Abstract:
In accordance with the invention, resonant quantum tunneling microscopy allows surface imaging while allowing characterization of physical properties associated with the surface.

Description:
BACKGROUND 
       [0001]    Scanning probe methods are being used to study an increasing range of surface phenomena. These surface phenomena include surface topography, dielectric and magnetic properties, molecular manipulation and many other effects that occur on the micrometer to the subnanometer scale. Scanning probe microscopes include scanning tunneling microscopes (STMs) and atomic force microscopes (AFMs). Both STMs and AFMs utilize a sharp probe tip to interact with the target surface to generate a topographic surface scan. The parameters used for image formation are different for STMs and AFMs. In an STM system, the modulations of the tunneling current are used to determine surface topography while in an AFM system, the varying force between the probe tip and the target surface is used to determine the surface topography. 
         [0002]    STM systems may also be employed to monitor the tunneling current versus the bias voltage at fixed locations on the target surface. The STM probe tip is positioned at a particular location above the target surface. A spectrum is produced by sweeping the bias voltage and measuring the tunneling current. This provides information regarding the density of surface states in the target surface proximate to the STM probe tip. 
         [0003]    The use of STM and AFM systems has enabled relatively routine imaging of surfaces at nanoscale resolutions, but more information may be desired about the surface. A desirable measurement modality is one that allows surface imaging of unaltered samples while at the same time providing a means for characterizing a physical property associated with the localized target surface. 
       SUMMARY 
       [0004]    In accordance with the invention, resonant quantum tunneling microscopy allows surface imaging while allowing characterization of physical properties associated with the surface. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0005]      FIGS. 1   a - b  show embodiments in accordance with the invention. 
           [0006]      FIG. 2  is a conceptual diagram in accordance with the invention. 
           [0007]      FIGS. 3   a - c  show the affect of various applied voltages on the Fermi level in accordance with the invention. 
           [0008]      FIG. 4   a  shows the tunneling current as a function of tunneling voltage in accordance with the invention. 
           [0009]      FIG. 4   b  shows the differential conductance as a function of tunneling voltage in accordance with the invention. 
           [0010]      FIGS. 5   a - c  show the effect of various ramp voltages on the Fermi level in accordance with the invention. 
           [0011]      FIG. 6  shows a one dimensional conceptual model of the physical system in accordance with the invention. 
       
    
    
