Open access peer-reviewed chapter

# Dynamic Resonant Tunneling

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Submitted: November 9th, 2015 Reviewed: February 3rd, 2016 Published: August 24th, 2016

DOI: 10.5772/62384

From the Edited Volume

## Research Advances in Quantum Dynamics

Edited by Paul Bracken

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## Abstract

The physics of dynamic resonant tunneling is investigated.

### Keywords

• Resonant Tunneling
• Dynamic Tunneling
• Vibrational Tunneling
• Odor detection
• Olfactory
• Forbidden activation
• Selected Activation

## 1. Introduction

Resonant tunneling is a fascinating quantum phenomenon. It manifests the ability of quantum particles to pass with high probability through an opaque barrier by traveling via a semibound state [13].

The high sensitivity of the current on the bound-state parameters suggested harnessing this effect to heterostructure devices in general and transistors in particular [410].

Resonant tunneling is usually described as a one-dimensional (1D) phenomenon; however, resonant tunneling in higher number of dimensions was also investigated (see, for example, [11, 12]). Nevertheless, because the main features of resonant tunneling appear in 1D, most of the research was concentrated on the simplest 1D systems.

Tunneling and resonant tunneling are rarely stationary processes. They are affected by thermal noises, and clearly, the accumulation of particles in the bound state varies the potential. It is well known that tunneling in the presence of an oscillating barrier can cause activation (higher energy) and therefore can increase substantially the tunneling current.

This phenomenon was investigated in electronics [13], nanotechnology [1416], the foundations of quantum mechanics [1733], and even biology and biochemistry [3439].

The resonant tunneling effect occurs when the incoming particle’s energy coincides with the eigenenergy of the quasi-bound state. In case the barrier is very opaque, the particle remains inside the well at the quasi-bound state for exponentially long time. Therefore, when the particle is quasi-trapped inside the well, its state has to vary with the changes in the well, and its energy varies with the eigenenergy of the quasi-bound state because it does not have the time to escape from the well. Therefore, it was conjectured (see refs. [17, 20]) that a decrease in the perturbation time-scale will enhance the activation. However, not in any energy the particle can remain within the well. Destructive interference can prevent particle trapping and therefore suppress particle activation [23, 33]. In the next several sections, we will elaborate on the delicate structure of these effects.

## 2. Stationary tunneling

Let us begin with the propagation of a quantum particle through an opaque but stationary barrier. The Schrödinger equation is then

2x2ψxt+Uxψxt=iψxttE1

Hereinafter for simplicity, we adopt the units, where the electron mass is half and the reduced Planck constant is unity (i.e., m=1/2 and =1).

In the stationary case (i.e., when the potential is time independent), there is no change in the incoming particle’s energy. For any incoming energy ω, the generic stationary solution looks like

ψxt=ϕωxexpiωtE2

where φω(x) are the solutions of the stationary Schrödinger equation:

2x2ϕωx+Uxωϕωx=0E3

Therefore, every solution can be written as a superposition of incoming [φω+(x)] and outgoing [φω(x)] solutions, where

φω+x=expiωx+rωexpiωxxtωexpiωxxE4
φωx=tωexpiωxxexpiωx+rωexpiωxx

Physically, φω+(x) and φω(x) stand for beams of particles coming from the left and right respectively. tω and rω are the transmission and reflection coefficients of the barrier for energy ω, respectively. In the WKB approximation (see, for example, ref. [19]), the transmission coefficient can be evaluated as

tω~expxLxRUxωdxE5

where xLand xRare the left and right boundaries of the barrier. For a rectangular barrier, an exact expression can be derived [19]:

tω=exp2ikLcosh2KL+12iK/kk/Ksinh2KLE6

where kωand KUω. Therefore, when the barrier is opaque [i.e., UωL>>1(high and/or wide)], the transmission is exponentially small.

