새로운 조정 가능한 이진 초격자로서의 이중 게이트 나노나선

아 A의 함수로서 접지와 첫 번째 부대역 사이의 밴드 갭 크기 _g (C _⊥ ) 빨간색 점선(녹색 점선)으로 표시됩니다(여기서는 B). _g =0.25. 다이어그램은 격리된 DQW 단위 셀과 고유 상태에 대한 다양한 섭동의 영향을 나타냅니다. ㄴ –ㄷ 2D 밀도 플롯을 통해 표시되는 첫 번째 및 두 번째 부대역 사이의 밴드 갭 크기는 다음과 같은 함수로 표시됩니다. ㄴ A _g 그리고 B _g C용 _⊥ =0 및 c A _g 및 C _⊥ 고정 B _g =0.25. ㄴ 인접한 등에너지 등고선은 0.17의 차이를 나타내며 \(A_{g}=\sqrt {B_{g}}\)에 대해 점선 빨간색 선으로 주어진 간격이 0인 반면 c 차이는 0.13이며 가장 작은 반원 윤곽(0.5,0)의 중심에 간격이 없습니다. 다이어그램은 격리된 DQW 및 고유 상태를 스케치합니다. 혼성화는 s 사이에서 발생하지 않습니다. -좋아요 및 p - b에서 공명과 같은 국지화된 개별 우물 상태 , 하지만 c에서 수행 전기장이 다른 장벽에 대해 한 장벽을 변경하기 때문에

에너지 대역 횡단

C를 유지하면 _⊥ =0 및 A 증가 _g , 처음에는 모든 축퇴가 해제되지만 이후의 더 높은 에너지 밴드는 Brillouin 영역의 중심과 가장자리 사이를 번갈아 교차하도록 가져옵니다(그림 3a에서 d까지 진행되는 파란색과 빨간색 점선 곡선이 번갈아 나타나는 동작 관찰). 물리적으로 우리는 단위 셀에서 국부적인 파동 함수의 상호 작용 측면에서 소실 밴드 갭을 이해할 수 있습니다. 비대칭 DQW 전위가 얕은 우물의 바닥 상태(s -like 궤도)는 깊은 우물(p)에서 첫 번째 여기 상태와 공명합니다. -유사 궤도), q에서 =0 어느 쪽 우물의 중심에 대한 반사 대칭으로 인해 이러한 상태의 반대 패리티는 그들 사이의 일반적인 터널 커플링을 방지하고 결과적으로 이러한 궤도에서 구성된 여기 상태가 일치합니다(그림 3b의 파란색 곡선). 이것은 소위 s를 연상시킵니다. -피 광학 격자의 공명 [73, 74]. 같은 이유로, 매개변수가 얕은 우물의 국부적인 바닥 상태가 동일한 패리티를 갖는 깊은 우물의 여기 상태와 공진하는 것과 같으면 |q |=1/2일 때 Bloch 상의 존재는 두 인접 국부 우물 상태 사이의 일반적인 혼성화를 완전히 억제하고 밴드 갭이 닫힙니다(그림 3c에서 두 번째 여기 상태로 접지 공진에 대해 표시됨). 주기적 잠재력에서 산란의 언어로; cos(φ에서 2차 브래그 산란 진폭의 완전한 상쇄 간섭으로 인해 밴드 갭이 닫힙니다. ) cos(2φ의 잠재적 및 1차 산란 진폭 ) 잠재력 [75–77].

식으로 돌아가서 Brillouin 영역의 중심과 가장자리 모두에서 에너지 밴드 교차(0 가로 전기장에 대한)의 존재를 정량적으로 나타낼 수 있습니다. 2, C일 때 휘태커-힐 방정식으로 인식할 수 있습니다. _⊥ =0 [78]. Bloch 기능 Eq. 4 꼬인 주기 경계 조건 ψ 준수 _n,q (φ +2π )=exp(2π iq )ψ _n,q (φ ). 특히 q일 때 =0 Eq.에 대한 공식 솔루션 2는 2π입니다. -주기적인 반면 |q |=1/2 해는 2π입니다. -반주기적(따라서 우리는 4π를 검색할 것입니다. -주기적인 솔루션). 구체적으로 식. 2와 C _⊥ =0은 Ince의 방정식 [79, 80]에 매핑될 수 있으며, 이는 Eq. 2 및 알 수 없는 함수 \(\psi _{n,q}(\varphi) =\exp \left [ -2\sqrt {B_{g}}\cos (\varphi)\right ]\Phi _{n, q}(\varphi)\),