     DETAILED DESCRIPTION 
       [0012]    An embodiment in accordance with the invention is shown in  FIG. 1   a.  Typically, two sharp conducting metal wires or conducting carbon nanotubes are used for tunneling electrodes  120  and  130  for Resonant Quantum Tunneling Scanning Microscope (RQTSM)  100 . Tunneling electrodes  120  and  130  are attached to tip support structure  150  which may be attached to, for example, insulating xyz scanner  160 , similar to that used in standard atomic force microscope systems. Typically the positioning performed by an xyz stage is actuated by a piezoelectric tube under the influence of applied elecric fields although other well-known positioning means may be used. Alternatively, tip support structure  150  may be attached to a fixed or z motion only structure while target surface  110  is positioned on an xyz stage or xy stage (not shown), respectively. The xyz or xy stage may be actuated by a piezoelectric tube while the z-motion structure, for example, may be actuated using a MEMS device such as a electrostatic stepper motor or an electrostatic comb drive. In accordance with the invention, target surface  110  is typically an insulating surface so that purely tunneling currents predominate over unwanted conduction processes. However, target surfaces with limited conductive properties may also be examined, with the level of allowable conductivity being determined by the measurement requirements. 
         [0013]    Tip support structure  150  functions as the base to which tunneling electrodes  120  and  130  are attached and can be made in a manner similar to the way that SPM support structures are manufactured. SPM support structures may be microfabricated from silicon in a manner similar to computer chips and produced to have a pyramidal shape. In order for tip support structure  150  to be insulating, the silicon can be oxidized prior to the attachment or growth of tunneling electrodes  120  and  130 . Other typical methods of fabricating appropriately dimensioned and insulating tip support structure  150  may also be used. Tunneling electrodes  120  and  130  may be attached to tip support structure  150  in a number of ways in accordance with the invention. For example, small metal precursor regions may be lithographically patterned and deposited on tip support structure  150 . The carbon nanotubes are then grown (e.g., M. Jose-Yacaman, M. Miki-Yoshida, L. Rendon, T. Santiesteban, Appl. Phys. Lett. 62, 202 (1993) incorporated herein by reference) on the small metal precursor regions to form tunneling electrodes  120  and  130 . Any other available technique to attach carbon nanotubes to tip support structure  150  may also be used to create tunneling electrodes  120  and  130 . Other structures may be manufactured and attached to tip support structure  150  to form tunneling electrodes  120  and  130  in accordance with the invention and includes structures typically used for STM tips. For example, a typical STM tip is made from tungsten polycrystalline wire, platinum wire or iridium wire which is electrochemically etched to provide a suitably sharp tip. Tip support structure  150  typically has a lateral dimension in the submicron range. Tunneling electrodes  120  and  130  typically have a separation on the order of nanometers or less at their furthest extent from tip support structure  150  to create adequate tunneling current, I T . 
         [0014]    Tunneling electrodes  120  and  130  are connected to variable voltage source  140  and current measuring device  145 . The variable voltage source  140  and current measuring device  145  are similar to those typically used in STMs. Electrical connections to variable voltage source  140  and to current measuring device  145  from tunneling electrodes  120  and  130  may be accomplished in a number of ways in accordance with the invention. For example, metal lines may be created lithographically from the attachment sites of tunneling electrodes  120  and  130  to larger metal bonding pads (not shown). This may be accomplished during the microfabrication of tip support structure  150 . 
         [0015]    In operation, xyz scanner  160  positions tunneling electrodes  120  and  130  proximate to target surface  110  and over scanned spot  135 . The tunneling voltage, V 1 , between tunneling electrodes  120  and  130  is ramped over a prescribed range, and the tunneling current, I T , is measured as a function of time using current measuring device  145 . The predominant physical mechanism for generating the tunneling current is resonant quantum tunneling (RQT). The tunneling differential conductance, σ=dI T /dV t , for RQT peaks sharply when two physical conditions are satisfied. First, the integrated magnitudes of the tunneling barriers between tunneling electrodes  120 ,  130  and scanned spot  135 , which functions as the intermediate well region, must be equal. Second, the Fermi energy in negatively biased tunnel electrode  120  must precisely match an energy level of scanned spot  135  which functions as the intermediate well region. The tunneling differential conductance, σ, may be enhanced by orders of magnitude when these two conditions are met. 
         [0016]    During operation, as tunneling electrodes  120  and  130  move to each scanned spot  135  on target surface  110 , the tunneling voltage, V t , is ramped over the prescribed range. The precise position and extent of scanned spot  135  is self-selected by the requirement that the integrated magnitudes of the tunneling barriers between tunneling electrodes  120 ,  130  and scanned spot  135  are equal. Typically, this implies that scanned spot  135  is approximately equidistant from tunneling electrodes  120  and  130 , although there may be some deviation, for example, if the electron emissivity of scanned spot  135  has inherent asymmetries and the effective barrier potential varies accordingly. Target surfaces with significant roughness, where the ratio of surface roughness to the spacing between tunneling electrodes  120  and  130  is less than unity will have a bias in the position of scanned spot  135 ; from being centered between electrodes  120  and  130  to being off-center. The spatial extent of scanned spot  135  is determined by how rapidly the tunneling probability decreases as the tunneling barriers with respect to tunneling electrodes  120  and  130  become asymmetric. Hence, even if tunneling electrodes  120  and  130  are spatially separated by a distance on the order of several nanometers, the spatial extent of scanned spot  135  is determined by the symmetry requirement for the tunneling barriers, i.e. only that limited spatial region with a symmetric barrier to tunneling electrodes  120  and  130  is interrogated. This typically results in sizes for scanned spot  135  on the order of several Angstroms. Tunneling current contributions from surface regions proximate to tunneling electrodes  120  and  130  that have asymmetric tunneling barriers contribute to background noise but this noise contribution is typically many orders of magnitude smaller than RQT currents. In directions perpendicular to the tunneling current flow, the size of scanned spot  135  is determined by tunneling suppression due to increased barrier width and not by the condition of tunneling barrier symmetry. 
         [0017]    As the tunneling voltage, V t , is ramped over the prescribed range during a measurement, the peaks in the tunneling differential conductance, σ, occur at those voltages where the Fermi energy of negatively biased tunnel electrode  120  matches an energy level of scanned spot  135 . The measurement provides detailed information about the electron energy spectrum of scanned spot  135  which can be used to determine local elemental or molecular composition and deviations from expected spectra measured from control regions. The magnitude of the tunneling current, I T , off-resonance can be used as a standard STM signal to obtain a measure of the separation of tunneling electrodes  120  and  130  from target surface  110  to obtain surface topography. The measured energy spectra are referenced to the potentials of tunneling electrodes  120  and  130  and surface charges or other sources of potential offsets at scanned spot  135  may produce measurable voltage shifts that may need to be considered in spectral identification. 
         [0018]    RQTSM  100  can be used to simultaneously obtain both topographical and spectral information about target surface  110 . The spatial resolution is typically substantially better than the separation distance between tunneling electrodes  120  and  130 , on the order of Angstroms. Operation of RQTSM  100  may in some cases be affected by thermal noise issues which may result in signal to noise issues. A typical solution to thermal noise issues is to cool a system to low temperature where thermal broadening is less than the energy spacing between the targeted differential conductance, σ, peaks (see  FIG. 4   b ), hence cooling of RQTSM  100  and target surface  110  may be required in some cases. Depending upon the application, this may require cooling below liquid helium temperatures. Another operational issue relates to the possibility of electrically charged or polarized contaminants in the air attaching themselves to tunneling electrodes  120  and  130  and effectively shorting out the tunneling current signal. If electrically charged or polarized contaminants are an issue, RQTSM  100  can be operated in a high vacuum environment to reduce the possibility of electrically charged or polarized contaminants. 
         [0019]      FIG. 1   b  shows an embodiment in accordance with the invention. Here, the tunneling voltage, V t , applied between tunneling electrodes  120  and  130  is kept constant and the ramped voltage, V R , is applied by ramped voltage source  190  between metal layer  175  beneath target surface  110  and voltage source  185 . The tunneling current, I T , and the differential conductance, σ, as a function of voltage provide the same information as in the embodiment shown in  FIG. 1   a.  In this embodiment, the energy levels of scanned spot  135  are varied by the applied ramped voltage, V R . 
         [0020]      FIG. 2  shows a simple model physical system that shows the relevant characteristics of an embodiment in accordance with the invention as shown in  FIGS. 1   a - b.    FIG. 2  shows a one-dimensional quantum mechanical representation of the physical system where the potential energy levels, V(x), represent the identified physical regions as shown. The energy levels μ f   (L)  and μ f   (R)  correspond to the Fermi level of electrodes  120  and  130 , respectively, and the height of potential barriers  215  and  225  roughly corresponds to the work function of the electrode material. The detailed shapes of potential barriers  215 ,  225  and quantum well  230  corresponding to scanned spot  135  are not important but the general characteristic of a quantum well with a distinct energy level spectrum that is separated by potential barriers from the tunneling electrodes is crucial. From straightforward quantum mechanical calculations it is well-known that for a double potential barrier as shown in  FIG. 2 , the transmission probability of a particle incident on potential barrier  215  is unity when its incident energy, E e , matches one of the bound state energies, E 1 , E 2 , E 3  . . . of quantum well  230  and the integrated magnitudes of potential barriers  215  and  225  are equal. 
         [0021]    In accordance with the invention as shown in  FIG. 1   a,  the tunneling voltage, V t , applied across tunneling electrodes  120  and  130  is ramped over a range that results in the Fermi level, μ f   (L)  of electrode  120  being scanned over the energy spectrum of scanned spot  135  which is represented by quantum well  230 .  FIGS. 3   a - c  show that at specific applied voltages, the Fermi level, μ f   (L) , will sequentially match the bound state energies E 1 , E 2 ,E 3  . . . of scanned spot  135  which typically results in a large increase in the tunneling current, I T . Specifically,  FIG. 3   a  shows the Fermi level, μ f   (L) , equal to the bound state energy, E 1 ;  FIG. 3   b  shows the Fermi level, μ f   (L) , equal to the bound state energy, E 2 ; and  FIG. 3   c  shows the Fermi level, μ f   (L) , equal to the bound state energy, E 3 , for three different tunneling voltages, V t , respectively.  FIGS. 4   a  and  4   b  show the expected behavior of the measured tunneling current, I T , and differential conductance, σ, respectively, as a function of tunneling voltage, V t , for the situation shown in  FIGS. 3   a - c.  The tunneling current, I T , increases dramatically and then plateaus as the Fermi level reaches and then passes each of the bound state energies. This is due to the fact that when the Fermi level reaches a bound state energy of quantum well  230 , there are now electrons in electrode  120  with energy that match this bound state energy, and resonant quantum tunneling can occur. As the Fermi level is increased further, resonant quantum tunneling can continue through this energy level, as there continue to exist electrons in electrode  120  available at the well bound state energy, and the tunneling current stays constant until an additional bound state energy is reached. The differential conductance, a which is equal to the derivative of the tunneling current I T , with respect to the tunneling voltage, V t , or σ=dI T /dV t , correspondingly has isolated peaks at these transition energies between tunneling current plateaus. 
         [0022]    The effects of a non-zero temperature for tunneling electrodes  120  and  130  are potentially significant. At zero temperature, the Fermi level denotes the energy below which all electron levels in the Fermi sea of electrodes  120  and  130  are occupied, and above which all electron levels in the Fermi sea of electrodes  120  and  130  are unoccupied. Thus, as the tunneling voltage, V t , is increased and the Fermi level passes through a narrow well bound state energy, a very sharp turn-on of the tunneling current, I T , occurs, as shown in  FIG. 4   a,  and the differential conductance, σ, is very sharply peaked. However, for non-zero temperatures, the occupation numbers of the energy levels in the Fermi sea of electrodes  120  and  130  are smeared out due to thermal effects, and a number of levels near the Fermi energy are statistically occupied according to a Fermi-Dirac distribution function. This smearing of the occupation numbers of the electrode Fermi sea near the Fermi energy causes current steps  410  shown in  FIG. 4   a  to be rounded, with a corresponding decrease and widening of differential conductance peaks  411 ,  412  and  413  shown in  FIG. 4   b.  It is to be expected that, for some applications, this decrease in spectral resolution may have a material impact. For these situations, the RQTSM is typically operated at low temperature, in a manner similar to standard STM and AFM systems. 
         [0023]    In an embodiment in accordance with the invention as shown in  FIG. 1   b,  the tunneling voltage, V t , applied across tunneling electrodes  120  and  130  is kept constant so that the Fermi level, μ f   (L) , does not change. Ramped voltage, V R , is applied between metal layer  175  below target surface  110  and tunneling electrode voltage source  185 . The effect of ramped voltage, V R , is to vary the energy levels of scanned spot  135  which is represented by quantum well  230 .  FIGS. 5   a - c  show that at specific ramped voltages, V R , the Fermi level, μ f   (L) . will sequentially match the modified bound state energies E 1 ,E 2 ,E 3  . . . of scanned spot  135  which typically results in a large increase in the tunneling current, I T . Specifically,  FIG. 5   a  shows a situation for a ramped voltage V R0 , and the Fermi level μ f   (L)  is equal to E 1 .  FIG. 5   b  shows the situation for ramped voltage, V R =E 2 −E 1 +V R0 , and the fixed Fermi level is equal to E 2 .  FIG. 5   c  shows the situation for ramped voltage, V R =E 3 −E 1 +V R0 , and the fixed Fermi level is now equal to E 3 . 
         [0024]    An examination of the one-dimensional quantum mechanical double-barrier transmission problem shows the conditions for resonant quantum tunneling (RQT). The Appendix shows the exact solution for the transmission problem for the one-dimensional quantum mechanical double-barrier and shows that the transmission probability only becomes unity when the energy, E e , of an incident particle matches a bound state energy level and the integrated magnitude of potential barriers  215  and  225  are equal as noted above. For the double-barrier structure shown in  FIG. 6 , the non-resonant tunneling probability has the form: 
         [0000]      T tot ˜T L. ·T R .   (1) 
         [0000]    where T L  and T R  are the quantum mechanical tunneling probabilities through left potential barrier  215  and right potential barrier  225 , respectively. The quantum mechanical tunneling probabilities, T L,R  have an exponential dependence on the integrated magnitude of the potential barriers: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       T 
                       
