Nanosensing Backed by the Uncertainty Principle

Possibility for a novel type of sensors for detecting nanosized substances (e.g., macromolecules or molecule clusters) through their effects on electron tunneling in a double nanoscale semiconductor heterostructure is discussed.We studied spectral distributions of localized/delocalized states of a single electron in a double quantum well (DQW) with relation to slight asymmetry perturbations. The asymmetry was modeled by modification of the dot shape and the confinement potential. Electron energy uncertainty is restricted by the differences between energy levels within the spectra of separated QWs. Hence, we established a direct relationship between the uncertainty of electron localization and the energy uncertainty. We have shown in various instances that a small violation of symmetry drastically affects the electron localization. These phenomena can be utilized to devise new sensing functionalities. The charge transport in such sensors is highly sensitive to minuscule symmetry violation caused by the detected substance. The detection of the electron localization constitutes the sensor signal.


Introduction
A new generation of nanosensors is underway and is expected to revolutionize modern engineering fields, including bioengineering and drug delivery systems in nanomedicine, to name a few.Semiconductor heterostructures, such as quantum dots and rings, are of high interest for the development of these nanosensors, as well as many other predicted nanodevices.Of particular importance is the electron tunneling that occurs between the nanosized elements of such devices.Double quantum systems facilitate the study of tunneling related to barrier penetration effects in double well potentials [1].When the elements of a quantum system get coupled, the energy barrier splitting of degenerate levels occurs, due to their common wave function.As a result, two nearly degenerate eigenstates are formed, which are a linear combination of the wave functions of the electron in isolated dots.Likewise, electron spectrum of double quantum dots formed by a set of quasidoublets is well described within the one-dimensional formalism given in [1] for double quantum wells.An example of theoretical analysis for double quantum dots has been presented in [2,3].These are supported by experimental studies of different aspects (spin effects, coupling distance, electron-phonon coupling, hybrid nanostructures, etc.) which are tightly connected to, and controlled by, charge tunneling in the DQDs.These aspects are extensively developing; see, for instance, [4][5][6].Recently, we have reported in [7] that single electron localized/delocalized states and the tunneling in DQWs and DQDs are highly sensitive to the violation of reflection symmetry of the dots.It should be noted that this aspect is relevant to quantum dot reality, as fabrication technologies produce dot arrays with imperfect shape and distributions.These imperfections inherently induce a chaotic behavior in the QDs.It has been demonstrated that chaos strongly influences charge transport and other properties of QDs [8,9].The relevance of electron tunneling and chaos in real QD arrays is obvious for next generation nanodevices, for instance, future single molecule nanosensors, quantum computing devices, and advanced solar cells, to name a few.
In the present work we study the spectral distribution of electron localized/delocalized states and tunneling in DQWs.The average coordinate is used to characterize the localization of a single electron.We discuss the case of identical QWs constituting a DQW, as an example, when the uncertainty principle is manifested.Uncertainty of electron localization occurs when the difference of electron energies in the left and right QW is very small; that is the case of almost identical