## 3. Resonant tunneling via a delta-function well

Let us introduce a delta-function well in the barrier (at x= x0; in Figure 1, x0=0). Then, the Schrödinger equation is

2x2ϕωx+Uxωf0δxx0ϕωx=0E7

It is convenient to use the outgoing Green function Gω+(x,x0), which is a solution of the equation:

2x2Gω+xx0+UxωGω+xx0=δxx0E8

with the boundary conditions:

Gω+xx0~expiωxx0xE9

Therefore, the outgoing Green function reads

Gω+xx0=ϕωx0ϕω+xϕω+x0ϕω'x0ϕωx0ϕω+'x0x>x0ϕωxϕω+x0ϕω+x0ϕω'x0ϕωx0ϕω+'x0x<x0E10

Using the Green function, we can easily construct a solution for the wave equation with the combined potential (the barrier with the delta-function well). In which case, the solution reads

ϕωx=ϕω+x+f0ϕω+x01f0Gω+x0x0Gω+xx0E11

which for x>x0 is

φωx=φω+x>x01f0Gω+x0x0=φω+x>x01f0/φω'x0/φωx0φω+'x0/φω+x0E120

In case of a rectangular barrier, i.e.,

Ux=UxL0x>LE12

then

φω+x>L=tωexpikxE13

where 2Lis the width of the barrier, and tω is taken from (6), i.e.,

φωx=iexpikx2LiarctanK2k2/2kK1+i1f02K/4kKUexp2KLE14

At the resonance ω=Ω*=Uf02/4, T=|tω|2=1, (see Figure 2).

Now, let us take a varying potential well,

ft/τ=λ0τexpt/τ2E15

In the adiabatic approximation, τ is longer than any other time-scale of the problem. The relevant time-scale (i.e., the longest one) is the resonance time or the dwelling time of the resonant state. Therefore, in the adiabatic approximation,

τU>>exp2UωLx0E16

the potential can be regarded as stationary, and the wavefunction simply reads

φωx=φω+xφω+x01τexpt/τ2/λ0Gω+x0x0Gω+xx0Gω+x0x0E17

The resonance energy of this system varies in time Ω*(t)=Uf2(t/τ)/4; therefore, large transmission occurs for ω=Ω*(t) and, in principle, can be as high as 1 when the well is located at the center of the barrier, i.e.,

When the lowest eigenenergy of the well

Ωmin*minΩ*t=Uf20/4=Uλ02/4τ2E18

is higher than the incoming energy ω, then the effect of the varying well is negligible; however, if the lowest eigenenergy of the well is lower than the incoming energy ω, then the eigenstate crosses the incoming energy twice. At the vicinity of the crossing time tt0=±τlnλ02Kτ, the solution has the Lorentzian shape:

ψxt=iexpikx2LiarctanK2k2/2kKiωt1it0tt0/2τ2kK/Uexp2KLE19

## 5. The general scenario

In principle, in the adiabatic approximation, the outgoing energy is equal to the incoming energy (i.e., ωoutin±τ−1), where τ−1 should be exponentially small τ1<<Uexp2UωLx0. However, in practice, the situation can be quite different due to the exponential decay of low energies in the tunneling process.

In general, the generic Schrödinger equation (see refs. [17, 33])

2x2ψxt+Uxft/τδxx0ψxt=iψxttE20

can be solved by a superposition of solutions of the type (11), namely,

ψxt=ϕΩxexpiΩt+dωaωGω+xx0expiωtE21

where it is taken that the incoming energy is Ω.

By substituting solution (21) into Eq. (20) (and after spatial integration), the solution can be reduced to the integral equation [17]:

12πϕΩx0fΩω+aω12πdω"fω"ωaω"Gω"+x0x0=0E22

where fω=dtft/τexpiωtis the Fourier transform of the perturbation’s amplitude.