                         L 
                         , 
                         R 
                       
                     
                     ~ 
                     
                        
                       
                         
                           - 
                           2 
                         
                          
                         
                           K 
                           
                             L 
                             , 
                             R 
                           
                         
                          
                         l 
                       
                     
                   
                    
                   
                       
                   
                    
                   where 
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
             
               
                 
                   
                     K 
                     
                       L 
                       , 
                       R 
                     
                   
                   = 
                   
                     
                       
                         2 
                          
                         
                           μ 
                            
                           
                             ( 
                             
                               
                                 V 
                                 
                                   L 
                                   , 
                                   R 
                                 
                               
                               - 
                               
                                 E 
                                 e 
                               
                             
                             ) 
                           
                         
                       
                       
                         ℏ 
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where l is the barrier width, μ is the charge carrier mass and (V L −E e ) is the energy difference between the top of the left potential barrier  215 , V L , and the incident particle energy, E e  and (V R −E e ) is the energy difference between the top of the right potential barrier  225 , V R , and the incident particle energy, E e . For an order of magnitude estimate, it can be assumed that a single potential barrier has a typical height of about one eV and the single potential barrier width is typically about one nanometer for scanned spot  135  so that a single potential barrier tunneling probability is T L,R ≈3×10 −5.  These small probabilities are the limiting factors for the expected magnitude of the tunneling current, I T . 
         [0025]    A property of the double-barrier structure shown in  FIG. 6  is that there can be large enhancements of the quantum mechanical transmission probability which results in a large enhancement of the tunneling current, I T , when the incident particle energy, E e  matches the bound state energies E 1 , E 2 ,E 3  . . . of quantum well  230 . Under certain conditions, the transmission probability becomes unity, independent of the magnitude of potential barriers  215  and  225 . This is resonant quantum tunneling (RQT). As derived in the Appendix, the total transmission probability, T tot , for the general double potential barrier structure such as that shown in  FIG. 2  or  FIGS. 3   a - c  has the form 
         [0000]    
       
         
           
             
               
                 
                   
                     T 
                     tot 
                   
                   ~ 
                   
                     1 
                     
                       [ 
                       
                         
                           
                             A 
                              
                             
                               ( 
                               
                                 E 
                                 e 
                               
                               ) 
                             
                           
                           
                             
                               T 
                               L 
                             
                              
                             
                               T 
                               R 
                             
                           
                         
                         + 
                         
                           
                             B 
                              
                             
                               ( 
                               
                                 E 
                                 e 
                               
                               ) 
                             
                           
                            
                           
                             
                               T 
                               L 
                             
                             
                               T 
                               R 
                             
                           
                         
                         + 
                         
                           
                             C 
                              
                             
                               ( 
                               
                                 E 
                                 e 
                               
                               ) 
                             
                           
                            
                           
                             
                               T 
                               R 
                             
                             
                               T 
                               L 
                             
                           
                         
                         + 
                         
                           
                             D 
                              
                             
                               ( 
                               
                                 E 
                                 e 
                               
                               ) 
                             
                           
                            
                           
                             T 
                             R 
                           
                            
                           
