409:. In certain materials the topological invariant can be changed when certain bulk energy bands invert due to strong spin-orbital coupling. At the interface between an insulator with non-trivial topology, a so-called topological insulator, and one with a trivial topology, the interface must become metallic. More over, the surface state must have linear Dirac-like dispersion with a crossing point which is protected by time reversal symmetry. Such a state is predicted to be robust under disorder, and therefore cannot be easily localized.
3602:(LCAO), see figure 5. In this picture, it is easy to comprehend that the existence of a surface will give rise to surface states with energies different from the energies of the bulk states: Since the atoms residing in the topmost surface layer are missing their bonding partners on one side, their orbitals have less overlap with the orbitals of neighboring atoms. The splitting and shifting of energy levels of the atoms forming the crystal is therefore smaller at the surface than in the bulk.
59:
81:
70:
2847:
1042:. Electronic band structure in the nearly free electron picture. Away from the Brillouin zone boundary the electron wave function has plane wave character and the dispersion relation is parabolic. At the Brillouin zone boundary the wave function is a standing wave composed of an incoming and a Bragg-reflected wave. This ultimately leads to the creation of a band gap.
1006:
3708:; in these experiments, periodic modulations in the surface state density, which arise from scattering off of surface impurities or step edges, are measured by an STM tip at a given bias voltage. The wavevector versus bias (energy) of the surface state electrons can be fit to a free-electron model with effective mass and surface state onset energy.
3122:
729:
200:
3779:
is a positive integer)? A well-accepted concept proposed by Fowler first in 1933, then written in Seitz's classic book that "in a finite one-dimensional crystal the surface states occur in pairs, one state being associated with each end of the crystal." Such a concept seemly was never doubted since
3569:
that are being cut by these rods allow states that penetrate deep into the crystal. One therefore generally distinguishes between true surface states and surface resonances. True surface states are characterized by energy bands that are not degenerate with bulk energy bands. These states exist in the
65:. Simplified one-dimensional model of a periodic crystal potential terminating at an ideal surface. At the surface, the model potential jumps abruptly to the vacuum level (solid line). The dashed line represents a more realistic picture, where the potential reaches the vacuum level over some distance.
2833:
all energies of the surface states fall into the forbidden gap. The complete solution is again found by matching the bulk solution to the exponentially decaying vacuum solution. The result is a state localized at the surface decaying both into the crystal and the vacuum. A qualitative plot is shown
2861:
can readily be generalized to the case of a three-dimensional crystal. Because of the two-dimensional periodicity of the surface lattice, Bloch's theorem must hold for translations parallel to the surface. As a result, the surface states can be written as the product of a Bloch waves with k-values
256:
is the wave number. The allowed wave numbers for a given potential are found by applying the usual Born–von Karman cyclic boundary conditions. The termination of a crystal, i.e. the formation of a surface, obviously causes deviation from perfect periodicity. Consequently, if the cyclic boundary
1243:. The solutions to the Schrödinger equation must be obtained separately for the two domains z < 0 and z > 0. In the sense of the nearly free electron approximation, the solutions obtained for z < 0 will have plane wave character for wave vectors away from the Brillouin zone boundary
2750:
768:
1046:
The nearly free electron approximation can be used to derive the basic properties of surface states for narrow gap semiconductors. The semi-infinite linear chain model is also useful in this case. However, now the potential along the atomic chain is assumed to vary as a cosine function
1841:
3783:
The investigation tries to understand electronic states in ideal crystals of finite size based on the mathematical theory of periodic differential equations. This theory provides some fundamental new understandings of those electronic states, including surface states.
2575:
2111:
1235:
2938:
594:
3273:
422:
A simple model for the derivation of the basic properties of states at a metal surface is a semi-infinite periodic chain of identical atoms. In this model, the termination of the chain represents the surface, where the potential attains the value
104:
2368:
272:
is an oversimplification which is mostly convenient for simple model calculations. At a real surface the potential is influenced by image charges and the formation of surface dipoles and it rather looks as indicated by the dashed line.
87:. Real part of the type of solution to the one-dimensional Schrödinger equation that corresponds to surface states. These states decay into both the vacuum and the bulk crystal and thus represent states localized at the crystal surface.