## 6. Adiabatic and slow variations

In general, due to the complex structure of the Green function, this is a complex integral equation; nevertheless, as long as the spectrum of the function is mainly concentrated near the incoming energy Ω, i.e., |a(|ω − Ω|τ > > 1)| < < |a(ω = Ω)|, the contribution to the integral of the components |ω − Ω|τ > > 1 is negligible; therefore, we can replace Gω+(0) with GΩ+(0). In this case, the integral equation reduces to

12πϕΩx0fΩω+aω12πdω"fω"ωaω"GΩ+x0x0=0E23

which is merely a convolution equation; therefore, the inverse Fourier transform of the solution

at=12πdωaωexpiωtE24

obeys [24, 33]

at=ϕΩx0ft/τexpiΩt1ft/τGΩ+x0x0E25

Clearly, resonance occurs when ft/τGΩ+x0x0=1. We will see in the next sections that nontrivial effects occur when f0GΩ+x0x0>1. In this regime, it is invalid to substitute GΩ+(x0,x0) for Gω+(x0,x0). However, in the f0GΩ+x0x0<1(where it is taken that f(0) = max{f(t/τ)}) regime, this approximation is still valid.

It is clear, for example, that, when f0GΩ+x0x0<<1, which is equivalent to the adiabatic regime,that

atϕΩx0ft/τexpiΩtE26

and thus

aω12πϕΩx0fωΩE290

which means that the spectrum broadening is exactly similar to the spectrum of the perturbation; in which case the solution is simply

ψxtϕΩxexpiΩtϕΩx02πdωfωΩGω+xx0expiωtE27

## 7. Activation

From Eq. (27), it is evident that elevation to higher energies is still a possibility even in the adiabatic and slowly varying cases, and because the Green function Gω+(x,x0) increases with the energy, there is still a possibility that the mean exit energy will be considerably higher than the incoming one. The outcome depends only on the specific functional shape of the perturbation spectrum.

More importantly, when the perturbation becomes more energetic and τ decreases so that f0GΩ+x0x0approaches 1 from below (i.e., there is still no intersection), the adiabatic Eq. (25) is still approximately valid. In this case, however, one can take

ft/τλ0/τ1t/τ2E28

Then, after substituting Eq. (28) in Eq. (25) and both of them in Eq. (24),

a(ω)=φΩ(x0)λ0τexp(i(ωΩ)t)1(λ0/τ)GΩ+(x0,x0)+(t/τ)2dt=φΩ(x0)λ0π2exp(1βτ|ωΩ|)E29

where βλ0/τGΩ+x0x0. Therefore,

ψxtφΩxexpiΩtφΩx0λ0π2dωexp1βτωΩGω+xx0expiωtE30

Now, because the Green function can be written (beyond the barrier) approximately as

Gω+x>L,x0~expUωLx0E31

then the exponent in the integrand, which can be regarded as an approximate evaluation of the spectrum of the outgoing wavefunction, consists of two main terms (two peaks):

exp1βτωΩUωLx0E32

The first peak occurs around the incoming energy Ω (suppressed activation) and the second one occurs around the barrier’s height U(activation).

Each one of these peaks has a different height. The higher one will determine whether activation will occur. Therefore, the probability to tunnel through the barrier and to exit with energy ωout is approximately [33]

PωoutaωoutGωout+x>L2exp21βτωoutΩ2UωoutLx0E33

which means that the activation probability peak is proportional to

PωoutUexp21βτUΩE34

whereas the inactivation probability peak (i.e., the probability for suppressed activation) goes like

PωoutΩexp2UΩLx0E35

In Figure 3, we demonstrate the fact that the spectrum is governed by two maxima (peaks) and that Eq. (33) is a good approximation to the numerical solution at the vicinity of these maxima. Therefore, activation occurs when

PωoutUexp21βτUΩ>PωoutΩexp2UΩLx0E36

This occurs approximately at the time scale

τTλ0+λ02+16L24UΩE37

or, for a given τ, the transition occurs for the following incoming energy:

ΩTU2λ0λ0+λ02+16L2+16L216τ2E38

Clearly, this energy is lower than the minimum resonance energy

ΩT<Ωmin*=Uλ024τ2E39

In Figure 4, the dependence of the spectrum on the transition time-scale τ is presented for three different values: below τT where activation prevails, above τT when simple tunneling wins, and when they are equal, and the outgoing particle’s spectrum has two equally probable outgoing energies.

It should be stressed that, because these peaks are exponentially narrow, the transition is extremely abrupt (i.e., the process resembles a phase transition). The identification of the process as a phase transition was first suggested by Azbel [17].

