Strongly anharmonic confinement
For a 2D ring of radius ({r}_{c}) and width W with width-diameter ratio (a=frac{W}{2{r}_{c}}), here we consider a continuous and bounded potential of strongly anharmonic confinement, (V={V}_{b}[Gamma (x+a)-Gamma left(x-aright)]), where ({V}_{b}) depicts the well depth and (Gamma (x)=1/(1+{e}^{x/upsigma })) is a step-like function of x. Here (x=r/{r}_{c}-1) is the relative radial coordinate of electron and is related to potential slope. Forthe laterconvenienceofcomparison, ({V}_{b}=frac{{hbar}^{2}{j}_{b}^{2}}{2m{r}_{c}^{2}}) is expressed to have the same form as the angular kinetic energy of an electron in 1D ring with m the electron mass, where the dimensionless parameter jb is equivalent to an angular quantum number. The potential has a reduced form
$$mathrm{V}left(xright)=-frac{{mathrm{V}}_{mathrm{b}}mathrm{sh}frac{a}{upsigma }}{mathrm{ch}frac{a}{upsigma }+mathrm{ch}frac{x}{upsigma }}.$$
(1)
For an experimental ring12 of rc=418nm and W=85nm, for example, (aapprox 0.1), ({V}_{b}approx 20meV) and ({V}_{a}/{V}_{b}approx -0.49331) at =0.04 and jb=300.
As represented in Fig.1, the potential has a static minimum of ({V}_{c}=-{V}_{b}mathrm{sh}frac{a}{upsigma }/(mathrm{ch}frac{a}{upsigma }+1)) at (r={r}_{c}) (x=0), and is bounded from its bottom ({V}_{c}) to zero far from (r={r}_{c}). At ring edge, (V(a)=-frac{1}{2}{V}_{b}thfrac{a}{sigma }={V}_{a}) at x=a. Both ({V}_{c}) and ({V}_{a}) depend sensitively on both and a. For large , the potential approximates a parabolic confinement within the ring region (a A continuous and bounded potential model for a nanoring of radius ({r}_{c}) and width W under j=0: (a) V(r) with various a and ; (b) V(y) with a=0.1 and =0.03, as well as its 2- and 4-order polynomial approximants, where ({V}_{a}approx -0.4987{V}_{b}) and ({E}_{F}={V}_{c}+0.09{V}_{b}approx -0.8411{V}_{b}) at jF=90 and jb=300, as shown by two horizontal dotted lines. Introducing (u=frac{1}{r}) and ({u}_{c}=frac{1}{{r}_{c}}), then (x =frac{{u}_{c}}{u}-1=-frac{y}{1+y}) with (u={u}_{c}(1+y)). The potential becomes a function of y only. For a larger (less slope), the potential may be well fitted by a parabolic approximation within the ring region. For a relatively small (larger slope), Fig.1b shows V(y) for a nanoring of a=0.1 and =0.03, as well as its 2- and 4-order polynomial approximations. The asymmetry with respect of y=0 can be clearly seen from Fig.1b at higher energies. The potential is evidently deviated from the parabolic curve, where the outside confinement (r>rc, y<0) is substantially harder than the inner one (r In the presence of magnetic flux piercing through the ring center, the Hamiltonian of non-interacting electron inspinless case is given by (H=frac{{p}_{r}^{2}}{2m}+{V}_{eff}), where ({p}_{r}=-Jfrac{du}{dvarphi }=-Jdot{u}) is a radial momentum and ({V}_{eff}=frac{{J}^{2}{u}^{2}}{2m}+V(y)) is an effective potential. Here (J=jhbar=(l+phi )hbar) is the generalized angular momentum, l is the angular quantum number, and (phi =Phi /{Phi }_{0}) is a dimensionless flux. Using extreme value condition on Veff, the dynamic equilibriumpoint is relocated by $${j}^{2}{u}_{0}^{3}=frac{{j}_{b}^{2}{u}_{c}^{3}}{2upsigma {w}_{0}^{2}}mathrm{sh}frac{a}{upsigma }mathrm{sh}frac{beta }{upsigma },$$ (2) at (x={u}_{c}/{u}_{0}-1=beta), where ({u}_{0}={u}_{c}/(1+beta )) and ({w}_{0}=chfrac{a}{sigma }+chfrac{beta }{sigma }). Then we get the extreme value ({V}_{0}=frac{{J}^{2}{{varvec{u}}}_{0}^{2}}{2m}-frac{{V}_{b}}{{w}_{0}}shfrac{a}{sigma }) at (u={u}_{0}). Byasimpleiteration, it follows that (beta approx {chi }^{2}{j}^{2}(1-3{chi }^{2}{j}^{2})), where (chi =frac{sqrt{2}sigma }{{j}_{b}}(chfrac{a}{sigma }+1)/{(shfrac{a}{sigma })}^{1/2}) is a structural factor, determined only by the characteristic parameters of , a and jb. Obviously, the value of is very tiny at large ({j}_{b}) and little (e.g., ({chi approx 5.47times 10}^{-4}ll 1) at =0.04, a=0.1 and jb=300). For a motion of small amplitude, the electron oscillates radially around u=u0, following in u=u0(1+y). The Hamiltonian can be expressed in a Taylor