                             T 
                             L 
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where A, B, C, D are factors of order unity that depend weakly on the incident particle energy, E e . For non-resonant tunneling, T L  and T R  are much less than unity as discussed above so that Eq. (4) for T tot  is dominated by the first term in the denominator giving Eq. (1) 
         [0000]      T tot ˜T L ·T R    (1) 
         [0000]    When the resonant quantum tunneling conditions are met, the coefficient A(E e ) in Eq. (4) goes to zero as shown in the Appendix. Eq. (4) is then dominated by a non-leading term. If potential barriers  215  and  225  are of equal integrated magnitudes, T L =T R , the total transmission probability, T tot , is of order unity. The Appendix shows that for equal potential barriers at resonance 
         [0000]      T tot =1   (5) 
         [0000]    However, if T L ≢T R , the transmission probability remains negligible. For example, if T L &lt;&lt;T R , the dominant term in Eq. (4) becomes 
         [0000]    
       
         
           
             
               
                 
                   
                     T 
                     tot 
                   
                   ~ 
                   
                     
                       T 
                       L 
                     
                     
                       T 
                       R 
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
         [0000]    which is much less than unity. Similarly, if T L &gt;&gt;T R , the dominant term in Eq. (4) becomes 
         [0000]    
       
         
           
             
               
                 
                   
                     T 
                     tot 
                   
                   ~ 
                   
                     
                       T 
                       R 
                     
                     
                       T 
                       L 
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
         [0000]    as expected from symmetry considerations. 
         [0026]    The enhanced spatial resolution provided in accordance with the invention using RQTSM can then be easily understood. For those regions on target surface  110  separated from scanned spot  135 , tunneling barriers  215  and  225  are unequal and the left and right tunneling probabilities T L  and T R , respectively, are typically of significantly different magnitudes due to the exponential dependence of T L  and T R  on the integrated barrier magnitudes as shown by Eq. (2). In this case, as shown by Eqs. (6) and (7), the gain in transmission probability, T tot  and tunneling current, I T , is small. Only for scanned spot region  135  is the large tunneling current enhancement due to RQT operative. 
       APPENDIX 
       [0027]      FIG. 6  shows the model physical system to be analyzed as one-dimensional quantum mechanical double-barrier structure  500 . Structure  500  is analyzed by solving the one-dimensional, time independent Schroedinger equation for a fixed energy incident particle and determining the transmission probability. The parameters used in the calculation are shown in  FIG. 6 . 
       A1. Double Barrier Solution 
       [0028]    It is assumed that the energy of the incident particle, E e  , is greater than the potential energy in all regions except for barrier regions 2 and 4. Under this condition, the general solutions, Ψ i , to the one-dimensional, time independent Schroedinger equation in each of the five regions shown in  FIG. 5  can be written as: 
         [0000]      Ψ 1 = A   1   e   ik     1     x   +B   1   e   −ik     1     x   (A1) 
         [0000]      Ψ 2 = A   2   e   ik     2     x   +B   2   e   −ik     2     x   (A2) 
         [0000]      Ψ 3 = A   3   e   ik     3     x   +B   3   e   −ik     3     x   (A3) 
         [0000]      Ψ 4 = A   4   e   ik     4     x   +B   4   e   −ik     4     x   (A4) 
         [0000]      Ψ 5 = A   5   e   ik     5     x   +B   5   e   −ik     5     x   (A5) 
         [0000]    where  hk   1,3,5 =√{square root over (2μ( E   e − E   e ).)}(A6)
 
and
 
         [0000]        hk   2,4 =√{square root over (2μ( V   2,4 − E   e ).)}  (A7) 
         [0029]    The solution specific solution is determined by matching Ψ and dΨ/dx at the region interfaces. This procedure may be accomplished as a pair of subproblems. Matching the boundary conditions across the first barrier allows the wavefunction coefficients in region  1 , A 1,  B 1  to be written in terms of the wavefunction coefficients in region  3 , A 3,  B 3    
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ( 
                       
                         
                           
                             
                               A 
                               1 
                             
                           
                         
                         
                           
                             
                               B 
                               1 
                             
                           
                         
                       
                       ) 
                     
                     = 
                     
                       
                         ( 
                         
                           
                             
                               
                                 M 
                                 11 
                               
                             
                             
                               
                                 M 
                                 12 
                               
                             
                           
                           
                             
                               
                                 M 
                                 21 
                               
                             
                             
                               
                                 M 
                                 22 
                               
                             
                           
                         
                         ) 
                       
                        
                       
                         ( 
                         
                           
                             
                               
                                 A 
                                 3 
                               
                             
                           
                           
                             
                               
                                 B 
                                 3 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                    
                   
                     
 
                   
                    
                   where 
                 
               
               
                 
                   ( 
                   A8 
                   ) 
                 
               
             
             
               
                 
                   
                     M 
                     11 
                   
                   = 
                   
                     
                       
                         
                           
                             
                               - 
                               
                                 
                                   i 
                                    
                                   
                                     ( 
                                     
                                       
                                         k 
                                         1 
                                         2 
                                       
                                       + 
                                       
                                         k 
                                         2 
                                         2 
                                       
                                     
                                     ) 
                                   
                                 
                                 
                                   1 
                                   / 
                                   2 
                                 
                               
                             
                              
                             
                               
                                 ( 
                                 
                                   
                                     k 
                                     2 
                                     2 
                                   
                                   + 
                                   
                                     k 
                                     3 
                                     2 
                                   
                                 
                                 ) 
                               
                               
                                 1 
                                 / 
                                 2 
                               
                             
                              
                             
                                
                               
                                 
                                    
                                    
                                   
                                     ( 
                                     
                                       
                                         k 
                                         1 
                                       
                                       + 
                                       
                                         k 
                                         3 
                                       
                                     
                                     ) 
                                   
                                 
                                  
                                 a 
                               
                             
                           
                         
                       
                       
                         
                           
                             ( 
                             
                               
                                  
                                 
                                   
                                     2 
                                      
                                     
                                       k 
                                       2 
                                     
                                      
                                     a 
                                   
                                   + 
                                   
                                      
                                      
                                     
                                       ( 
                                       
                                         
                                           φ 
                                           2 
                                         
                                         + 
                                         
                                           φ 
                                           3 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                               - 
                               
                                  
                                 
                                   
                                     
                                       - 
                                       2 
                                     
                                      
                                     
                                       k 
                                       2 
                                     
                                      
                                     a 
                                   
                                   - 
                                   
                                      
                                      
                                     
                                       ( 
                                       
                                         
                                           φ 
                                           2 
                                         
                                         + 
                                         
                                           φ 
                                           3 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                     
                       4 
                        
                       
                         k 
                         1 
                       
                        
                       
                         k 
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   A9 
                   ) 
                 
               
             
             
               
                 
                   
                     M 
                     12 
                   
                   = 
                   
                     
                       
                         
                           
