A Novel Energy Filter Using Semiconductor
Superlattices and its Application
to Tunneling
Time
Calculations
Hsin-Han Tung and Chien-Ping
Lee, Senior Member, ZEEEAbstract-An artificial quantum-mechanical filter using super- lattice structures is proposed in this paper. By gradually changing the barrier widths of a superlattice according to a Gaussian function, a broad-band and almost zero sidelobe transmission profile can be obtained. The WKB approximation is applied to
demonstrate the phenomena of abrupt change of transmission profile. The proposed structure allows the incident electrons to be nearly totally transmitted when the impinging electron energy is in the passband. On the other hand, a complete reflection occurs when the impinging energy is in the stopband. By adjusting the structure parameters, the desired passband and stopband of such a filter can be obtained. Time evolution of an electron wavepacket moving through the structure is calculated by numerically solv- ing the time-dependent Schrodinger equation. Numerical results clearly demonstrate the characteristics of total transmission and reflection. By simulating the movement of a totally transmitted wavepacket, ambiguity results from the nature of the wavepacket in the determination of electron tunneling time can be avoided. The generalized concept of matched quantum-mechanical wave impedance (QMWI) analogous to transmission line theory is presented to explaine the occurrence of total transmission of the proposed structure. The tunneling time (QMWI) calculated based on the concept of QMWI is compared with the accurate tunneling time obtained by our simulation.
I. INTRODUCTION
ECENTLY, we have proposed a novel superlattice struc-
R
ture for energy bandpass filter (EBPF) , [l]. Such astructure with a Gaussian superlattice potential profile allows the incident electrons to be nearly totally transmitted when the impinging electron energy is in the passband. On the other hand, a complete reflection occurs when the impinging energy is in the stopband. The structure can be considered as a regular superlattice with the potential modulated by a Gaussian function. A similar technique has already been used to suppress the sidelobe for the optical grating filter [2]. With the advances in the technology of molecular beam expitaxy
(MBE), it is possible to control the film growth in a monolayer scale. Investigation of the propagation of electron waves in such a layered medium is of particular interest because the transport property is governed by the minibands formed in the superlattices. Multiple quantum barriers that enhance carrier blocking [ 3 ] , [4] and resonant tunneling structures that enhance Manuscript received March 25, 1996; revised July 2, 1996. This work was supported by the National Science Council of the Republic of China under Contract NSC84-2215-E009-039.
The authors are with the Department of Electronics Engineering, National Chiao Tung University, Hsin Chu, Taiwan, R.O.C.
Publisher Item Identifier S 0018-9197(96)08657-5.
carrier transmission [5] are good examples that demcnstrate the current manipulating capability of these structures. In this paper, another broad-band EBPF with similar transmission characteristics is presented. The bands are flat and their positions and bandwidths are adjustable. The new structure, unlike the previously proposed EBPF, has constant barrier heights but a variable barrier width, so it can be much more easily realized by epitaxial growth techniques. Details of the superlattice structure and transmission characteristics are given in Section 11.
In Section 111, the WKB approximation, which takes ad- vantage of the fact that the local minibands formed in the graded structure are changing slowly, is applied to demonstrate the abrupt change of the transmission profile. In Section IV, we present results on numerical simulations of a wavepacket moving through such a structure to demonstrate the special band-passing characteristics of EBPF. At the same time, we also investigate the electron tunneling time. The physical meaning and difficulty of trying to extract the tunneling time from wavepacket simulation are discussed. The special transmission characteristics of EBPF make it possible for precise determination of the tunneling time.
In Section V, the theory of quantum-mechanical wave impedance (QMWI) is used to compute the transmission spectrum and explain the occurrence of total transmission of the proposed structure. It also calculates the electron tunneling time using the equation derived from the concept of QMWI. Finally, a conclusion is given in Section VI.
11. STRUCTURE OF EBPF AND TRANSMISSION SPECTRUM Fig. 1 shows the schematic conduction-band diagram of the proposed structure, where U and b are the widths of
the potential barriers and the wells, respectively, and V, is the potential barrier height. The period, a
+
b , is kept constant throughout the structure but the ratio a l a+
b is varied according to a modulated Gaussian function. Consider one of the examples with the structure shown in Fig. 1. The superlattice is composed of 61 layer-pairs with a barrier height V, = 0.25 eV. The period a+
b is equal to 50A,
and the total length of the structure L is 3050A.