series of y, $$H={mathrm{V}}_{0}+frac{{mathrm{J}}^{2}{u}_{0}^{2}{dot{mathrm{y}}}^{2}}{2m}+frac{{mathrm{J}}^{2}{u}_{0}^{2}}{mbeta }(frac{1}{2}{gamma }_{1}{mathrm{y}}^{2}-frac{1}{3}{gamma }_{2}{mathrm{y}}^{3}+frac{1}{4}{gamma }_{3}{mathrm{y}}^{4}+dots ),$$ (3) with ({gamma }_{1}approx 1+4beta), ({gamma }_{2}approx 3+6beta +beta frac{{w}_{0}-6}{2{sigma }^{2}{w}_{0}}), and ({gamma }_{3}approx 6+frac{{w}_{0}-6}{6{sigma }^{2}{w}_{0}}+beta (10+frac{3}{2{sigma }^{2}}-frac{15}{{sigma }^{2}{w}_{0}})). Upon the initial condition of y(0)=0 and (dot{y})(0)=, the electronic energy E is simply given by $$E={mathrm{V}}_{0}+frac{{mathrm{J}}^{2}{u}_{0}^{2}}{2m}{upeta }^{2},$$ (4) depending sensitively on the initial conditions of both J and . The quantized solutions for follow in a semi-classical description from the Bohr-Sommerfeld quantization rules22,30, $$oint {p}_{r}mathrm{dr}=oint frac{mathrm{J}{dot{y}}^{2}mathrm{dvarphi }}{{left(1+yright)}^{2}}=2pi nhbar,$$ (5) with n the radial quantum number. Then we get from Eq.(4) the quantized energy levels En,l((phi)), each carrying a current in,l (=-frac{1}{{Phi }_{0}}frac{partial {E}_{n,l}}{partial phi }). The total current Itot is finally obtained by $${I}_{tot}=sum {i}_{n,l}=-frac{1}{{Phi }_{0}}sum frac{partial {E}_{n,l}}{partialphi },$$ (6) summing over En,l below Fermi energy EF at T=0. Fourier harmonics Ak of a current I are derived by ({A}_{k}={int }_{-1/2}^{1/2}dphi Isin(2pi kphi )). To examine the quantized solutions for (eta), the rest key task is to find the radial function y under its initial conditions. From Newtons law, to the first order of , we get the equation of motion, $$upbeta ddot{y}+{gamma }_{1}mathrm{y}=f={gamma }_{2}{mathrm{y}}^{2}-{gamma }_{3}{mathrm{y}}^{3}+cdots ,$$ (7) where f acts as a nonlinear driving force. In a regular iterative method31, it is noticed that the even-order terms develop a constant average force (zero frequency), leading to a dynamic displacement, while the odd-order terms contain a base-frequency component, giving rise to a resonant divergence. That is, only if f~sin, y~cos becomes divergent with . It is physically evident that the magnitude of the oscillation cannot increase of itself in a closed system with no external source of energy32. For the strong nonlinearity, it is a huge challenge to solve Eq.(7) analytically. One needs to develop a fully nonlinear and nonperturbative approach. To avoid the resonant divergence, we express the solution y as y=yi in a series of all-order trial solutions yi, meeting the initial conditions y1(0)=0 and ({dot{y}}_{1})(0)= while ({y}_{i}(0)={dot{y}}_{i})(0)=0 at i>1. Equation(7) is then rewritten into (sum (beta {ddot{y}}_{i}+{gamma }_{1}{y}_{i})={f}_{2}+{f}_{3}+dots), where f is classified by the power series into f2, f3, , with ({f}_{2}={gamma }_{2}{y}_{1}^{2}) and ({f}_{3}=2{gamma }_{2}{y}_{1}{y}_{2}-{gamma }_{3}{y}_{1}^{3}). Taking into account the nonlinear contributions of both the constant average force and the base-frequency component, we consider a generally trial solution of ({y}_{1}=varepsilon +frac{eta }{gamma }singamma varphi -varepsilon cosgamma varphi =varepsilon +Asintheta) with ({mathrm{y}}_{1})(0)=0 and ({dot{y}}_{1})(0)=. Here (A={(frac{{eta }^{2}}{{gamma }^{2}}+{varepsilon }^{2})}^frac{1}{2}), (theta =gamma varphi +{theta }_{0}), and (tan{theta }_{0}=-varepsilon gamma /eta). Two preset parameters of both and are introduced for the dynamic displacement and the frequency shift, which can be conveniently obtained by the order-by-order self-consistent approach. At the linear approximation ((fapprox) 0), it simply follows that (upvarepsilon =0), (gamma ={gamma }_{0}=sqrt{frac{{gamma }_{1}}{beta }}) and (yapprox {y}_{1}=frac{eta }{{gamma }_{0}}sin{gamma }_{0}varphi). For small (({gamma }_{0}gg 1)), the solution exhibits a low-amplitude and high-frequency oscillation. From this, the amplitude of high-order terms above the fifth in Eq.