                             
                               - 
                               
                                 
                                   i 
                                    
                                   
                                     ( 
                                     
                                       
                                         k 
                                         1 
                                         2 
                                       
                                       + 
                                       
                                         k 
                                         2 
                                         2 
                                       
                                     
                                     ) 
                                   
                                 
                                 
                                   1 
                                   / 
                                   2 
                                 
                               
                             
                              
                             
                               
                                 ( 
                                 
                                   
                                     k 
                                     2 
                                     2 
                                   
                                   + 
                                   
                                     k 
                                     3 
                                     2 
                                   
                                 
                                 ) 
                               
                               
                                 1 
                                 / 
                                 2 
                               
                             
                              
                             
                                
                               
                                 
                                    
                                    
                                   
                                     ( 
                                     
                                       
                                         k 
                                         1 
                                       
                                       - 
                                       
                                         k 
                                         3 
                                       
                                     
                                     ) 
                                   
                                 
                                  
                                 a 
                               
                             
                           
                         
                       
                       
                         
                           
                             ( 
                             
                               
                                  
                                 
                                   
                                     2 
                                      
                                     
                                       k 
                                       2 
                                     
                                      
                                     a 
                                   
                                   + 
                                   
                                      
                                      
                                     
                                       ( 
                                       
                                         
                                           φ 
                                           2 
                                         
                                         - 
                                         
                                           φ 
                                           3 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                               + 
                               
                                  
                                 
                                   
                                     
                                       - 
                                       2 
                                     
                                      
                                     
                                       k 
                                       2 
                                     
                                      
                                     a 
                                   
                                   - 
                                   
                                      
                                      
                                     
                                       ( 
                                       
                                         
                                           φ 
                                           2 
                                         
                                         - 
                                         
                                           φ 
                                           3 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                     
                       4 
                        
                       
                         k 
                         1 
                       
                        
                       
                         k 
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   A10 
                   ) 
                 
               
             
             
               
                 
                   
                     M 
                     22 
                   
                   = 
                   
                     M 
                     11 
                     * 
                   
                 
               
               
                 
                   ( 
                   A11 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       M 
                       21 
                     
                     = 
                     
                       M 
                       12 
                       * 
                     
                   
                    
                   
                     
 
                   
                    
                   and 
                 
               
               
                 
                   ( 
                   A12 
                   ) 
                 
               
             
             
               
                 
                   
                     φ 
                     2 
                   
                   = 
                   
                     a 
                      
                     
                         
                     
                      
                     
                       tan 
                        
                       
                         ( 
                         
                           
                             k 
                             2 
                           
                           / 
                           
                             k 
                             1 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   A13 
                   ) 
                 
               
             
             
               
                 
                   
                     φ 
                     3 
                   
                   = 
                   
                     a 
                      
                     
                         
                     
                      
                     
                       
                         tan 
                          
                         
                           ( 
                           
                             
                               k 
                               2 
                             
                             / 
                             
                               k 
                               3 
                             
                           
                           ) 
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   A14 
                   ) 
                 
               
             
           
         
       
     
         [0030]    Similarly, matching the boundary conditions across the second barrier allows the wavefunction coefficients in region  3 , A 3,  B 3  to be written in terms of the coefficients in region  5 , A 5,  B 5    
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ( 
                       
                         
                           
                             
                               A 
                               3 
                             
                           
                         
                         
                           
                             
                               B 
                               3 
                             
                           
                         
                       
                       ) 
                     
                     = 
                     
                       
                         ( 
                         
                           
                             
                               
                                 N 
                                 11 
                               
                             
                             
                               
                                 N 
                                 12 
                               
                             
                           
                           
                             
                               
                                 N 
                                 21 
                               
                             
                             
                               
                                 N 
                                 22 
                               
                             
                           
                         
                         ) 
                       
                        
                       
                         ( 
                         
                           
                             
                               
                                 A 
                                 5 
                               
                             
                           
                           
                             
                               
                                 B 
                                 5 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                    
                   
                     
 
                   
                    
                   where 
                 
               
               
                 
                   ( 
                   A15 
                   ) 
                 
               
             
             
               
                 
                   
                     N 
                     11 
                   
                   = 
                   
                     
                       
                         
                           
                             
                               - 
                               
                                 
                                   i 
                                    
                                   
                                     ( 
                                     
                                       
                                         k 
                                         3 
                                         2 
                                       
                                       + 
                                       
                                         k 
                                         4 
                                         2 
                                       
                                     
                                     ) 
                                   
                                 
                                 
                                   1 
                                   / 
                                   2 
                                 
                               
                             
                              
                             
                               
                                 ( 
                                 
                                   
                                     k 
                                     4 
                                     2 
                                   
                                   + 
                                   
                                     k 
                                     5 
                                     2 
                                   
                                 
                                 ) 
                               
                               
                                 1 
                                 / 
                                 2 
                               
                             
                              
                             
                                
                               
                                 
                                   
                                     - 
                                      
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       k 
                                       3 
                                     
                                      
                                     
                                       ( 
                                       
                                         a 
                                         + 
                                         L 
                                       
                                       ) 
                                     
                                   
                                 
                                 + 
                                 
                                    
                                    
                                   
                                       
                                   
                                    
                                   
                                     k 
                                     5 
                                   
                                    
                                   b 
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             ( 
                             
                               
                                  
                                 
                                   
                                     2 
                                      
                                     
                                       k 
                                       4 
                                     
                                      
                                     b 
                                   
                                   + 
                                   
                                      
                                      
                                     
                                       ( 
                                       
                                         
                                           φ 
                                           4 
                                         
                                         + 
                                         
                                           φ 
                                           5 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                               - 
                               
                                  
                                 
                                   
                                     
                                       - 
                                       2 
                                     
                                      
                                     
                                       k 
                                       4 
                                     
                                      
                                     b 
                                   
                                   - 
                                   
                                      
                                      
                                     
                                       ( 
                                       
                                         
                                           φ 
                                           4 
                                         
                                         + 
                                         
                                           φ 
                                           5 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                     
                       4 
                        
                       
                         k 
                         3 
                       
                        
                       
                         k 
                         4 
                       
                     
                   
                 
               
               
                 
                   ( 
                   A16 
                   ) 
                 
               
             
             
               
                 
                   
                     N 
                     12 
                   
                   = 
                   
                     
                       
                         
                           
                             
                               - 
                               
                                 
                                   i 
                                    
                                   
                                     ( 
                                     
                                       
                                         k 
                                         3 
                                         2 
                                       
                                       + 
                                       
                                         k 
                                         4 
                                         2 
                                       
                                     
                                     ) 
                                   
                                 
                                 
                                   1 
                                   / 
                                   2 
                                 
                               
                             
                              
                             