The barrier widthis varied gradually according to a Gaussian function a; =
[(u+b)/2] e ~ p [ - ( i - 3 1 ) ~ / ~ : ] , where a, is the width of the ith
barrier, and cz is chosen to be 10. The calculated transmission probability by the transfer-matrix method [6] is plotted as 0018-9197/96$05.00 0 1996 IEEE
TUNG AND LEE: A NOVEL ENERGY FILTER AND ITS APPLICATION TO TUNNELING TIME CALCULATIONS 2123 b a xo a+b H
Fig. 1. Schematic conduction band-edge diagram of the proposed superlat- tice structure which is used for numerical calculation of electron tunneling time later. @(x, 0) stands for the initial Gaussian wavepacket and @(z, t f )
is the transmitted wavepacket. The dashed line represents the function VM(Z)
of the first forbidden band-edge in the superlattices.
a function of the electron energy and is shown in Fig. 2. This figure shows nearly total transmission when the electron energy is in the range between 0.5 V, to 0.8 V, and above
1.8 V,, while the transmission probability is zero when the electron energy lies outside these ranges. Flat passbands and stopbands result. These characteristics are quite different from those of a conventional distributed Bragg reflector (DBR), in which the transmission bands are never totally flat. Since in our structure the barrier width is changed gradually, the minibands formed within the superlattice are expected to be changed in the same way. From the basic quantum theory
[7], we know that when electrons are moving in a region
where the spatial variation of potential is small compared with the electron wavelength, the reflection is expected to be small. By combining a slowly varying potential with a periodic superlattice which exhibits miniband structures, we thereby obtain a broad-band transmission profile. This superlattice structure behaves like an energy band-pass filter, which is similar to the frequency band-pass filter in an electronic circuit. This is quite different from the transmission characteristics of the conventional superlattice structures. If the structure parameters are changed to V, = 0.15 eV and a
+
b = 60&the calculated transmission profile is shown in Fig. 3, both the band position and bandwidth change. One can get the desired transmission probability spectrum by adjusting the parameters such as Vo,a
+
b, etc.111. THE WKB APPROXIMATION
The novel feature of the present structure is that its layer thickness ratio r (= a l a
+
b ) gradually changes along the stacking direction. It is convenient to treat such inhomoge- neous material system in terms of the graded minibands which are associated with the ratio r and the barrier height V,. The graded minibands can be constructed by continuously joining the local minibands formed in each individual section. It was shown in [8] that the height of the graded barrier of the first forbidden band varies linearly with the layer thickness ratio r and reaches its maximum about & / 2 when r is 0.5, or'I
I"0 0.5 1 1.5 2
Normallzed Incident Electron Energy, (EW,,)
Fig 2 Plot of the transmission probability as a funchon of normalized incident electron energy for the structure shown in Fig 1, where a
+
b =50 &V, = 025 eV, and periods = 61
0 0.5 1 1 . 5 2 2.5
Normalized Incident Electron Energy, (Ell',)
Fig. 3. Plot of the transmission probability as a function of the normalized
incident electron energy for the structure shown in Fig. 1, where a
+
b = 60A,V, = 0.15 eV, and periods = 61.
when the thicknesses of a and b are equal. In the present case, the calculated barrier edge V M ( z ) also varies according to the modulated Gaussian function and is shown in Fig. 1. For the convenience of calculation, we approximate VM (x) by a truncated parabolic function
, - L / 2 < x < L / 2 otherwise.
The WKB approximation is then used to calculate the trans- mission probability for electron incident energies E less than Vola. The transmission probability T is calculated by the
Normahzed Incident Electron Energy, (E/Vo)
Fig. 4. Plot of the transmission probability T ( E ) with the application of the WKB approximation for the structure shown in Fig. 1. The structure parameter is the same as that used in Fig. 2.
formula [9]
4
T = ( 2 0
+
where
Substituting VM (z) into the above equation and evaluating the integral, we get
Using the structure parameters in Fig. 2, the calculated T ( E ) is shown in Fig. 4. As we can see, the transmission probability increases rapidly with E and is very sensitive to E when the incident energy is near Vo/2. Ninety percent of the change in the transmission coefficient takes place when the incident energy is between 0.49 to 0.5 of V,. The derived analytic expression for T ( E ) gives a reasonable explanation why the EBPF has an abrupt transition region between the stopband and passband.