여기서 보조 매개변수 \(\xi =8\sqrt {B_{g}}\), η를 정의했습니다. _n,q =4ε _n,q +8나 _g , \(-p \xi =8A_{g}+8\sqrt {B_{g}}\) 및 Φ _n,q (φ ) 각 솔루션의 필요한 꼬인 주기성을 유지합니다(여기 p 아님 나선 피치). 또한 여기에서 초격자 전위는 φ 변환에서 불변하므로 →− φ , q 솔루션 =0 및 q =1/2는 다음과 같은 삼각 시리즈

공식 솔루션을 다루고 q에 대한 솔루션은 =−1/2는 q와 동일합니다. =1/2. 여기에서 위 첨자 e 그리고 오 함수에 각각 짝수 및 홀수로 레이블을 지정하고 n 여전히 n을(를) 참조합니다. n이기도 한 th 서브밴드 지정된 q에 대한 고유 상태 가치. 이것을 식에 대입합니다. 6은 푸리에 계수에 대한 3항 재귀 관계를 나타냅니다. q =0 짝수 솔루션 산출량

및 q에 대한 홀수 솔루션에 대한 해당 재귀 관계 =0은

q =1/2 짝수 솔루션은

and the q =1/2 odd solution gives

Consider then Eqs. (8c) and (9b) for the q =0 solutions. The series solutions (7a) and (7b) can clearly be made to terminate if p is 0 or an even positive integer. The resulting polynomials are referred to as Ince polynomials. The remaining solutions for higher eigenvalues are simultaneously double degenerate and correspond to the energy crossings observed at q =0 for certain parameters. The existence of these degeneracies can be seen by looking at the diagonalizable matrices describing the recursion relations for a _l 그리고 b _l :

그리고

각기. Either of the above tridiagonal matrices can be broken into tridiagonal sub-matrices if a leading off-diagonal matrix element is equal to zero, i.e. if p is an even number. The matrices will decompose into two tridiagonal blocks, one smaller finite matrix \(\boldsymbol {\mathcal {A}_{1}}\) (\(\boldsymbol {\mathcal {B}_{1}}\)) and a remaining infinite matrix \(\boldsymbol {\mathcal {A}_{2}}\) (\(\boldsymbol {\mathcal {B}_{2}}\)). From the theory of tridiagonal matrices the corresponding eigenvalue spectra for each matrix is then \(\eta (\boldsymbol {\mathcal {A}}) =\eta (\boldsymbol {\mathcal {A}_{1}}) \cup \eta (\boldsymbol {\mathcal {A}_{2}})\) and \(\eta (\boldsymbol {\mathcal {B}}) =\eta (\boldsymbol {\mathcal {B}_{1}}) \cup \eta (\boldsymbol {\mathcal {B}_{2}})\). The smaller finite matrices are analytically diagonalizable in principle, giving exact eigenvalues, and their corresponding finite length eigenvectors define the fourier coefficients yielding Ince polynomials via Eq. 7. We can see that for a given even integer p , the remaining infinite tridiagonal matrices are the same \(\boldsymbol {\mathcal {A}_{2}}=\boldsymbol {\mathcal {B}_{2}}\equiv \boldsymbol {\mathcal {D}}\) which results in the double degenerate eigenvalues. To be clear, we provide an example of when p =2 in the Appendix.