                               
                                 ( 
                                 
                                   
                                     k 
                                     4 
                                     2 
                                   
                                   + 
                                   
                                     k 
                                     5 
                                     2 
                                   
                                 
                                 ) 
                               
                               
                                 1 
                                 / 
                                 2 
                               
                             
                              
                             
                                
                               
                                 
                                   
                                     - 
                                      
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       k 
                                       3 
                                     
                                      
                                     
                                       ( 
                                       
                                         a 
                                         + 
                                         L 
                                       
                                       ) 
                                     
                                   
                                 
                                 + 
                                 
                                    
                                    
                                   
                                       
                                   
                                    
                                   
                                     k 
                                     5 
                                   
                                    
                                   b 
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             ( 
                             
                               
                                 - 
                                 
                                    
                                   
                                     
                                       2 
                                        
                                       
                                         k 
                                         4 
                                       
                                        
                                       b 
                                     
                                     + 
                                     
                                        
                                        
                                       
                                         ( 
                                         
                                           
                                             φ 
                                             4 
                                           
                                           + 
                                           
                                             φ 
                                             5 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                               + 
                               
                                  
                                 
                                   
                                     
                                       - 
                                       2 
                                     
                                      
                                     
                                       k 
                                       4 
                                     
                                      
                                     b 
                                   
                                   - 
                                   
                                      
                                      
                                     
                                       ( 
                                       
                                         
                                           φ 
                                           4 
                                         
                                         + 
                                         
                                           φ 
                                           5 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                     
                       4 
                        
                       
                         k 
                         3 
                       
                        
                       
                         k 
                         4 
                       
                     
                   
                 
               
               
                 
                   ( 
                   A17 
                   ) 
                 
               
             
             
               
                 
                   
                     N 
                     22 
                   
                   = 
                   
                     N 
                     11 
                     * 
                   
                 
               
               
                 
                   ( 
                   A18 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       N 
                       21 
                     
                     = 
                     
                       N 
                       12 
                       * 
                     
                   
                    
                   
                     
 
                   
                    
                   and 
                 
               
               
                 
                   ( 
                   A19 
                   ) 
                 
               
             
             
               
                 
                   
                     φ 
                     4 
                   
                   = 
                   
                     a 
                      
                     
                         
                     
                      
                     
                       tan 
                        
                       
                         ( 
                         
                           
                             k 
                             4 
                           
                           / 
                           
                             k 
                             3 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   A20 
                   ) 
                 
               
             
             
               
                 
                   
                     φ 
                     5 
                   
                   = 
                   
                     a 
                      
                     
                         
                     
                      
                     
                       
                         tan 
                          
                         
                           ( 
                           
                             
                               k 
                               4 
                             
                             / 
                             
                               k 
                               5 
                             
                           
                           ) 
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   A21 
                   ) 
                 
               
             
           
         
       
     
         [0031]    The full expression connecting the wavefunction coefficients of region  1  with those of region  5  is then determined by the linear transformation using Eqs. (A8) and A(15) 
         [0000]    
       
         
           
             
               
                 
                   
                     ( 
                     
                       
                         
                           
                             A 
                             1 
                           
                         
                       
                       
                         
                           
                             B 
                             1 
                           
                         
                       
                     
                     ) 
                   
                   = 
                   
                     
                       ( 
                       
                         
                           
                             
                               M 
                               11 
                             
                           
                           
                             
                               M 
                               12 
                             
                           
                         
                         
                           
                             
                               M 
                               21 
                             
                           
                           
                             
                               M 
                               22 
                             
                           
                         
                       
                       ) 
                     
                      
                     
                       ( 
                       
                         
                           
                             
                               N 
                               11 
                             
                           
                           
                             
                               N 
                               12 
                             
                           
                         
                         
                           
                             
                               N 
                               21 
                             
                           
                           
                             
                               N 
                               22 
                             
                           
                         
                       
                       ) 
                     
                      
                     
                       
                         ( 
                         
                           
                             
                               
                                 A 
                                 5 
                               
                             
                           
                           
                             
                               
                                 B 
                                 5 
                               
                             
                           
                         
                         ) 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   A22 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The full transmission coefficient is determined by applying the boundary condition 
         [0000]    
       
         
           
             
               
                 
                   
                     ( 
                     
                       
                         
                           
                             
                               A 
                               1 
                             
                             = 
                             1 
                           
                         
                       
                       
                         
                           
                             B 
                             1 
                           
                         
                       
                     
                     ) 
                   
                   = 
                   
                     
                       ( 
                       
                         
                           
                             
                               M 
                               11 
                             
                           
                           
                             
                               M 
                               12 
                             
                           
                         
                         
                           
                             
                               M 
                               21 
                             
                           
                           
                             
                               M 
                               22 
                             
                           
                         
                       
                       ) 
                     
                      
                     
                       ( 
                       
                         
                           
                             
                               N 
                               11 
                             
                           
                           
                             
                               N 
                               12 
                             
                           
                         
                         
                           
                             
                               N 
                               21 
                             
                           
                           
                             
                               N 
                               22 
                             
                           
                         
                       
                       ) 
                     
                      
                     
                       ( 
                       
                         
                           
                             
                               A 
                               5 
                             
                           
                         
                         
                           
                             
                               
                                 B 
                                 5 
                               
                               = 
                               0 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   A23 
                   ) 
                 
               
             
           
         
       
     
         [0000]    which corresponds to an incident wave of unit amplitude from the left (A 1 =1) in  FIG. 5  and no wave incident from the right (B 5 =0) in  FIG. 5 . Hence, the total transmission probability is given by 
         [0000]    
       
         
           
             
               
                 
                   
                     T 
                     tot 
                   
                   = 
                   
                     
                       
                         k 
                         5 
                       
                       
                         k 
                         1 
                       
                     
                      
                     
                       
                         
                            
                           
                             1 
                             
                               
                                 
                                   M 
                                   11 
                                 
                                  
                                 
                                   N 
                                   11 
                                 
                               
                               + 
                               
                                 
                                   M 
                                   12 
                                 
                                  
                                 
                                   N 
                                   21 
                                 
                               
                             
                           
                            
                         
                         2 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   A24 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Explicit evaluation of Eq. (A24) and collecting and grouping terms gives 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       T 
                       tot 
                     
                     = 
                     
                       
                         
                           
                             2 
                             6 
                           
                            
                           
                             k 
                             1 
                           
                            
                           
                             k 
                             2 
                             2 
                           
                            
                           
                             k 
                             3 
                             2 
                           
                            
                           
                             k 
                             4 
                             2 
                           
                            
                           
                             k 
                             5 
                           
                         
                         
                           
                             ( 
                             
                               
                                 k 
                                 1 
                                 2 
                               
                               + 
                               
                                 k 
                                 2 
                                 2 
                               
                             
                             ) 
                           