Iv. TIME EVOLUTION OF WAVEPACKET THROUGH EBPF AND TUNNELING TIME CALCULATIONS
A. Wavepacket Simulations
Time evolution of an electron wavepacket propagating through the proposed structure is calculated by numerically solving the Schrodinger equation. Our work follows closely that of Goldberg et al. [lo]. The one-dimensional time- dependent wave equation is transformed into a set of difference
I I I
3201)
1
i i l
12000 15200
Fig 5 Time evolution of a Gaussian wavepacket scattering from the pro- posed structure with the layer parameters used in Fig 2 The average incident energy is 1 25 V, Numbers in each configurahon denote the time in the unit of 2 316 x s The region enclosed by rectangle is the superlattices as shown in Fig 1
equations, which are solved with the initial condition 9 ( z , O ) = exp(ikoz) exp[-(z - z 0 ) ~ / 2 a $ (see Fig. 1). We
see that this packet is centered at z = z g with a spread
in z governed by 00. The factor exp(ik0e) indicates our
initial wavepacket move to the positive direction with an average momentum hko, and ko is equal to
-I??,,
where E is the average energy of the electron wavepacket. The electron effective mass is assumed to be 0.067mo throughout the structure. Fig. 5 shows the time evolution of a Gaussian wavepacket impinging upon a structure, which is the same as that used for Fig. 2. The average incident energy of the wavepacket is chosen to be 1.25 V, and the spread 00 of the packet is 400A,
which corresponds to an energy uncertainty of about 0.03 eV. From Fig. 2, we h o w that the incident energy of the wavepacket lies within the stopband of the transmission spectrum. Fig. 5 shows clearly that the wavepacket is totally reflected although the average incident energy is greater than the potential barrier height. If the initial wavepacket moves to the right with an incident energy E = 0.65 V, and a same width of 400A
(this corresponds to an energy uncertainty about 0.02 eV), i.e., when the incident energy lies in the passband of the transmission spectrum, the time evolution of the wavepacket, shown in Fig. 6 (the solid line), is totally different. Complete transmission happens and no noticeable reflection is detected. From the numerical results presented here, it is clear that the proposed structure can really serve as energy bandpass filters for electrons.B. Tunneling Time Calculations
The question of “how long does it take for an electron to tunnel through the classically forbidden region?” has been a long-standing controversy in quantum mechanics. It is funda- mental and important to quantify the electron traversal time since it limits the maximum operating speed of many modern electronic devices. Several approaches have been proposed
TUNG AND LEE: A NOVEL ENERGY FILTER AND ITS APPLICATION TO TUNNELING TIME CALCULATIONS 2125
Fig. 6. Time evolution of a Gaussian wavepacket scattering from the pro- posed structure with the layer parameters used in Fig. 2. The average incident energy is 0.65 V,. The free wavepacket is presented by the dotted line.
[ 111-[ 171. Different approaches lead to different results and the debates still exist.
The attempts to extract electron tunneling time by following the transmitted wavepacket is, however, not so successful. Generally, there are two ways to determine the tunneling time of an electron from wavepacket simulation.
1) Follow the peak of the transmitted wavepacket (i.e., the maximum of the square of the wavepacket) to determine the location z of the packet at time t. Actually, this is the same as the original idea of phase time calculation
[ 111, but the transmitted packet may be considerably deformed and defining its peak may become ambiguous.
2) Evaluating the expectation value of the position from the transmitted packet (i.e., the center of mass) [18],
[19]. This seems to give a more reliable result, but the transmission spectrum usually shows an asymmetrical profile and the mean energy of the transmitted packet may be shifted toward higher energies (the reflected packet is shifted to lower energies because of energy conservation), which makes the derivation of tunneling time somewhat ambiguous.
So, using either method, it is difficult to obtain the mean transmission time from the wavepacket simulation.
