Note on a Floquet/Bloch-band fusion phenomenon in scattering by truncated periodic multi-well potentials
NNote on a Floquet/Bloch-band fusion phenomenon inscattering by truncated periodic multi-well potentials
K-E Thylwe
Kl¨overv¨agen 16, 387 36 Borgholm, Sweden(Retired from Department of Mechanics, KTH- Royal Institute of Technology)E-mail: [email protected]
Abstract.
A transmission phenomenon for a quantal particle scattered through a multi-well potential in one dimension is observed by means of an amplitude-phase method. Thepotential model consists of n identical potential cells, each containing a symmetric well. Typicaltransmission bands contain n − n energies of total transmission. A fusion phenomenon of this type of bandwith a typical neigboring band is also found. As the transmission gap between them collapseand disappear, a resulting fused single band is seen to contain 2 n − Keywords: Hill’s equation, Stability theory, Quantal transmission, Floquet/Bloch theory,Transmission bands, Amplitude-phase method
1. Introduction and presentation of results
Basic quantum physics related to truncated periodic potentials focuses on transmission andreflection properties. A free particle wave enters the potential region from one side (here from theright side); part of the wave is reflected (here in the direction x → + ∞ ), while the remaining partof the wave is transmitted through the potential region (here in the direction x → −∞ ). Fromintensities of wave component one obtains the so-called reflection and transmission coefficients.The present approach is similar to that for wave scattering by a single potential barrier orwell [1]. A main difference in this study is that the interaction has a finite periodic structure ofpotential ’cells’, where each cell contains a symmetric well. This part of the potential is treatedas described in [2].Quantal transmissions through multi-barrier/well systems in one dimension [3]-[12] provideimportant theoretical notions for analyzing tunneling in solids [13], chemical selection ofgas components [14], electronic properties of material structures such as graphene [15]-[20].Generalizations involve 2 D models, relativistic corrections, and spin-coupling effects [21]-[26]. a r X i v : . [ c ond - m a t . m e s - h a ll ] J u l loquet/Bloch-band fusion phenomenon − − − x / (cid:47) V ( x ) Figure 1.
Illustration of a multi-well potential and traveling wave components. Theincoming wave component (thick arrow) enters from the right.
Transmission effects caused by various non-vanishing exterior potentials are also in progress[27]. The truncated periodic potential is assumed vanishing outside an interval 0 ≤ x ≤ nπ ,where π is the dimensionless unit length of the period (or cell) and n is the number of suchcells. The potential satisfies V ( π ) = V (0) = 0 and its derivative with respect to x satisfies V (cid:48) ( π ) = V (cid:48) (0) = 0. The analytic form of the multi-well potential used for numerical illustrationsis V ( x ) = V sin ( x ) , ≤ x ≤ nπ, V ( x ) = 0 , x < , x > nπ, (1)where V < ≤ x ≤ π . Arrows indicate directions of propagating quantal waves.The time-independent Schr¨odinger equation with dimensionless parameters is expressed asd F ( x )d x + 2 [ E − V ( x )] F ( x ) = 0 , (2)as if presented in atomic units. The symbol E ( >