Effective Model for
InAs/GaAs Quantum Dots We consider quantum dots composed of InGaAs on a GaAs substrate.The fabrication of such kind of quantum dots is reported in [10,11], for example.In practice, QDs have average lateral size and height of 41 and 1.6 nm, respectively, with variations within 23% and 28%, respectively.Hence, in our model the heterostructured QD dimensions were varied within these limits.The QDs were laterally distributed (two-dimensional array) for minimizing the computational recourses.The underlying quantum problem is modeled utilizing the kp-perturbation single subband approach, which is mathematically formulated by the Schrödinger equation as follows [12]: Here Ĥkp is the single band kp-Hamiltonian operator Ĥkp = −∇(ℎ 2 /2 * )∇,  * =  * (r) is the electron effective mass, which depends on the position of the electron, and   (r) is the band gap potential, which is null inside the QW,   (r) = 0, and constant, equal to   , outside the QW.The value of   is defined by the conduction band offset for the bulk.The band gap potential for the conduction band is chosen as   = 0.594 eV [13].The magnitude of   is calculated as   = ( ,2 −  ,1 ), where  ,1 and  ,2 are the band gaps of QD and the substrate, respectively.The coefficient  can be different for the conduction and valence bands.Here, the dimensionless constant  values are taken from [14].For the conduction band (CB) = 0.54, and for the valence band (VB) = 0.46.Using experimental values  ,1 = 0.42 eV and  ,2 = 1.52 eV,   = 0.594 eV was obtained for the conduction band.The bulk effective masses of InAs and GaAs are  * 1 = 0.024  0 and  * 2 = 0.067  0 , respectively [15], where  0 is the free electron mass.  (r) is the effective potential simulating the strain effect; it is attractive and acts inside the QW.The magnitude of the potential can be chosen [12] to reproduce experimental data.In presented work, the magnitude of   for the conduction band is 0.21 eV.With this value, the results of the 8th band kp-calculations of [16] are well reproduced [17].

Wave Function of Two Level System
Single electron spectrum of a two-level system is defined as a set of quasidoublets [1].The one-dimensional wave functions of each quasidoublet can be expressed as follows [1]: V Relation (2) shows that the wave function decomposed onto the basis set ( 1 ,  2 ).The parameter  is a coupling coefficient of the quantum system elements.It depends on the wave function overlap for the "unperturbed states"  1 and  2 of the left and right quantum dots considered to be separated.Δ 12 =  1 −  2 is energy difference of separated QWs.It is defined for each energy level in the spectrum.An illustration of the band gap model with the effective potential is presented in Figure 1.In (2), we neglected the phase shift as it does not play a role in the present case.The energy difference Δ 12 coincides with the quasidoublet splitting Δ =  + −  − for separated QWs.
To evaluate the electron localization, we analyze the single electron average coordinate ⟨⟩, calculated as ⟨⟩  = ⟨||⟩ for  = 1, 2, which are associated, respectively, with QW  ,  = 1, 2, considered to be separated.The -coordinate origin is the midpoint of the two QWs.The average coordinate ⟨⟩ + (⟨⟩ − ) of the electron in Ψ + state (Ψ − ) can be written as for corresponding levels of a quasidoublet spectrum (i.e., the DQW spectrum).Due to the dependence on Θ, single electron spectrum is constituted of three parts: delocalized states (Θ ≈ /2), localized states (Θ ≈ 0), and states with different probability for localizations in left and right QWs.The equation Θ = arctg(/Δ 12 ), derived from (3), provides the relationship between Θ parameter and localization/delocalization likelihood of electron in DQW. Figure 2 illustrates that relationship and shows the domain ranges where location and delocalization occur.
According to relation ( 3), the electron localization in DQW is extremely sensitive to small violations of DQW symmetry, when Δ 12 ≈ 0. By differentiation of relation ( 3), one can obtain an estimate of the sensitivity of the terms in (2) to small variations of the energy difference Δ 12 : Due to the square in the denominator, small variations of Δ 12 cause large variations of the prefactors in relations (2) and (4).Hence, for weakly coupled QWs the uncertainty of single electron energy is minimal, while the uncertainty for electron localization is large.This fact is in line with the uncertainty principle.