In the same way, when p is a positive odd integer the series solutions (7c) and (7d) can be made to terminate, and the matrices corresponding to \(\widetilde {a}_{l}\) and \(\widetilde {b}_{l}\) share eigenvalues resulting in the closing of higher subbands at the edge of the Brillouin zone q =± 1/2. From the definitions of the auxiliary parameters in Eq. 6, we have

which defines the condition for exactly-solvable solutions for the lower lying solutions and simultaneously the existence of higher double degenerate eigenvalues above the p th subband, with p =0 or an even positive integer corresponding to crossings at the centre of the Brillouin zone, while crossings at the edge require p to be an odd positive integer. Figure 4b plots the size of the band gap between the first and second subbands \(\Delta \varepsilon _{12}^{(q=0)}/\varepsilon _{0}(\rho)\) as a function of A _g 그리고 B _g , with the dot-dashed red contour line corresponding to Eq. 14 for p =0. The schematic indicates the appropriate eigenstates of the isolated DQW at the p =0 resonance.

The application of a small transverse field C _⊥ breaks the reflection symmetry of the system, permitting hybridization of the localized well states of the isolated DQW which results in a significant change at points of degeneracy, as can be seen by comparing the schematic depicted in Fig. 4b with that in c (see also inset of Fig. 3b). We plot in Fig. 4c the behaviour of the band gap between the first and second subbands as a function of A _g 및 C _⊥ . Here we see that the band gap is more sensitive to C _⊥ due to the significant change in the isolated DQW eigenstates by lowering one barrier with respect to the other. This behaviour is notably the converse of the parameter sensitivity for the band gap between the ground and first subbands. By degenerate perturbation theory, it can be shown that this induced band gap is linear in C _⊥ for the lowest crossing bands when p =0, and to higher order with increasing p . Finally, within the vicinity of the crossings, e.g. for small q about q =0 in Fig. 3a, the dispersions could be approximated as a quasi-relativistic linear dispersion yielding Dirac-like physics, which could permit superfluiditiy [81] for example. The advantage in using nanohelices lies in introducing such phenomena to portable nanostructure based devices, while also exhibiting unusual responses of the charge carriers to circularly polarized radiation [44, 45, 82–85] (or indeed magnetic fields [86, 87]) due to the helical spatial confinement.

Optical transitions

In order to understand how our double-gated nanohelix system interacts with electromagnetic radiation, we study the inter-subband momentum operator matrix element \(T^{g\rightarrow f}_{j} =\langle {f}|\boldsymbol {\hat {j}} \cdot \boldsymbol {\hat {P}}_{j} |{g}\), which is proportional to the corresponding transition dipole moment, and dictates the transition rate between subbands ψ _f and ψ _g . Here, \(\boldsymbol {\hat {j}}\) is the projection of the radiation polarization vector onto the coordinate axes (j =x,y ,z ) and the respective self-adjoint momentum operators are [44, 45, 82–84]

In terms of the dimensionless position variable φ , we are required to evaluate \(T^{g\rightarrow f}_{j} =\rho \int _{0}^{2\pi N}\psi _{f}^{\ast } P_{j} \psi _{g} d\varphi \), and upon substituting in from Eq. 4 we find

We see from Eqs. 16a and 16b that light linearly polarized transverse to the helix axis couples coefficients with angular momentum differing by unity Δ 나 =± 1, whereas from Eq. 16c, linear polarization parallel to the helix axis couples only Δ 나 =0. In Fig. 5, we plot the absolute square of the momentum operator matrix element between the lowest three bands for linearly polarized light propagating perpendicular to the helix axis (i.e. with z -편극). Initially, for A _g =C _⊥ =0, transitions between the ground and first bands are forbidden (as is to be expected for a unit cell with two equivalent wells resulting in a doubling of the first Brillouin zone, so it is in fact the same band). As the strength of the doubled period potential A _g is increased with respect to B _g , transitions become allowed away from q =0 as can be seen from Fig. 5a (following behaviour from the dotted red curve through to the solid blue curve). The parameters are swept through a resonance as we go from the solid to the dashed blue curve, wherein the situation changes drastically. To understand this behaviour, we must consider the special case of q =0. As we traverse this resonance, the energy of the Bloch function with q =0 constructed from the first excited state of the deeper well in the DQW unit cell (p -like) passes below the Bloch function constructed from the ground state in the shallower well (s -like). Consequently, the parity with respect to φ (which is a good quantum number only for q =0 or |q |=1/2) of the two excited states is exchanged resulting in the rapid switch from forbidden to allowed at q =0, wherein the z -polarized inter-subband matrix element becomes non-zero due to the operator \(\boldsymbol {\hat {P}}_{z}\) (see Eq. 15c) now coupling the even ground state with the odd first excited state. We therefore see the opposite behaviour for transitions between the ground and second band in Fig. 5b about q =0. While initially increasing A _g allows transitions at q =0 between the ground state and the second excited state when it is p -like, beyond resonance (when the order of the s -like and p -like excited states are swapped) transitions are suppressed. See for example Ref. [88] for a clear picture of this interchange between the ordering of the even and the odd parity excited states. For transitions between the first and second band (Fig. 5c), we observe a large transition centred about q =0 due to the lifting of the m =± 1 degenerate states of the field-free helix by the superlattice potential. The presence of symmetry-breaking C _⊥ ruins the pristine parities of the states at the centre of the Brillouin zone and all transitions are allowed, as shown in the insets of Fig. 5.