The difficulties mentioned above can be overcome if we use the EBPF to study the electron tunneling time. The special transmission characteristics of this structure make it possible for precise determination of the tunneling time. Since the wavepacket with a finite space spread Ax and a wave vector spread Ak can be 100% transmitted even when the incident energy is smaller than the barrier height, there would be no uncertainty in the evaluation of electron tunneling time. We again use the EBPF with the transmission characteristics shown in Fig. 2 in our calculation. The intrinsic tunneling time is obtained by substrating the time of free motion outside the energy filter from the total propagation time. As shown in
I
11000]I
1300011
15000Fig. 7. Time evolution of a Gaussian wavepacket scattering from the pro- posed structure with the layer parameters used in Fig. 2. The average incident energy is 2.25V0. The free wavepacket is presented by the dotted line.
Fig. 1, it is calculated as
where xf is the final position of the transmitted wavepacket, t f is the total simulation time, and v =
d
m
is the free-electron velocity. We first consider the situation that the impinging electron energy E = 0.65V,, which is smaller than the barrier height. The time evolution of the wavepacket is shown in Fig. 6. Notice the number in each frame denotes the time in the unit of 2.316 x s. The calculated wavepacket tunneling time rWp is 0.4072 ps. In Fig. 6, we also show the time evolution of the wavepacket in the absence of the filter (dashed line). The calculated free traveling wavepacket time qree (= L / v ) is 0.3277 ps, which is smaller than the tunneling time in the presence of the EBPF. Although the scattering strength is strong when the wavepacket is moving within the superlattices, there is no distortion for the transmitted wave except the spreading of the wavepacket. This is of great advantage since the peak, stationary-phase point, and the center of mass of the transmitted packet all correspond to the same position. The ambiguity and the difficulty in determining the location of the transmitted packet are avoided. In the case of E>
V, with E = 2.25V0, the time series of the moving wavepacket is shown in Fig. 7. The calculated T~~ is 0.1891ps, which is again greater than TfPf,,, (= 0.1762 ps). It is noticed from Figs. 6 and 7 that not only is the tunneling time longer, but the wavepacket emerged from the EBPF is also wider than the free traveling wavepacket. From the time delay and the broadening of the transmitted wavepacket compared with the free moving wavepacket, we can see that tunneling through an EBPF is like a free particle moving across a region with an effective length longer than the barrier region, and the extra length is due to the electron bouncing back and forth within the energy filter.
V. QUANTUM-MECHANICAL WAVE IMPEDANCE (QMWI) AND TQMWI
A. Quantum-Mechanical Wave Impedance Matching
The occurrence of broad-band transmission can be also explained using the generalized concept of matched quantum- mechanical wave impedance (QMWI) [20]. This concept is analogous to the impedance in the well-developed transmission line theory. The QMWI at any plane x can be defined as
2h, W ( x ) Z ( x ) = - ~
Ilm W X )
where Z ( x ) is the wave impedance looking into the positive z direction, j =
fl,
and Q(z) and Q'(x) are the electron wave function and its spatial derivative, respectively, for the problem interested. For an arbitrary-shaped potential, we can approximate the potential and effective mass by multistep functions with a sequence of N segments. Thus, if x, and x,+1 are the boundaries of segmenti,
the QMWI at x, can be calculated by Z(x,+l) cosh(y,l,) - 20, sinh(y,l,) Z ( x 2 ) = 2 0 % 20% c o s h ( ~ t L ) -z ( ~ + i )
sinh(yz4) where 7%= g J G F E q
1, = & + l - x, andis the characteristic impedance of the medium. The above equations express the QMWI at x, in terms of the QMWI at %,+I, and y,, I,, and 20,. Once the Z ( x z ) is calculated, we can repeat the process for segment i - 1 to calculate Z(z,-,) using ~ ~ - 1 , Z t - l , and Zot-l. Repeatedly using these equaitons, we can evaluate the total input impedance of the proposed structure and treat the whole superlattices as an equivalent load impedance
2s.
Thus, the reflection coefficient p(E) for the wave amplitude can be calculated aswhere
20
is the characteristic QMWI of the uniform semicon- ductor, and the transmission probability is given byWe have used the above equation to calculate the transmission probability for an EBPF with the identical structure parameters used in Fig. 2. The result is the same as that calculated by the transfer-matrix method, so we will not show it here.