Identical QWs.
The experimentally demonstrated circular disc shaped InAs/GaAs quantum dots constitute double quantum wells.We consider in this section identical quantum wells, with equal radii  1 =  2 = 33 nm.An example of experimental realization of this system is reported in [10,11], where the grown QDs are highly symmetric and free from any lateral elongation.
The electron average coordinate ⟨⟩ in this DQW system has been calculated for each level of the single electron spectrum, according to the above described model.The results are presented in Figure 3.In the DQW quasidoublet spectrum, the spectral distribution appears symmetric relative to the ⟨⟩ = 0 axis, due to relation (4).Furthermore, the energy spectrum of single electron can be sorted according to electron localization probability, as a set of localized states (in the left QW or right QW, for which ⟨⟩ = ±|⟨⟩|), delocalized states (i.e., ⟨⟩ ∼ 0), and states with different probabilities in left and right QWs.Competition between values of Δ 12 and the coupling coefficient  in (2)-( 3) defines the type of electron localization in the system.For the ideal case of identical QWs, that is, when Δ 12 = 0, the wave functions of delocalized states are expressed as Ψ + = ( 1 +  2 )/ √ 2, and It should be noted that numerical discretization of the Schrodinger equation on a finite mesh leads to significant computational errors.Thus the ideally symmetric system cannot be solved numerically, and computer modeling rather leads to solutions pertaining to dissymmetric QWs.As can be seen in Figure 3, there are strong deviations of the spectral distribution for the average coordinate ⟨⟩, calculated for two interdot distances which differ for small values of .This instability is stronger for the intermediate region of the spectrum, where the levels with different probability for localization in left and right QWs are found.
To evaluate the accuracy of the calculations we applied different meshes.Figure 4 shows the sensitivity of spectral distribution of ⟨⟩ to tuning the discretization meshes.The results are unstable relative to mesh variations.In particular, the calculation for states with different localization probability (in left and right QWs) is unstable due to the uncertainty of Δ 12 (artificially due to computational errors) and relation (3).One can use the quasidoublet splitting energy Δ =  + −  − at the ground state, as a reference, to evaluate the effect of discretization.Figure 4 shows ⟨⟩ for three meshes and the inset gives the resulting Δ, energy splitting of the quasidoublet for the ground state.It appears that the finer mesh gives the smaller Δ.Hence, we assume that the results of calculations presented below for identical QW have a numerical error of about 10 −7 eV.Visualization of the localized-delocalized states dynamics in the InAs/GaAs DQW is given in Figure 5, where the average coordinate ⟨⟩ is presented as a function of the interdot distance  (see also [7]).When the distance between QWs is large, the electron is localized in one of the QWs; in that case the parameter Θ ≈ 0. When the interdot distance decreases, the electron tunneling increases and the wave function spreads over the whole double system, 0 ≤ Θ ≤ .
All states become delocalized for interdot distances less than 17 nm.The localized states occur for interdot distances larger than 25 nm.Additionally, as can be seen in ( 5), the electron localization demonstrates extreme sensitivity to small variations of QW shapes, which also violate the DQW left-right symmetry.Dissymmetry of dot shape will be discussed in the following section.