0) represents the total scattering energy.Equation (2) is a special case of a so called Hill equation [28]-[33] for V ( x ) in (1). Scatteringboundary conditions for the wave function F ( x ) are F ( x ) ∼ t √ k exp( − i kx ) , x → −∞ , (3 a ) F ( x ) ∼ √ k exp( − i kx ) + r √ k exp(i kx ) , x → + ∞ , (3 b )where t and r are the transmission and reflection amplitudes, respectively. The reduced (angular)wave number is k = √ E. (4)The complex-valued amplitudes t and r determine the transmission and reflection coefficientsfrom T = | t | , R = | r | . (5) loquet/Bloch-band fusion phenomenon T ( E ) = (cid:16) | Λ( E ) | (cid:17) − , (6)containing an imaginary quantityΛ( E ) = i J ( E ) sin nα ( E ) . (7) α ( E ) is the real Floquet/Bloch phase as function of energy E , an intrinsic phase defined by thewave propagation across a single cell. This phase is independent of which exact method is used.The energy dependent factor J ( E ) is independent of n and is singular at band edges. Totaltransmission occurs at zeros of Λ( E ), i.e. by either of the conditions J ( E J ) = 0 , (8) α njν = ( j + ν/n ) π, α njν = α ( E njν ) . (9)A band number is represented by j = 0 , , · · · , and ν = (0) , , · · · , n − J ( E ), for E = E J , is assigned an index ’ J ’ instead of a ν -number. J ( E ) is not related to the much ’faster’Floquet/Bloch phase α , as function of energy. Zeros of J ( E ) are not present in all bands. Singlezeros occur in particular bands.The case ν = 0 is usually forbidden in a typical j -band. There, the phase α ( E ) is confinedto an interval jπ ≤ α ( E ) ≤ ( j + 1) π . Phase values α = jπ and α = ( j + 1) π correspondto band edges, where J ( E ) is singular and sin nα ( E ) is zero. Note that the product of thesequantities, and Λ( E ), is still finite. If two neighboring bands, e.g. the j - and ( j + 1)-bands,fuse, the case ν = 0 becomes valid for the ( j + 1)-band. Then, α is real in the larger interval jπ ≤ α ( E ) ≤ ( j + 2) π . This fusion phenomenon is illustrated in figures 1 and 2.Figure 1 shows transmission coefficients as functions of energy for n = 6 and three potentialparameter values V = − , −
8, and -9. The two first bands, j = 0 ,
1, have negative energies.The top subplot ( V = −
7) shows two separated transmission bands corresponding to j = 2 and j = 3. The number of energy peaks of total transmission is 5 in the band j = 2. Two peaks inthe band j = 3 are less sharp. A detailed analysis shows that the peak at E ≈ .
85 is due tothe single zero of J ( E ). A close-lying peak at E ≈ .
72 is due to the phase factor sin nα ( E ) inequation (7). There are 6 peaks of total transmission in band j = 3.The bottom subplot ( V = −
9) in Figure 1 shows that the band with j = 3 has 5 peaks oftotal transmission. For j = 2, two peaks near the band edge are unclear. A detailed analysisshows that the peak at E ≈ .
12 is due to the single zero of J ( E ). A close-lying peak at E ≈ . nα ( E ) in equation (7). Other peaks of total transmission are due to thephase factor sin nα ( E ). loquet/Bloch-band fusion phenomenon E T ( E ) Figure 2.
Energy behaviors of T for potential parameter values V = − j = 2 and3 are separated in the top and bottom subplots. In the middle subplot, the fused bandwith j = (2 , − J ( E ) − − E Figure 3.
Energy behaviors of J ( E ) for potential parameter values V = − J ( E ) has a zero passage near E = 2 .
85 for j = 3 in the top subplot, and near E = 1 .
12 for j = 2 in the bottomsubplot. In the middle subplot, the fused band j = (2 , J < In the middle subplot ( V = −
8) all peaks of total transmission are due to zeros of sin nα ( E ). J ( E ) has no zeros. The total number of peaks is 2 n −
1. The way a gap between such bandsvanishes is explained in detail in [27].Figure 2 illustrates the energy behaviors of J ( E ) in the complete band zones j = 2 and j = 3. J ( E ) is imaginary in gap zones and is not illustrated. The top subplot shows that J ( E )has a zero and changes sign within band j = 3. The bottom subplot shows the same thingexcept that the zero occurs in band j = 2. The middle subplot shows the fused band, where loquet/Bloch-band fusion phenomenon ( n, j, ν ) E n,j,ν ( V = − , ,
1) 0.4310069(6 , ,
2) 0.6020104(6 , ,
3) 0.8594292(6 , ,
4) 1.1829404(6 , ,
5) 1.5592963(6 , ,
0) 1.9804654(6 , ,
1) 2.4414865(6 , ,
2) 2.9392161(6 , ,
3) 3.4715098(6 , ,
4) 4.0368108(6 , ,
5) 4.6336285
Table 1.