                            
                           
                             ( 
                             
                               
                                 k 
                                 2 
                                 2 
                               
                               + 
                               
                                 k 
                                 3 
                                 2 
                               
                             
                             ) 
                           
                            
                           
                             ( 
                             
                               
                                 k 
                                 3 
                                 2 
                               
                               + 
                               
                                 k 
                                 4 
                                 2 
                               
                             
                             ) 
                           
                            
                           
                             ( 
                             
                               
                                 k 
                                 4 
                                 2 
                               
                               + 
                               
                                 k 
                                 5 
                                 2 
                               
                             
                             ) 
                           
                         
                       
                        
                       
                         1 
                         F 
                       
                     
                   
                    
                   
                     
 
                   
                    
                   where 
                 
               
               
                 
                   ( 
                   A25 
                   ) 
                 
               
             
             
               
                 
                   F 
                   = 
                   
                     
                       
                          
                         
                           
                             2 
                              
                             
                               γ 
                               2 
                             
                           
                           + 
                           
                             2 
                              
                             
                               γ 
                               4 
                             
                           
                         
                       
                        
                       
                         
                           sin 
                           2 
                         
                          
                         
                           ( 
                           
                             
                               φ 
                               1 
                             
                             - 
                             
                               φ 
                               3 
                             
                             - 
                             
                               φ 
                               4 
                             
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                          
                         
                           2 
                            
                           
                             γ 
                             2 
                           
                         
                       
                        
                       
                         cos 
                          
                         
                           ( 
                           
                             2 
                              
                             
                               φ 
                               5 
                             
                           
                           ) 
                         
                       
                        
                       
                         ( 
                         
                           
                             - 
                             
                               cos 
                                
                               
                                 ( 
                                 
                                   2 
                                    
                                   
                                     φ 
                                     4 
                                   
                                 
                                 ) 
                               
                             
                           
                           + 
                           
                             cos 
                              
                             
                               ( 
                               
                                 
                                   2 
                                    
                                   
                                     φ 
                                     1 
                                   
                                 
                                 - 
                                 
                                   2 
                                    
                                   
                                     φ 
                                     3 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                          
                         
                           2 
                            
                           
                             γ 
                             4 
                           
                         
                       
                        
                       
                         cos 
                          
                         
                           ( 
                           
                             2 
                              
                             
                               φ 
                               2 
                             
                           
                           ) 
                         
                       
                        
                       
                         ( 
                         
                           
                             - 
                             
                               cos 
                                
                               
                                 ( 
                                 
                                   2 
                                    
                                   
                                     φ 
                                     3 
                                   
                                 
                                 ) 
                               
                             
                           
                           + 
                           
                             cos 
                              
                             
                               ( 
                               
                                 
                                   2 
                                    
                                   
                                     φ 
                                     1 
                                   
                                 
                                 - 
                                 
                                   2 
                                    
                                   
                                     φ 
                                     4 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                          
                         
                           
                             2 
                              
                             
                               γ 
                               2 
                             
                           
                           - 
                           
                             2 
                              
                             
                               γ 
                               4 
                             
                           
                         
                       
                        
                       
                         
                           sin 
                           2 
                         
                          
                         
                           ( 
                           
                             
                               φ 
                               1 
                             
                             - 
                             
                               φ 
                               3 
                             
                             - 
                             
                               φ 
                               4 
                             
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                          
                         
                           
                             2 
                              
                             
                               γ 
                               4 
                             
                           
                           - 
                           
                             2 
                              
                             
                               γ 
                               2 
                             
                           
                         
                       
                        
                       
                         
                           sin 
                           2 
                         
                          
                         
                           ( 
                           
                             
                               φ 
                               1 
                             
                             + 
                             
                               φ 
                               3 
                             
                             - 
                             
                               φ 
                               4 
                             
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                          
                         0 
                       
                        
                       
                         cos 
                          
                         
                           ( 
                           
                             
                               2 
                                
                               
                                 φ 
                                 2 
                               
                             
                             - 
                             
                               2 
                                
                               
                                 φ 
                                 5 
                               
                             
                           
                           ) 
                         
                       
                        
                       
                         ( 
                         
                           
                             - 
                             
                               cos 
                                
                               
                                 ( 
                                 
                                   2 
                                    
                                   
                                     φ 
                                     1 
                                   
                                 
                                 ) 
                               
                             
                           
                           + 
                           
                             cos 
                              
                             
                               ( 
                               
                                 
                                   2 
                                    
                                   
                                     φ 
                                     3 
                                   
                                 
                                 - 
                                 
                                   2 
                                    
                                   
                                     φ 
                                     4 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                          
                         0 
                       
                        
                       
                         cos 
                          
                         
                           ( 
                           
                             
                               2 
                                
                               
                                 φ 
                                 2 
                               
                             
                             + 
                             
                               2 
                                
                               
                                 φ 
                                 5 
                               
                             
                           
                           ) 
                         
                       
                        
                       
                         ( 
                         
                           
                             - 
                             
                               cos 
                                
                               
                                 ( 
                                 
                                   2 
                                    
                                   
                                     φ 
                                     1 
                                   
                                 
                                 ) 
                               
                             
                           
                           + 
                           
                             cos 
                              
                             
                               ( 
                               
                                 
                                   2 
                                    
                                   
                                     φ 
                                     3 
                                   
                                 
                                 + 
                                 
                                   2 
                                    
                                   
                                     φ 
                                     4 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                          
                         
                           
                             - 
                             2 
                           
                            
                           
                             γ 
                             4 
                           
                         
                       
                        
                       
                         cos 
                          
                         
                           ( 
                           
                             2 
                              
                             
                               φ 
                               2 
                             
                           
                           ) 
                         
                       
                        
                       
                         ( 
                         
                           
                             - 
                             
                               cos 
                                
                               
                                 ( 
                                 
                                   2 
                                    
                                   
                                     φ 
                                     3 
                                   
                                 
                                 ) 
                               
                             
                           
                           + 
                           
                             cos 
                              
                             
                               ( 
                               
                                 
                                   2 
                                    
                                   
                                     φ 
                                     1 
                                   
                                 
                                 + 
                                 
                                   2 
                                    
                                   
                                     φ 
                                     4 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                          
                         
                           
                             - 
                             2 
                           
                            
                           
                             γ 
                             2 
                           
                         
                       
                        
                       
                         cos 
                          
                         
                           ( 
                           
                             2 
                              
                             
                               φ 
                               5 
                             
                           
                           ) 
                         
                       
                        
                       
                         ( 
                         
                           
                             - 
                             
                               cos 
                                
                               
                                 ( 
                                 
                                   2 
                                    
                                   
                                     φ 
                                     4 
                                   
                                 
                                 ) 
                               