Asymmetric DQW.
Let us consider two circular QWs that are nonidentical.The asymmetry parameter for the DQW is defined as  =  2 / 1 .We chose very small asymmetry (e.g.,  ∼ 0.999975 = 39.999/40), as nearly identical QWs lead to relevant quantum effects and mimic the reality of QW arrays.In this study, we vary the interdot distance.The calculation results are presented in Figure 6, where the average coordinate ⟨⟩ along the single electron spectrum is shown for four interdot distances.The interdot distance is gradually decreased from large distances, when the QWs are separated, to contacting QWs.
Comparing with the previously discussed identical DQWs, for asymmetric DQW, the interdot distance that causes the quantum states to become delocalized, over the entire spectrum, is smaller.For example, when  = 0.999975, such interdot distance is less than 7 nm, whereas it is equal to 17 nm for identical QWs.
It has been previously found [7] that extremely small difference in QW radii in a DQW (ratio  =  2 / 1 slightly smaller than 1) drastically affects the spectral distribution of localized/delocalized states.Such violation of DQW shape symmetry defines small variation of Δ 12 , according to (3).Also, according to (4), small change of Δ 12 can induce significant changes of the coefficients of linear combination (2), thereby drastically changing the wave functions.Typical picture for the spectral distribution of average coordinate ⟨⟩ is shown in Figure 7(a), which was obtained for interdot distance  = 10nm.The spectrum readily divided in localized states (upper spectral levels), states with different localization probability in left and right QW (transitional states), and delocalized states (low-lying states).The relations in (4) explain the spectral distribution of the average coordinate shown in Figure 7(a).
The matrix element  (which is also proportional to the quasidoublet energy splitting Δ [18]) can be described using the following relation:  ∼ , where  is the overlap integral, approximated by  = ∫ Σ  2 (, ) , with the integration domain Σ being the area between the QWs; see inset in Figure 7(b).In the overlap integral, the wave functions (, ) are normalized.The distribution of  along the spectrum is shown in Figure 7(b).The magnitude of the wave function overlap depends on both the distance between quantum objects and the spreading of the wave function over the QWs.Generally, the matrix elements  are larger for upper levels in comparison with those for low-lying levels.This is due to the fact that the upper levels of the spectrum have broader wave functions.The large values of  are associated with the large Θ values (i.e., Θ ≤ ) for such levels; in such cases, the electron takes a localized state, as can be seen in Figure 7(a).
The symmetry breaking in DQW can be also made by variations of the confinement strength in one of the QWs.In Figure 8, the results of such variations are presented for QWs with identical shape.The magnitude of the DQW asymmetry is increased by increasing the potential from   = 0.21 eV to 0.21005 eV, to 0.2101 eV, and to 0.211 eV.With the last value of   all states in the spectrum become localized.Note that initially the states were delocalized, due to the chosen short interdot distance ( = 4 nm).Again we see that such quick transformation of electron localization (due to symmetry breaking) is possible under the condition of minimal uncertainty of electron energy.The pace and the sensitivity of the transformation are high for almost identical QWs.
One more illustration of symmetry breaking effect is given in Figure 9, where the DQW asymmetry is made by a small cut of the right quantum well (QW 2 ).The average coordinate ⟨⟩ plotted along electron spectrum for different depths of the cut ( 0 = 0.5 nm and 1 nm) is given in  The interpretation of this data can be done by noticing that the decrease of the asymmetry (through shape variation or through decrease of energy uncertainty limits) leads to increasing uncertainty of electron position in the system.For larger depth of the cut, the states of the DQW are mainly localized states.An additional illustration of the symmetry breaking effect is presented in Figures 10 and 11, where we compare the tunneling for two cases: small ( = 0.99375) and relatively large ( = 0.875) asymmetry of the DQW with fixed interdot distance.This is composed of a QW circular in shape with a radius  1 = 40 nm and an elliptical QW with the major axis along the -axis ( 2 ≤  2 ).The values Δ 12 and  are shown in Figure 10(a).Again, the number of the delocalized levels appears larger for the case of small asymmetry.The "random" distribution of delocalized states in the spectrum relates to the distribution of the matrix element  is reflected here in the distribution of the  integral.The  parameter varies up to two degrees for different spectral levels.The levels with minimal  directly relate to delocalized states of the spectrum, as visible in Figure 10(b).One can note that if the value of  is relatively large and Δ 12 is relatively small, the corresponding state can be delocalized.Hence, we can divide the ranges of Δ 12 and  in mainly localized states and mainly delocalized states regions (see vertical and horizontal lines).The number of delocalized states decreases when the asymmetry increases; these can be evaluated by comparing Figures 10(b) and 11.It is clear that when the asymmetry becomes increasingly larger, the states with ⟨⟩ ∼ 0 disappear.The violation of DQW shape symmetry largely effects the spectral distribution of localized/delocalized states by varying the energy level splitting, manifested by a large energy difference Δ 12 .This effect can favorably be applied to make a nanosensing functionality.We present in Figure 12 results of modeling an asymmetric DQW with a protuberance at the central edge of QW 2 , as is shown in Figure 12(a).The protuberance plays the role of either a defect grown with the quantum dot or a foreign material adsorbed on the surface of a dot that was initially identical to its peer.
It appears from data in Figure 12 that the degree of delocalization is a direct function of the size of the protuberance, which as indicated above could be a defect or an adsorbed substance.This offers the possibility to sense the protuberance size.Calibration of this sensing functionality can be done through our modeling and by varying the size of the defect, respectively, the adsorbed substance.