However, it was wrongly assumed that, if the particle’s incoming energy matches the quasi-eigenstate energy, then an eigenstate-assisted activation (EAA) effect occurs (i.e., if Ω>Ωmin*, then activation will definitely increase). In fact, it will be shown that this process is more complicated, and at some energies (above Ωmin*), activation is “totally” suppressed.

## 8. Selected elevations and forbidden activations

For Ω>Ωmin*, the spectrum’s shape becomes more complicated. Instead of only two peaks, it has a more complex structure. There is a clear difference between the ωact<Ω (i.e., the underactivated regime) and the ωact>Ω (i.e., the activated one). The former oscillates as a function of ωact but almost independent of the incoming Ω, whereas the latter oscillates as a function of the incoming Ω but has a mild dependence on the outgoing ωact. In Figure 5, there is an illustration of this behavior, where a small change in the incoming particle’s energy has an enormous effect on the activated regime. In Figure 6, a numerical example illustrates this phenomenon where a ~6.7% change in the incoming energy made a dramatic change from full activation to suppressed one.

As a consequence, it is clear that, for specific incoming particle’s energies, the entire activated part of the spectrum is suppressed. To illustrate this point, we define the mean activated energy

ωactωactPωactΩdωactPωactΩdωactE40

where PactΩ)is the probability of an incoming particle with energy Ω to exit the barrier with the energy ωact.

In Figure 7 the mean activation energy <ωact> is plotted as a function of the perturbation time scale τ, and in Figure 8, <ωact> is plotted as a function of the incoming particle’s energy Ω. It is clearly seen that activation (<ωact> ≅ U) occurs mainly below τ<τT. However, even below this time-scale, there are specific values of τ, for which activation is suppressed (i.e., <ωact> ≅ Ω). Similarly, activation occurs <ωact> ≅ Umainly above Ω>ΩT; however, even in the activation regime, there are specific energies for which <ωact> ≅ Ω (i.e., suppressed activation).

When τ<τT, two important things occur: (1) At two specific times, the particle’s incoming energy is equal to the eigenenergy of the quasi-bound state of the varying well. (2) The well varies quickly enough so the particle has no time to escape from the well.

As a consequence of these two, the particle’s state changes with the well’s eigenstate; therefore, it is easier to excite the particle energetically. That was the logic that led Azbel to predict the EAA effect. Indeed, this effect does occur, and it is clearly seen (see Figure 4) that, when τ<τT, then, for most values of τ, the spectrum’s energy is concentrated around the barrier’s height U. However, this process cannot last if the particle cannot dwell inside the quasi-bound state. This event occurs when there is destruction interference inside the well.

Had it been a stationary eigenstate with an eigenenergy Ω0 the eigenstate would accumulate a linear phase [i.e., exp(−0t)].

However, because the quasi-bound state evolves in time, it gains the integral

expidt'Ω*t'E41

When the incoming energy is above the minimum eigenenergy (i.e., Ω>Ωmin*=U−λ02/4τ2), there are two times, in which Ω=Ω*(t1)=Ω*(t2) (see Figure 9), and due to the temporal symmetry of the perturbation t1=−t2. Therefore, the particle has two options to be temporally bounded to the quasi-eigenstate: it can either begin at t1 and gain the phase expit1tdt'Ω*t'or at t2 and gain the phase expiΩt2t1it2tdt'Ω*t'. If the two components are out of phase and a destructive interference occurs [33], i.e.,

t1t2dt'Ω*t'Ω=π2m+1E42

the particle cannot survive within the well, and activation is frustrated.

In our case, at the vicinity of the parabola peak,

Ω*t=Uλ02τ2exp2t2τ2Uλ02τ212t2τ2E43

After substituting (43) into (42), the values of the forbidden energies Ωm, for which destructive interference occurs and the activation is suppressed, are directly given

ΩmΩmin*+123λ0τ2m+12π2/3E44

In each one of these energies, the activation is suppressed.