                             
                           
                           + 
                           
                             cos 
                              
                             
                               ( 
                               
                                 
                                   2 
                                    
                                   
                                     φ 
                                     1 
                                   
                                 
                                 + 
                                 
                                   2 
                                    
                                   
                                     φ 
                                     3 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                          
                         
                           
                             
                               - 
                               2 
                             
                              
                             
                               γ 
                               2 
                             
                           
                           - 
                           
                             2 
                              
                             
                               γ 
                               4 
                             
                           
                         
                       
                        
                       
                         
                           sin 
                           2 
                         
                          
                         
                           ( 
                           
                             
                               φ 
                               1 
                             
                             + 
                             
                               φ 
                               3 
                             
                             + 
                             
                               φ 
                               4 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   A26 
                   ) 
                 
               
             
           
         
       
     
         [0000]    and the terms of F are listed in descending powers of the large “barrier suppression factors” and where 
         [0000]      Φ 1 ≡ k   3   L   (A27) 
         [0000]      γ 2 ≡2 k   2   a   (A28) 
         [0000]      γ 4 ≡2 k   4   b .  (A29 ) 
       A2. Resonance Condition 
       [0032]    Assuming the potential baffiers are strong impediments to particle transmission, i.e. e 2γ     2   , e 2γ     4   &gt;&gt;1, for general non-resonant conditions the total transmission is dominated by the leading term in Eq. (A26) giving 
         [0000]        T   tot ˜ e   −2γ     2     −2γ     4     ˜T   l   T   R .  (A30) 
         [0000]    However, for the situation when 
         [0000]      γ 1 −γ 3 −γ 4 =   nπ   (A31) 
         [0000]    the coefficients of the first three terms in Eq. (A26) are zero. If the two potential baffiers are of equal integrated magnitudes, i.e. γ 2 =γ 4 , then the leading term in Eq. (A26) is of the order unity, and the total transmission coefficient can be shown to approach unity. This is the condition for resonant quantum tunneling and exhibits the property of total transmission through a double potential barrier structure, regardless of the strength of the individual bafflers as long as the potential barriers are of equal integrated magnitudes. 
         [0033]    It is important to understand the physical significance of the resonance condition of Eq. (A3 1). To simplify the discussion, the completely symmetric case will be considered where 
         [0000]      Φ 3 = a tan( k   2   lk   3 )= a tan( k   4 k 3 )=Φ 4   (A32) 
         [0000]    which gives 
         [0000]      sin(Φ 1 −2Φ 3 =0.  (A33) 
         [0000]    Using trigonometric identities and substituting the definitions for Φ 1  and Φ 3 , allows Eq. (A33) to be rewritten as 
         [0000]    
       
         
           
             
               
                 
                   
                     tan 
                      
                     
                       ( 
                       
                         
                           k 
                           3 
                         
                          
                         L 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           
                             
                               V 
                               2 
                             
                             - 
                             
                               E 
                               e 
                             
                           
                         
                          
                         
                           
                             
                               E 
                               e 
                             
                             - 
                             
                               V 
                               3 
                             
                           
                         
                       
                       
                         
                           E 
                           e 
                         
                         - 
                         
                           
                             ( 
                             
                               
                                 V 
                                 2 
                               
                               + 
                               
                                 V 
                                 3 
                               
                             
                             ) 
                           
                           / 
                           2 
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   A34 
                   ) 
                 
               
             
           
         
       
     
         [0000]    If the arbitrary baseline for the potential energy is chosen to be V 3 ≡0 and V 2  is renamed V 0 , Eq. (A34) becomes 
         [0000]    
       
         
           
             
               
                 
                   
                     tan 
                      
                     
                       ( 
                       
                         
                           k 
                           3 
                         
                          
                         L 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           
                             
                               V 
                               0 
                             
                             - 
                             
                               E 
                               e 
                             
                           
                         
                          
                         
                           
                             E 
                             e 
                           
                         
                       
                       
                         
                           E 
                           e 
                         
                         - 
                         
                           
                             V 
                             0 
                           
                           / 
                           2 
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   A35 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Eq. (A35) is the eigenvalue equation for the energy levels of a quantum square well potential with the parameters as defined above. This shows why the phenomenon of total transmission through a double-well structure is called resonant quantum tunneling. The condition for resonant quantum tunneling is that the energy, E e , of the incident particle matches any of the bound state energy levels of the quantum square well. Whenever the incident energy, E e , matches any of the bound state energy levels, the total transmission probability increases dramatically, as long as the double potential barriers are symmetric. 
       A3. Tunneling Current on Resonance 
       [0034]    As discussed above, for a symmetric potential barrier structure, the transmission probability becomes unity when the incident energy, E e , passes through a bound state energy level of the quantum square well. However, the situation is different for a double potential barrier structure that has asymmetric potential barriers. For the general asymmetric potential barrier structure on resonance, it follows from Eq. (A26) that the dominant terms have the form 
         [0000]        F˜e   2 γ     2     −2 γ     4   sin 2 (Φ 1 +Φ 3 −Φ 4 ).  (A36) 
         [0000]    This implies that for the situation where the left potential baffler in FIG. 5 is larger than the right potential barrier in  FIG. 5  (γ 2 &gt;&gt;γ   4 ) 
         [0000]    
       
         
           
             
               
                 
                   
                     T 
                     tot 
                   
                   ∼ 
                   
                      
                     
                       
                         2 
                          
                         
                           γ 
                           4 
                         
                       
                       - 
                       
                         2 
                          
                         
                           γ 
                           2 
                         
                       
                     
                   
                   ∼ 
                   
                     
                       T 
                       L 
                     
                     
                       T 
                       R 
                     
                   
                 
               
               
                 
                   ( 
                   A37 
                   ) 
                 
               
             
           
         
       
     
         [0000]    and for the reverse situation (γ 4 &gt;&gt;γ 2 ) 
         [0000]    
       
         
           
             
               
                 
                   
                     T 
                     tot 
                   
                   ∼ 
                   
                      
                     
                       
                         2 
                          
                         
                           γ 
                           2 
                         
                       
                       - 
                       
                         2 
                          
                         
                           γ 
                           4 
                         
                       
                     
                   
                   ∼ 
                   
                     
                       
                         T 
                         R 
                       
                       
                         T 
                         L 
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   A38 
                   ) 
                 
               
             
           
         
       
     
         [0000]    This shows the different resonant tunneling behavior for the asymmetric double potential barrier structure . If the potential baffler structure is highly asymmetric, there is very little gain in the tunneling probability as the resonance condition is approached. Only under the condition of double potential baffler symmetry is resonant quantum tunneling in effect.