Conclusion
We studied the spectral distribution of localized/delocalized states in DQW.The electron localization in DQW appeared extremely sensitive to small violations of symmetry in DQWs due to relation (5) for Δ 12 nearing zero.The extreme sensitivity to symmetry violations, found through our modeling, can be technologically important.
The observed electron behavior in double quantum structures can be interpreted in terms of the uncertainty principle.We have shown that the spectral distribution of the electron average coordinate ⟨⟩ is distinctly different for various types of DQW symmetry violations.It appeared that the ideal case of identical QWs cannot be reliably modeled numerically, since the closer the DQW to perfectly symmetric shape, the higher the effect of numerical discretization on tunneling properties.We demonstrated that the DQW is a quantum system suitable for sensing tiny substances adsorbed on one of the quantum dots constituting the DQW or by a defect in one of the dots.The protuberance volume capable of breaking the symmetry, changing the quantum states of the DQW, and provoking tunneling was estimated to contain one to several thousand atoms, which is in the range of the size of polymer macromolecules and biomolecules.

Figure 2 :
Figure 2: Relation between the parameter Θ and the QW coupling coefficient in a DQW.The ranges for localized and delocalized states of the electron in DQW are shown.

Figure 3 :
Figure 3: Effects of the interdot distance, , on the spectrum of a DQW made of symmetric circular InAs/GaAs dots ( 1 =  2 = 40 nm).(a) Schematic representation of geometry parameters of DQW.(b) The average coordinate ⟨⟩ calculated for each level of single electron DQW spectrum.

Figure 4 :Figure 5 :
Figure 4: Spectral distributions for the average coordinate ⟨⟩ for a circular DQW.The symbols correspond to discretization of the Schrödinger equation on various numerical meshes.The meshes differ by the elements minimum size.The energy splitting (Δ) of the quasidoublet of the ground state for each discretization of a mesh is given in the inset.The finer mesh corresponds to smaller Δ.Interdot distance  is kept the same and equal to 26 nm.

Figure 8 :
Figure 8: Single electron average coordinate ⟨⟩ along the energy spectrum for asymmetric DQW under different confinement potentials   in left QW (shown in the inset).The potential in right QW is not changed (  = 0.21 eV).The interdot distance is kept constant and equal to 4 nm.

Figure 9 (
Figure 9(b).The interpretation of this data can be done by noticing that the decrease of the asymmetry (through shape variation or through decrease of energy uncertainty limits) leads to increasing uncertainty of electron position in the system.For larger depth of the cut, the states of the DQW are mainly localized states.An additional illustration of the symmetry breaking effect is presented in Figures10 and 11, where we compare the tunneling for two cases: small ( = 0.99375) and relatively large ( = 0.875) asymmetry of the DQW with fixed interdot distance.This is composed of a QW circular in shape with a

Figure 12 :
Figure 12: (a) Semicircle shaped DQW, where the asymmetry is generated by a rectangular defect in the right QW.The defect height, , and width, , were varied, as indicated in the inset, while the radii and the dot interdistance are maintained constant ( 1 =  2 = 40 nm and  = 16 nm).(b) Electron average position ⟨⟩ plotted against the electron spectrum of the asymmetric DQWs.
Figure 1: Band gap model for one-dimensional DQW. 1 and  2 are widths of the QWs,  1 ̸=  2 .Δ 12 is the difference between homolog energy levels pertaining to left and right QWs, considered separated.The notations are corresponding to the theoretical model described above. .. is ground state energy of an electron.