To determine these energies more accurately, we take advantage of the fact that, at the vicinity of the minimum Ωmin*, the instantaneous resonance energy has a parabola shape; therefore, any varying potential with the same parabola should have approximately the same suppressed energies. Therefore, we replace the Gaussian with a parabolic function, that is, we choose Eq. (28) for the perturbation, namely, f(t/τ) ≅ (λ0/τ)(1 − t2/τ2), then

fωλ0τ1t2τ2expiωtdt=2πλ0τδω+1τ2δ''ωE45

Therefore, the integral equation

12πϕΩx0fΩω+aω12πdω"fω"ωaω"Gω"+x0x0=0E46

reduces to the differential equation

d2sndn2+sn11Gω"+x0x0τλ0=φΩx0λ0δnE47

where we used the dimensionless parameters n=(ω−Ω)τ and .

After linearization of the Green function, Eq. (47) can be approximated to

d2sndn2+sn12Kτλ0+nλ0K=φΩx0λ0δnE48

where again KUΩ.

The solution that maintain the boundary conditions that s(n → ∞) → 0 is

sξ=iπφΩx0λ02λ0K1/3AiξAiξ0+iBiξ0forξ<ξ0Aiξ0Aiξ+iBiξforξ>ξ0E49

where ξ≡(n + K(λ0/τ − 2K)τ)/(2λ0K)1/3, ξ0≡(K(λ0/τ − 2K)τ)/(2λ0K)1/3 and Ai and Bi are the Airy functions [40].

Therefore, it is clear that activation is suppressed when

Aiξ0=0E50

which, in the slowly varying approximation (i.e., large τ), correspond to (see [40])

cos22312Kλ0τ3/2Kλ0π4=0E51

Therefore, the incoming energies for which Ωact ≅ Ω, and thus no activation occurs, are approximately (see Figures 10 and 11)

Ωm=Ωmin*+123λ0τ2m+34π2/3E52

Therefore, Eq. (42) should be rewritten more accurately as

t1t2dt'Ω*t'Ω=π2m+32E53

like destructive interference condition in the WKB approximation (see, for example, ref. [19]). Eq. (53) can be applied to any varying potential whose temporal shape has minima.

Because (52) was derived for any potential, which can be approximated by a parabola [Eq. (28)], then the same conclusions and the same suppression of activation are valid to periodic potentials of the type

ft/τ=λ0τcos2t/τδxx0E54

See ref. [23] for a more extensive study on these potentials; nevertheless, the suppressed activation energies (52) and (53) are still valid for these potentials.

## 9. Instantaneous changes

Lastly, we are going to investigate the scenario in which the perturbation appears instantaneously. In this case, the Schrödinger equation reads

2ψx2f0utδxx0ψ+Uxψ=iψtE55

where ut=1t00t<0is the Heaviside step function.

It was shown in ref. [41] that the effect of such a perturbation has a universal pattern in the short time domain and in fact depends only on the product of the strength of the delta-function (f0 in this case) and the local value of the initial wavefunction. The idea is [41] that an instantaneous delta-function perturbation is equivalent to a discontinuity in the wavefunction, and it was proven elsewhere [42, 43] that such a discontinuity has, in the short time, a universal pattern even in the presence of potentials [44]. Therefore, if the initial state was

ψx,t<0=ϕΩ+xexpiΩtE56

then, after the instantaneous perturbation, the short time dynamics is simply [41]

ψxtϕΩ+xexpiΩt2iπt3/2xx02expixx024tf0ϕΩ+0E57

Because the first term is exponentially smaller than the second one, then very quickly the second term becomes dominant. In fact, it becomes dominant as early as

t3xx04Uf02UΩexpKLx0>1E58

If the delta-function well is turned on instantaneously but only for a period of 2τ, then the Schrödinger equation can be written as

2ψx2λ02τut+τutτδxx0ψ+Uxψ=iψtE59

and the solution is

ψxtϕΩ+xexpiΩtiπλ0ϕΩ+0τxx02t+τ3/2expixx024t+τtτ3/2expixx024tτE60

which can be further simplified (provided τ<<t) to

ψxtϕΩ+xexpiΩt+iiπt1/2λ0ϕΩ+02expixx024tsincτxx024t2E61

where sinc(x)≡ sin(x)/x.

