for the tunneling cases of Figure 4(a) are shown in Figures 4(b) and (c). Figure 5 shows the existence of the adjacent wells has no effect on the term Im m since m is completely specified by the main waves number ( x) between the two turning points and as a result, the WKB tunneling probabilities show no differences. Its numerical results indicate that
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the electron does not “see” the existence of the attractive potential wells. However, our elaborative calculations show this resonance-like tunneling across a barrier with two adjacent wells as a result of the scattered subwaves. In Figure 4 the resonant-like structure of tunneling probability indeed follows the oscillated variation trend of Im s (see Figures 4(a) and (c)). As a second illustration, we consider a band-pass filter based on a Gaussian modulated superlattice. The Gaussian function is given by V0 exp ( x L 2)2 s2 , where V0, L
Figure 3 The ATM tunneling probabilities for three cases: a smooth barrier with no well, with one adjacent well and with two adjacent wells. WKB approximation tunneling probabilities, and the parameters are V0=6.5, V1=3.3, x0=0.4, x2=22x0, and a=b=0.1. Units in calculation are 2 m 1.
Figure 4 (a) The tunneling probabilities for two wells case with a variable parameter V0 and the other parameters are the same as in Figure 3, (b) Im m and (c) Im s .
Figure 5
in Figure 3.
Im s and Im m corresponding to the same parameters used
and s are constant. Parameters a and b are the widths of the potential barriers and the wells, respectively. A schematic diagram of this superlattice structure is shown in Figure 6. The ATM and WKB tunneling probabilities as a function of normalized incident electron energy are both plotted in Figure 7 when the barrier heights are V0=0.45 eV and V0=0.3 eV. In calculation, the electron effective mass is assumed to be 0.067m0 and m0 is the electron mass. The tunneling probabilities calculated by the ATM tunneling expression agree well with the numerical calculation [2]. Since Im m Im s and Im m Im s Im s 1,
Figure 6 Schematic diagram of the superlattice structure of 40 layerpairs with a=b=3.2 nm, L=256 nm, s=L/4, and V0=0.45 eV.
Figure 7 ATM and WKB (eq. (2)) tunneling probabilities as a function of normalized incident electron energy for the barrier height V0=0.45 eV (solid curve) and V0=0.3 eV (dotted dash curve).
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6
7
8
9 10
Figure 8
Im m Im s and Im m as a function of normalized incident
electron energy for the barrier height V0=0.45 eV (solid curve) and V0=0.3 eV (dotted dash curve).
11
12
in order to display the details of Im s , in Figure 8 we plot Im m Im s instead. Similar to the first example, the same conclusion can also be obtained here: the tunneling probability increases and decreases synchronously with the variation trend of Im s (i.e., with a similar structure).
3
Conclusions
In conclusion, we show that for a variety of potentials, tunneling probability has the same resonant or band structure as that of the imaginary part of the phase shift which results from the subwaves. Consequently, the definition of scattered subwaves is not at all trivial. The conclusion developed in this paper helps designing optoelectronic devices.
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20 This work was supported by the State Key Laboratory of Advanced Optical Communication Systems and Networks (Grant No. 2008SH05).
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