Fig. 8 shows the QMWI,
ZS,
and 20 as functions of energy for the structure used in Fig. 2. From the transmission line theory [21], we know that matched condition occurs when the load impedance is equal to the characteristic impedance of the transmission line. Under the matched condition, the reflection coefficient is zero and the transmission coefficient is0 5 1 1.5 2
Normalized Incident Electron Energy, (E/V,)
5
Fig. 8.
ized incident electron energy with the structure parameters used in Fig 2.
The absolute values of QMWI 2s and 20 as a functlon of normal-
equal to 1. Fig. 8 clearly demonstrates that when the incident energy lies in the passband, 2s matches 2 0 completely, while outside the passband, large impedance mismatch exists. The result is in good agreement with that calculated in Fig. 2. This explains why the proposed structure has the property of total transmission for a Gaussian wavepacket.
B. Tunneling Time Calculated by QMWI in an integral form [22]:
The tunneling time based on the QMWI can be expressed
TQMwI = 2
iL
6
where R ( x ) is the real part of the QMWI, Z ( x ) , and L is the total tunneling length. The R ( x ) can be obtained using the method described in Section V-A. Thanikasalam
et al. [17] have used this method and derived the analytical expressions for tunneling time through single and double barrier structures. The QMWI approach is unique and it can be easily implemented with a numerical method for arbitrary-shaped potential barriers [22]. Since this approach does not involve the nature of the wavepacket, ambiguity results from the wavepacket can be avoided. The EBPF proposed in this paper provides us a chance to examine the accuracy of the tunneling time calculated by this method and compare it with the accurate rwwp presented above. We use the Simpson's rule to evaluate the integral for TQMWI. Comparison of rwp, TQMWI, and qre, (the EBPF is the same as that used in Fig. 2) is shown in Table I. When the incident energy E is 0.65V0 (i.e., E
<
Vo), we find that TQMWI is 0.4485 ps, which is greater than real tunneling time, rwp (= 0.4072 ps). It is interesting to note [17] that the calculated TQMWI is always greater than those obtained from other theoretical approaches. Although TQMWI is not equal to rwp, they are all greater than the free-particle transit time qree (= 0.3277 ps). The tunneling time obtained from some approaches may be below qree for a range of incident energies [17], which lead us to questionTUNG AND LEE: A NOVEL ENERGY FILTER AND ITS APPLICATION TO TUNNELING TIME CALCULATIONS 2127 TWP TOMWI %ee
E
= 0.65Vo
E
= 2.25Vo
0.4072 ps
0.1891
ps 0.4485 ps 0.1982 ps 0.3277 ps 0.1762 psthe physical content of these approaches. When the incident energy E is 2.25V0, (i.e., E
>
Vo), the calculated T Q M W I is 0.1982 ps, which is greater than rwp (= 0.1891 ps), but the difference is small. Both TQMWI and rwp are also greater than qree (= 0.1762 ps). It is reasonable since the electron is moving in a classically allowed region. It was shown thatT Q M W I , which is based on the concept of group velocity, gives
physically meaningful results in the limits of zero and infinite incident electron energy [17]. From our numerical results, we believe the free particle time qree is always less than the traversal time with the presence of potential barriers for all incident energies.
VI. CONCLUSION
An artificial quantum-mechanical energy bandpass filter using a superlattice structure with a gradually changing barrier and well width has been proposed. Adjustable flat transmission bands and reflection bands are obtained by properly choosing the layer parameters. When an electron impinges upon the proposed structure, it is completely reflected or transmitted depending on whether the incident energy lies in the stopband or passband of the transmission spectrum. Application of the WKE3 approximation gives a reasonable explanation of the abrupt change of the transmission profile. We have performed the numerical simulations of wavepacket propagating through an energy bandpass filter to obtain the electron tunneling time under the condition of total transmission, and the ambiguity results from the nature of the wavepacket can be totally avoided in this way. The phenomenon of total transmission can be successfully explained by using the concept of wave impedance analogous to that in the transmission line the- ory. A comparison of the tunneling time rwp with the free wavepacket traveling time rfree and the tunneling time TQTQMwI based on the QMWI method has been presented. rwp is found to be always greater than pree. The extra time is due to the electron bouncing back and forth within the barrier system.
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Hsin-Han Tung, for photograph and biography, see p. 512 of the March
issue of this JOURNAL.
Chien-Ping Lee (M’8OPSM’94), for photograph and biography, see p. 512 of the March issue of JOURNAL.