Energies E n,j,ν of total transmission for V = − n, j, ν ) E n,j,ν ( V = − E n,j,ν ( V = − , ,
1) 0.5864576 0.2720599(6 , ,
2) 0.7839609 0.4232600(6 , ,
3) 1.0683904 0.6575990(6 , ,
4) 1.4144690 0.9593766(6 , , J ) −− , ,
5) 1.8083837 1.3158913(6 , ,
1) 2.7205677 2.1769876(6 , , J ) 2.8508686 −− (6 , ,
2) 3.2275311 2.6622808(6 , ,
3) 3.7680369 3.1848282(6 , ,
4) 4.3405175 3.7418530(6 , ,
5) 4.9438598 4.3311594
Table 2.
Energies E n,j,ν and E n,j,J of total transmission for V = − V = −
9. Thesymbol ’ J ’ is used for peak energies caused by J ( E ). J ( E ) < n − j = 2 an 3, are used. The quantum number ν = 0 appears for j = 3. Allenergies are due to the Floquet/Bloch phase condition in (9).Table 2 contains energy peaks of total transmission seen in Figure 1 for the top and bottomsubplots. Here, the transmission bands j = 2 and 3 are separated by a gap zone. Peaks of totaltransmission due to J ( E ) occur. The total number of peaks for j = 2 and 3 is 11, the same asin Table 1, but the reasons are different.If exterior potentials are added to the trucated potential considered , the fusion phenomenon loquet/Bloch-band fusion phenomenon
2. Derivations
To obtain t and r in (3 a ) and (3 b ), particular amplitude-phase solutions in each characteristicregion are introduced. There are two asymptotic regions and the region of n identical cells.Two independent solutions of (2) are defined in terms of a positive amplitude function A ( x )and a related real phase function p ( x ) as [2]Ψ ( ± ) ( x ) = A ( x ) exp( ± i p ( x )) , (10) p (cid:48) ( x ) = A − ( x ) ( > , (11)where (cid:48) = d / d x . Due to the relation (11), the Wronskian determinant of the two solutions (10)is independent of x [1]. Any amplitude function satisfies a nonlinear Milne-Pinney equation[35]-[34] d A ( x )d x + 2 [ E − V ( x )] A ( x ) = A − ( x ) . (12)Amplitude functions differ by their boundary conditions [39, 40]. For any choice of A ( x ) onehas two independent exact solutions Ψ ( ± ) ( x ). An amplitude function is known to be more orless oscillatory due to different choices of its boundary conditions. Several amplitude functionsmay be used to represent a given linear wave function. Different representations of a linear wavefunction can be expressed in terms of the others by linear combinations.Equation (12) is re-written for computational purposes as a first-order differential equationas A ( x ) A (cid:48) ( x ) p ( x ) (cid:48) = A (cid:48) ( x ) A − ( x ) − E − V ( x )) A ( x ) A − ( x ) . (13)The integration starts from boundary conditions of the amplitude function. The phase functionneeds a specified integration constant.Amplitude-phase solutions A ( x ) of (12) are used locally, in each characteristic region of x [39, 40]. Firstly the two exterior regions are considered. The two exterior solutions, withamplitude functions A L ( x ) = A R ( x ) = k − / of (12), are:Ψ ( ± ) L ( x ) = k − / exp( ± i kx ) . (14 a )Ψ ( ± ) R ( x ) = k − / exp( ± i kx ) exp( ∓ i knπ ) , (14 b ) x = 0 and x = nπ , are reference points for the respective phases. Fundamental solution matricesconsist of Ψ ( ± ) L,R ( x ) in the upper row and Ψ (cid:48) ( ± ) L,R ( x ) in the lower row. The exterior fundamentalsolutions satisfy Ψ L (0) = (cid:32) k − / k − / i k / − i k / (cid:33) , Ψ R ( nπ ) = (cid:32) k − / k − / i k / − i k / (cid:33) . (15) loquet/Bloch-band fusion phenomenon A p ( x ). Such an amplitude function is defined byparticular boundary conditions at the first cell boundary point x = 0, chosen as [2] A p (0) = u p , A (cid:48) p (0) = 0 . (16)The corresponding phase satisfies p (cid:48) p ( x ) = A − p ( x ), and the phase reference point is taken at x = 0. Particular phase values are p p (0) = 0 , p p ( π ) = α ; p p ( nπ ) = nα. (17)A principal fundamental solution matrix is defined by the solutions S ( x ) = A p ( x ) /u p sin p p ( x ) , C ( x ) = A p ( x ) u p cos p p ( x ) . (18)as Ψ ( x ) = (cid:32) C ( x ) S ( x ) C (cid:48) ( x ) S (cid:48) ( x ) (cid:33) , det Ψ ( x ) = 1 , (19)satisfying Ψ (0) = (cid:32) (cid:33) , Ψ ( π ) = (cid:32) cos α u p sin α − u − p sin α cos α (cid:33) , Ψ ( nπ ) = (cid:32) cos nα u p sin nα − u − p sin nα cos nα (cid:33) . (20)Note that Ψ ( nπ ) = Ψ n ( π ). Calculations of u p and α require knowledge of a single cell of theperiodic part of the potential. Details of how to compute u p and α are found in reference [2].A connection between two fundamental solutions of the Schr¨odinger equation is formulatedby a matrix equation involving a constant matrix. For example, the two fundamental ’exterior’solutions Ψ L,R ( x ) are related by Ψ L ( x ) = Ψ R ( x ) Ω , (21)where Ω is an x -independent matrix. Ω can be determined at any matching point, say x = nπ .This gives the relation Ψ L ( x ) = Ψ R ( x ) (cid:104) Ψ − R ( nπ ) Ψ L ( nπ ) (cid:105) . (22)The matrix value Ψ L ( nπ ) can be expressed in terms of Ψ ( x ) in (19) for the periodic part of thepotential. A matching between Ψ L ( x ) and Ψ ( x ) at x = 0 yields Ψ L ( x ) = Ψ ( x ) (cid:104) Ψ − (0) Ψ L (0) (cid:105) . (23)By combining (22) and (23) with the use of (15) and (20), one finds the matrix Ω as Ω = (cid:32) k − / k − / i k / − i k / (cid:33) − (cid:32) cos nα u p sin nα − u − p sin nα cos nα (cid:33) (cid:32) k − / k − / i k / − i k / (cid:33) , (24)i.e. Ω = (cid:32) ∆ ∗ ΛΛ ∗ ∆ (cid:33) , det Ω = 1 , , | ∆ | = 1 + | Λ | , (25) loquet/Bloch-band fusion phenomenon (cid:18)(cid:16) ku p (cid:17) − − ku p (cid:19) sin nα, ∆ = cos nα − i2 (cid:18)(cid:16) ku p (cid:17) − + ku p (cid:19) sin nα. (26)The quantity Λ is expressed with the use of J ( E ) = (cid:18)(cid:16) ku p (cid:17) − − ku p (cid:19) / Ω , scattering boundary conditions (3 a ) and (3 b ) can be re-interpreted in termsof amplitude-phase quantities. In the left asymptotic region of x the amplitude-phase solutionΨ ( − ) L ( x ) behaves as [1]Ψ ( − ) L ( x ) ∼ √ k e − i kx , as x → −∞ . (27)Ψ ( − ) L ( x ) corresponds, via equation (21), to an equivalent expression in terms of Ψ ( ± ) R ( x ), givenby Ψ ( − ) L ( x ) = ΛΨ (+) R ( x ) + ∆Ψ ( − ) R ( x ) , (28)where Ψ ± R ( x ) ∼ √ k e ∓ i knπ e ± i kx , x → + ∞ . (29)From (28) and (29) followsΨ ( − ) L ( x ) ∼ Λ √ k e − i knπ e i kx + ∆ √ k e i knπ e − i kx , x → + ∞ . (30)Normalizing (30) to agree with condition (3 b ), the transmission and reflection amplitudes appearas t = e − i knπ , r = e − knπ Λ∆ . (31)The transmission and reflection coefficients defined in (5) can be expressed in terms of Λ as T = 11 + | Λ | , R = | Λ | | Λ | . (32)This brief derivation is generalized to include exterior potentials with more general amplitude-phase methods in [27] (to be published elsewhere).