(3) can be roughly estimated by ({y}_{1}^{5}/beta sim {beta }^{3/2}sim {chi }^{3}), which may be very small and thus may be neglected. This means that we can obtain better accuracy only by considering the first few items. As a reference, the quantized solution for is analytically obtained by (frac{{}^{2}}{{gamma }_{0}^{2}}=1-1/{(1+frac{n}{{gamma }_{0}j})}^{2}approx frac{2n}{{gamma }_{0}j}) at the linear approximation. The quantized energy levels are then given by ({E}_{n,l}approx {V}_{0}+frac{nJ{u}_{0}^{2}}{m}sqrt{frac{{upgamma }_{1}}{upbeta }}), in approximately proportional to n, which is similar to that in a 2D parabolic potential27,28,29. For the higher-order approximation, the detailed derivations are given in Supplementary Information file. Neglecting the higher-order terms, to the third-order approximation, the nonlinear driving force of (fapprox {f}_{2}+{f}_{3}) involves not only the orbital-coupling-like effect ((mathrm{e}.mathrm{g}., 2{gamma }_{2}{y}_{1}{y}_{2})) but also the self-energy-like effect from the odd-order terms ((mathrm{e}.mathrm{g}., -{gamma }_{3}{mathrm{y}}_{1}^{3})), both contributing to the average force and the base frequency. Defining (mathrm{z}={upgamma }^{2}/{upgamma }_{0}^{2}), both and (i.e., z) are exactly derived by $$varepsilon =frac{1}{2}frac{{upeta }^{2}}{{gamma }_{0}^{2}}frac{{gamma }_{2}}{{gamma }_{1}},$$ (8) $$(mathrm{z}-1)(mathrm{z}-frac{1}{4})(mathrm{z}-frac{1}{9})=frac{1}{3}mu frac{{upeta }^{2}}{{gamma }_{0}^{2}}frac{{gamma }_{2}^{2}}{{gamma }_{1}^{2}}.$$ (9) The dimensionless coefficients and are given by $$kappa=frac{1}{z}(z-frac{1}{4})(z-frac{1}{9})/left[{(z-frac{1}{4})}^{2}(z-frac{1}{9})+frac{5}{6}{Omega }_{0}frac{{upeta }^{2}}{{gamma }_{0}^{2}}right],$$ (10) $$upmu =1+2(mathrm{z}-frac{1}{4})frac{{{gamma }_{1}gamma }_{3}}{{gamma }_{2}^{2}}-frac{9}{4}{}^{2}{mathrm{z}}^{2}(mathrm{z}-frac{1}{4})(mathrm{z}-frac{1}{9}),$$ (11) with ({Omega }_{0}=frac{1}{2}frac{{gamma }_{2}^{2}}{{gamma }_{1}^{2}}+frac{{gamma }_{3}}{{gamma }_{1}}(z-frac{1}{4})), both of which are only determined by variable z. Only if (eta ne 0), it is necessitated that (zne 1), (zne 1/4), and (zne 1/9), so that (kappa ne 0), and (mu ne 0). This means that the frequency shifts, the dynamic displacement, and even a series of new energy levels and new energy states can be expected due to the nonlinear resonance levels in such a confinement32, with no regular resonance divergence. In essence, Eq.(9) is reducibleto a ninth-order equation of z, which cannot be solved analytically. For a tiny amplitude of /0, using an iterative approximation, we can solve Eq.(9) for z (i.e., ) separately by sub-region at about z~1, (zsim frac{1}{4}),and (zsim frac{1}{9}). Furthermore, can be obtained from Eqs.(8) and (10). The radial function of (yapprox {y}_{1}+{y}_{2}+{y}_{3}) is specified by (yapprox {uplambda }_{0}+{uplambda }_{1}Asintheta +{uplambda }_{2}{A}^{2}cos2theta +{uplambda }_{3}{A}^{3}sin3theta), of which the parameters of both A and ({uplambda }_{mathrm{0,1},mathrm{2,3}}) depend on and (or z) and thus on /0. Ignoring higher-order effects, can be simply quantized by Eq.(5), and the quantized energy is then given by Eq.(4). The total current can be further decomposed into three partial currents I1,2,3, originated from the levels contributions respectively at about z~1, (zsim frac{1}{4}),and (zsim frac{1}{9}). The first current I1 just corresponds to PC in a parabolic potential, and the latter two currents I2,3 are induced by the newly found nonlinear resonance levels at about z=1/4 and z=1/9. While itisdifficulttodistinguish one from another, experimentally, three partial currents are measurable as a whole. Theoretically, the signs and the relative sizes of three partial currents will reveal the intrinsic magnetic response mechanism, which is different from that in the 1D ring, 2D square well, and parabolic potential. In even higher approximations, nonlinear oscillations may also appear at other frequencies. As the degree of approximation increases, however, the oscillating strength decreases so rapidly that in practice only the first lower-order contribution can be observed31. Read the original here: Nonperturbative approach to magnetic response of an isolated ... - Nature.com