This result again suggests that, if the variation occurs quickly enough, there is no dependence on the incoming particle’s energy Ω. In fact, the variation term vanishes for

τxx024t2=mπform=1,2,3,...E62

Moreover, when τxx024t2<<1, the second term has a totally universal pattern, which is even independent of τ:

ψxtϕΩ+xexpiΩt+iiπt1/2λ0ϕΩ+02expixx024tE63

## 10. Applications

(A) The effect of controlled activation can be used in dynamic heterostructures, which can be used as frequency effect transistors. A schematic presentation of the system is presented in Figure 12. The barrier can be constructed by a semiconductor, and its potential shape can be controlled by the gate’s voltage. In such a device, the current from source to drain will be governed by the frequency of the gate and therefore can be much more sensitive than any other transistor (because frequency, unlike voltage or currents, can be determined with great accuracy).

In Figure 13, the current j ∝ | ' (xt)ψ*(xt) − cc.| [where ψ(x,t) is the solution of Eq. (21)] of such a device is plotted as a function of the incoming energy Ω and τ. The forbidden activations values are clearly seen by the white stripes. They are exponentially narrow; therefore, the device’s current can be controlled by small variations in τ.

(B) As mentioned in the Section 1, there are evidences that odor detection is governed by dynamic resonant tunneling. It is known that odor receptors are, in some sense, like resonant tunneling devices—only specific molecules, with the right chemical properties, “activate” the receptor, which sends a signal to the brain. However, it was recently recognized that the olfactory system could distinguish between two molecules, which have the same chemistry but have different mechanical properties. That is, in cases where one of the atoms in a molecule is replaced with one of its isotopes (e.g., deuterium instead of hydrogen), a different odor is detected. It was therefore suggested that the receptor actually operates as a “dynamic” resonant tunneling device [38], which is sensitive on the molecule vibrations. If this is indeed the case, an enigma still remains: how humans can distinguish between approximately 10,000 different smells while they have only few hundred receptors. If every receptor is calibrated to a specific molecule with a specific vibration, then only several hundred odors should have been detected. The dynamics, which is presented by us in this chapter by the forbidden activation energy, suggests that each molecule has a different fingerprint, and every molecule can activate several receptors in a specific combination, which characterizes only the specific molecule. From this specific combination (instead of a single specific receptor), the brain can identify the specific detected molecule (see Figure 14).

## 11. Summary

Resonant tunneling is a fascinating quantum phenomenon. In this chapter, we have focused on the transmission of a quantum particle through an opaque (mostly rectangular) barrier via a delta-function potential. We have discussed all dynamics regimes—stationary resonant tunneling, adiabatic changes, activation, forbidden activation, and instantaneous changes. The generic dynamic problem is highly complex. Each one of these regimes has its specific characteristics.

The main results are related to the domain in which the incoming particle’s energy is higher than the minimum eigenenergy of the changing well and the changes are fast enough to prevent particle escape. Unlike previous predictions, the fact that the particle can be trapped in the well does not mean by itself that it will necessarily be activated. In fact, in some cases, the particle’s wavefunction experiences “destructive interference” inside the well, and as a consequence, activation is suppressed. We show that all the cases in which the potential’s temporal minima has a parabola shape have the same forbidden energies, and for the first time, we present a generic solution for this case.

Furthermore, the instantaneous case is also discussed for the first time. In this case, the dependence on the specific incoming energy vanishes, and a generic universal pattern appears.

Finally, we suggest testing this effect as a frequency effect transistor, which have the potential to be used as a highly accurate transistor.

Moreover, it is suggested that the effect of forbidden activation energies may explain the reason that humans can distinguish between 10,000 different odors while they have only several hundred odor receptors. According to this suggestion, every molecule can trigger different receptors, and only the combination of the activated ones creates the perception of the right smell. In any case, this research shows that dynamic resonant tunneling, in general (and forbidden activation, in particular), has a major, and totally nontrivial, role in the olfactory mechanism.

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