Square of the dimensionless momentum operator matrix element between the g th and f th subbands in the first Brillouin zone as a function of the dimensionless wave vector q of the electrons photoexcited by linearly z -polarized radiation and for a variety of parameter combinations spanning the first incident of resonance. The different blue curves keep A _g =0.5 and C _⊥ =0 fixed and vary B _g =0.1, 0.2, and 0.3 corresponding to dot-dashed, dashed, and solid. The different red curves keep B _g =0.3 and C _⊥ =0 fixed while varying A _g =0.05, 0.1 and 0.3 as dot-dashed, dashed, and solid, while the dotted blue (dotted red) plots the limiting case A _g =0.5 &B _g →0 (A _g →0 &B _g =0.3). 아 Transitions between the ground and first bands. The inset plots the behaviour for fixed A _g =0.5 and changing B _g crossing the resonant condition at B _g =0.25 (see text) in a reduced q -range, ranging from upper blue B _g =0.245, lower blue B _g =0.249, upper purple B _g =0.251, to lower purple B _g =0.255. The dashed green curves are for small non-zero transverse field C _⊥ =0.05 ranging from B _g =0.245 (upper curve) to B _g =0.255 (lower curve) in increments of 0.05. ㄴ Plots transitions between the ground and second bands, the inset plots the behaviour close to resonance when A _g =0.5; blue is B _g =0.249, purple is B _g =0.251, and dark green is at resonance with C _⊥ =0.05. ㄷ Plots transitions between the first and second bands, the parameters for the inset are the same as those in (b )

In Fig. 6, we plot the absolute square of the momentum operator matrix element for right-handed circularly polarized light which propagates along the helix axis, given by |T _x +iT _와 | ² . Notably, we observe a large anisotropy between the two halves of the first Brillouin zone, while the result for left-handed polarization is a mirror image to what we see in Fig. 6. Physically, this can be attributed to the conversion of the photon angular momenta to the translational motion of the free charge carriers projected onto the direction of the helix axis, with an unequal population of the excited subband in a preferential momentum direction controlled by the relative handedness of both the helix and the circular polarization of light. An intuitive mechanical analogue would be the rotary motion of Archimedes’ screw being converted into the linear motion of water along the direction of the screw axis dictated by the handedness of the thread. As such, our system of a double-gated nanohelix irradiated by circularly polarized light exhibits a photogalvanic effect, whereby one can choose the net direction of current by irradiating with either right- or left-handed circularly polarized light [44, 45, 89]. This differs from conventional one-dimensional superlattices, wherein the circular photogalvanic effect stems from the spin-orbit term appearing in the effective electron Hamiltonian and is consequently a weaker and hard-to-control phenomenon [90, 91]. The electric current induced by promoting electrons from the ground subband to an excited subband f via the absorption of circularly polarized light can be understood from the equation for the electric current contribution from the f th subband