3. Concluding remarks
A formally exact amplitude-phase approach is explored in the context of one-dimensionalscattering and Floquet/Bloch bands. The relevance of Floquet/Bloch theory is illustrated for amulti-well potential with n = 6. Band/gap structures for a given potential explains structuresof transmission bands for multi-well potentials even for n = 6. Band types can be classified byan intrinsic quantity J ( E ), which may have a zero or not in the band. Energy peaks of totaltransmission are caused by two intrinsic quantities: J ( E ), a slowly varying functions of energy;and the intrinsic phase α ( E ), a rapidly vaying function of energy. loquet/Bloch-band fusion phenomenon References [1] K.-E. Thylwe, J. Phys. A: Math. Gen. (2005) 235.[2] K.-E. Thylwe, Phys. Scr. 94 (2019) 065201; https://doi.org/10.1088/1402-4896/ab40d3.[3] D. J. Griffiths and C. A. Steinke, American Journal of Physics 69 (2001) 137;https://doi.org/10.1119/1.1308266.[4] M. Dharani, and C. S. Shastry, AIP Conference Proceedings 1731, 110017 (2016); https://doi-org./10.1063/1.4948038[5] Z. Shao and W. Porod, Phys. Rev. B 51 (1995) 1931.[6] G.-Y. Oh, arxiv.org/abs/cond-mat/9902181[7] F. Maiz, Physica B: Condensed Matter, 463 (2015) 93.[8] J. Nanda, P. K. Mahapatra, C. L. Roy, Physica B: Physics of Condensed Matter, Vol.383(2) (2006) 232[9] S. Mukhopadhyay, R. Biswas, C. Sinha, Physics Letters A, Vol.376(15) (2012) 1306.[10] K. W. Yu, Computers in Physics 4, 176 (1990), https://doi.org/10.1063/1.168361.[11] D. W. L. Sprung, Hua Wu, and J. Martorell, American Journal of Physics 61 (1993) 1118,https://doi.org/10.1119/1.17306.[12] D. Bar, International Journal of Theoretical Physics, Vol. 44, (2005) 1281, DOI: 10.1007/s10773-005-4686-x[13] C. B. Duke, Tunneling in Solids (Academic, New York and London, 1969).[14] S. Mandr`a, J. Schrier, M. Ceotto, The journal of physical chemistry. A, Vol.118(33) (2014) 6457.[15] M. Dragomana, D. Dragoman, Progress in Quantum Electronics 33 (2009) 165214; A. H. Castro Neto, F.Guinea, N. M. R. Peres, K. S. Novoselov and A. K. Geim, Rev. Mod. Phys. 81 (2009) 10962.[16] A. Zubarev and D. Dragoman, Physica E 44 (2012) 1687.;A. Zubarev and D. Dragoman, J. Phys. D: Appl.Phys. 47 (2014) 425302.[17] D. S. Daz-Guerrero, L. M. Gaggero-Sager, I. Rodrguez-Vargas and O. Sotolongo-Costa, Panchadhyayee,Pradipta Philosophical Magazine, (2013) 1.[18] L. A. Cury, N. Studart, Superlattices and Microstructures, Vol.4(2) (1988) 245.[19] G. Karavaev, N. Chuprikov, Russian Physics Journal, Vol.36(8) (1993) 749.[20] S. Kumar; S. Kumari Int. J. of Nanoparticles, Vol 10 (2018) 92.[21] P. Pyykk¨o, Chem. Rev. 88 (1988) 563.[22] P. Pereyra, J. Phys. A 31 (1998) 4521 .[23] Siddhant Das, American Journal of Physics 83 (2015) 590; https://doi.org/10.1119/1.4916834.[24] R. Zhao, Y. Zhang, Y. Xiao, and W. Liu, J. Chem. Phys. 144 (2016) 044105.[25] C L Roy, Journal of Physics: Condensed Matter, Vol.5(41) (1993) 7701.[26] C.H. Chen, P. Tseng, W.J. Hsueh, Physics Letters A 380 (2016) 2957.[27] K.-E. Thylwe, http://arxiv.org/pdf/2005.11695.[28] W. Magnus and S. Winkler 1979
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