Square of the dimensionless momentum operator matrix element between the g th and f th subbands in the first Brillouin zone as a function of the dimensionless wave vector q of the electrons photoexcited by right-handed circularly polarized radiation |T _x +iT _와 | ² and for a variety of parameter combinations spanning the first incident of resonance. 아 The blue curves denote transitions between the ground and first band while the red curves denote transitions between the ground and second band, both with the following parameters:A _g =0.5 and B _g =0.3 for solid curves, A _g =0.5 and B _g =0.1 for dashed curves, A _g =0.3 and B _g =0.3 for dot-dashed curves, and A _g =0.01 and B _g =0.3 for dotted curves (as A _g →0 the maximum of the 0→2 increases rapidly as it approaches q =− 1/2). The inset plots the behaviour as B _g is tuned through resonance for A _g =0.5; dotted is B _g =0.24, dot-dashed is B _g =0.25, and dashed is B _g =0.26. The solid purple (orange) curve denotes transitions between the ground and first (second) band at resonance with C _⊥ =0.05 applied. ㄴ Plots transitions between the first and second bands. The different blue curves keep A _g =0.5 fixed and vary B _g =0, 0.2, and 0.3 corresponding to dotted, dot-dashed, and solid. The different red curves keep B _g =0.3 fixed while varying A _g =0.05, 0.1, and 0.3 as dotted, dot-dashed, and solid. We have omitted plots for C _⊥ ≠0 here as it yields no great qualitative change to the matrix elements

where \(v_{g,f}(q)=(\rho /\hbar)\partial \varepsilon _{g,f}/\partial q \) is the antisymmetric electron velocity v (q )=− v (− q ) (which we can deduce from the symmetric dispersion curves), τ _g,f (q ) is a phenomenological relaxation time, and \(\Gamma _{CP}^{g \rightarrow f}(q)\) is the transition rate resulting from the optical perturbation of the electron system. Given that \(\Gamma _{CP}^{g \rightarrow f}(q) \propto |T_{x}^{g \rightarrow f} + i T_{y}^{g \rightarrow f}|^{2}\) for right-handed circularly polarized light where T _x 그리고 T _와 are given by Eqs. 16a and 16b, respectively. The anisotropy present in Fig. 6a enters Eq. 17 to yield a non-zero photocurrent. This current flows in the opposite direction for left-handed polarization. Such a circular photogalvanic effect is also exhibited in chiral carbon nanotubes under circularly polarized irradiation [92, 93], although tunability predominantly stems from manipulating the nanotube physical parameters, which are hard to control. The double-gated nanohelix system offers superior versatility by fully controlling the landscape of the superlattice potential, which can be used to tailor the non-equilibrium asymmetric distribution function of photoexcited carriers (as shown in Fig. 6 for inter-subband transitions between the three lowest subbands).

On a side note, we expect that (as with chiral carbon nanotubes [93–95]) the application of a magnetic field along the nanohelix axis can take up the role played by circularly polarized radiation, whereby the current is induced by a magnetic-field-induced asymmetric energy dispersion—which in turn produces an anisotropic electron velocity distribution across the two halves of the Brillouin zone.

결론

In summary, we have shown that the system of a nanohelix between two aligned gates modelled as charged wires is a tunable binary superlattice. The band structure for this system exhibits a diverse behaviour, in particular revealing energy band crossings accessible via tuning the voltages on the gates. The application of an electric field normal to the plane defined by the gates and the helix axis introduces an additional parameter with which to open a band gap at these crossings. Engineering the band structure in situ with the externally induced superlattice potential along a nanohelix provides a clear advantage over conventional heterostructure superlattices with a DQW basis [96, 97]. Both systems can be used as high-responsivity photodetectors, wherein tailoring the band structure (to the so-called band-aligned basis [98–100]) can lead to a reduction in the accompanying dark current. Here control over the global depth of the quantum wells also permits versatility over the detection regime, which can lie within the THz range. We have also investigated the corresponding behaviour of electric dipole transitions between the lowest three subbands induced by both linearly and circularly polarized light, which additionally allows this system to be used for polarization sensitive detection. Finally, the ability to tune the system such that a degenerate excited state is optically accessible from the ground state, along with the inherent chirality present in the light-matter interactions, may make this a promising system for future quantum information processing applications [101]. It is hoped that with the advent of sophisticated nano-fabrication capabilities [102], fully controllable binary superlattice properties will be realized in a nanohelix and will undoubtedly contribute to novel optoelectronic applications.

Appendix

Touching energy bands at Brillouin zone boundary when A _g =C _⊥ =0

Here, we show using matrix algebra that in the picture of a binary superlattice pairs of subbands touch at the Brillouin zone edges if A _g =C _⊥ =0 and B _g ≠0, as seen from the solid blue curves in Fig. 3a. Equation 5 is equivalent to the following N -by-N pentadiagonal matrix Hamiltonian with zeros on the leading sub- and superdiagonals:

(18)

Let us consider q =1/2 (we could alternatively take q =− 1/2) which makes the leading diagonal symmetric. We can then express this matrix Hamiltonian \(\widetilde {\mathcal {\boldsymbol {H}}}_{N} \equiv \boldsymbol {\mathcal {H}}_{N,q=1/2} \) in block form as

(19)

어디

(20)

are both of dimension N /2-by- N /2, and J is the exchange matrix. We may construct a matrix via permuting \(\boldsymbol {\mathcal {H}}_{N}\) with the N -by-N permutation matrix \(\boldsymbol {\mathcal {P}}_{N}\),

$$ \boldsymbol{\mathcal{P}}_{N} =\left[ \begin{array}{cccccc} 1 &0 &\hdots &\hdots &\hdots &0 \\ 0 &\hdots &\hdots &\hdots &\hdots &1 \\ 0 &1 &\hdots &\hdots &\hdots &0 \\ 0 &\hdots &\hdots &\hdots &1 &0 \\ \vdots &&&&&\vdots \\ 0 &\hdots &\hdots &1 &\hdots &0 \\ \end{array} \right], $$ (21)

such that the permutation-similar matrix is

(22)

Hence, the eigenvalues of \(\boldsymbol {\mathcal {P}}_{N}^{-1}\widetilde {\boldsymbol {\mathcal {H}}}_{N} \boldsymbol {\mathcal {P}}_{N}\), which are the same as the eigenvalues \(\widetilde {\boldsymbol {\mathcal {H}}}_{N}\), are double degenerate with the values given by the eigenvalue spectrum of the tridiagonal matrix C

(23)

which can also be expressed succinctly in terms of the previously defined matrices via \(\mathbf {C}=\boldsymbol {\mathcal {P}}_{N/2}^{-1}\mathbf {A}\boldsymbol {\mathcal {P}}_{N/2}+\boldsymbol {\mathcal {P}}_{N/2}^{-1}\mathbf {B}\mathbf {J}\boldsymbol {\mathcal {P}}_{N/2}^{4}\). We can see that applying C _⊥ ≠0 (inset of Fig. 3a) or both C _⊥ 그리고 A _g ≠0 (inset of Fig. 3b) ruins the symmetry in the matrix Hamiltonian and prevents the existence of eigenvalues with multiplicity beyond unity, resulting in the appearance of band gaps.

Energy crossing at centre of Brillouin zone between third and forth subbands

As an example, let us specifically consider the case where p =2, wherein the matrices (12) and (13) become:

(24)

그리고

(25)

This case corresponds to the crossings of the blue curves at the edge of the Brillouin zone in Fig. 3d (whereas p =0 results in crossings at q =0 in Fig. 3b). The lower eigenvalues are found exactly by diagonalizing each of the two finite matrices and they interlace, yielding \(\eta _{0,1,2} =2-\sqrt {4+4\xi ^{2}}, 4, 2+\sqrt {4+4\xi ^{2}}\). The infinite lower-right-hand block tridiagonal matrices coincide, thus the remaining double degenerate eigenvalues are found by approximately or numerically solving Det[D −η I ]=0.

데이터 및 자료의 가용성

The data for the figures all stem from numerically diagonalizing the matrix described by Eq. 5 and can readily be achieved in any numerical software package. With this in mind, the datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.

유한 요소 찢김 및 상호 연결 방법을 통한 나노 규모의 광학 트래핑의 효율적인 예측 및 분석 Her2 기능 골드 나노쉘 자기 하이브리드 나노 입자:유방암의 이중 모드 영상 및 광열 치료를 위한 치료 물질