41:
of materials. They are formed due to the sharp transition from solid material that ends with a surface and are found only at the atom layers closest to the surface. The termination of a material with a surface leads to a change of the
583:
1480:
1001:{\displaystyle {\begin{aligned}\Psi (z)&=&\left\{{\begin{array}{cc}Bu_{-k}e^{-ikz}+Cu_{k}e^{ikz},&{\textrm {for}}\quad z<0\\A\exp \left,&{\textrm {for}}\quad z>0\end{array}}\right.,\end{aligned}}}
2586:
2247:
400:
All materials can be classified by a single number, a topological invariant; this is constructed out of the bulk electronic wave functions, which are integrated in over the
Brillouin zone, in a similar way that the
284:
The first type of states (see figure 2) extends into the crystal and has Bloch character there. These type of solutions correspond to bulk states which terminate in an exponentially decaying tail reaching into the
288:
The second type of states (see figure 3) decays exponentially both into the vacuum and the bulk crystal. These type of solutions correspond to surface states with wave functions localized close to the crystal
76:. Real part of the type of solution to the one-dimensional Schrödinger equation that corresponds to the bulk states. These states have Bloch character in the bulk, while decaying exponentially into the vacuum.
2831:
2853:. Atomic like orbitals of a Pt-atom. The orbitals shown are part of the double-zeta basis set used in density functional calculations. The orbitals are indexed according to the usual quantum numbers (n,l,m).
257:
conditions are abandoned in the direction normal to the surface the behavior of electrons will deviate from the behavior in the bulk and some modifications of the electronic structure has to be expected.
1593:
313:
of the projected band structure of metals. It can be shown that the energies of these states all lie within the band gap. As a consequence, in the crystal these states are characterized by an imaginary
1613:
246:
750:, where at the domain boundary (z=0) the usual conditions on continuity of the wave function and its derivatives are applied. Since the potential is periodic deep inside the crystal, the electronic
354:. Shockley states are thus states that arise due to the change in the electron potential associated solely with the crystal termination. This approach is suited to describe normal metals and some
3172:
2943:
2591:
2384:
2286:
2142:
1869:
1618:
1373:
773:
599:
450:
109:
2930:
2379:
1864:
1052:
3642:. These states include states originating from reconstructed surfaces, where the two-dimensional translational symmetry gives rise to the band structure in the k space of the surface.
3117:{\displaystyle {\begin{aligned}\Psi _{0}({\textbf {r}})&=&\psi _{0}(z)u_{{\textbf {k}}_{||}}({\textbf {r}}_{||})e^{-i{\textbf {r}}_{||}\cdot {\textbf {k}}_{||}}\end{aligned}}}
724:{\displaystyle {\begin{aligned}V(z)=\left\{{\begin{array}{cc}P\delta (z+la),&{\textrm {for}}\quad z<0\\V_{0},&{\textrm {for}}\quad z>0\end{array}}\right.,\end{aligned}}}
358:. Figure 3 shows an example of a Shockley state, derived using the nearly free electron approximation. Within the crystal, Shockley states resemble exponentially-decaying Bloch waves.
1036:
3563:
3494:
3167:
3613:
hybrid in Si or Ge, it is strongly affected by the presence of the surface, bonds are broken, and the remaining lobes of the orbital stick out from the surface. They are called
3429:
3356:
3318:
3534:
3465:
195:{\displaystyle {\begin{aligned}\Psi _{n{\boldsymbol {k}}}&=\mathrm {e} ^{i{\boldsymbol {k}}\cdot {\boldsymbol {r}}}u_{n{\boldsymbol {k}}}({\boldsymbol {r}}).\end{aligned}}}
3599:
338:. There is no strict physical distinction between the two types of states, but the qualitative character and the mathematical approach used in describing them is different.
1278:
2281:
4630:"Sagittal acoustic waves in finite solid-fluid superlattices: Band-gap structure, surface and confined modes, and omnidirectional reflection and selective transmission"
1520:
1360:
3834:
1323:
2256:
which lies in the allowed band. As in the case for metals, this type of solution represents standing Bloch waves extending into the crystal which spill over into the
3159:
3855:
3805:
3391:
3927:
1015:. It is an extended Bloch wave within the crystal with an exponentially decaying tail outside the surface. The consequence of the tail is a deficiency of negative
3737:
3875:
3777:
3757:
3579:
445:
3887:). Numerical calculations have confirmed such findings. Further, these behaviors have been seen in different one-dimensional systems, such as in.
1847:
1523:
1368:
2745:{\displaystyle {\begin{aligned}E&={\frac {\hbar ^{2}}{2m}}\left\pm V{\sqrt {1-\left({\frac {\hbar ^{2}\pi q}{maV}}\right)^{2}}}\end{aligned}}}
2137:
330:
In the discussion of surface states, one generally distinguishes between
Shockley states and Tamm states, named after the American physicist
268:, of the lattice while close to the surface it has to somehow attain the value of the vacuum level. The step potential (solid line) shown in
377:(LCAO). In contrast to the nearly free electron model used to describe the Shockley states, the Tamm states are suitable to describe also
3650:
surface states are usually defined as states not originating from a clean and well ordered surface. Surfaces that fit into the category
2758:
3894:
The fundamental property of a surface state is that its existence and properties depend on the location of the periodicity truncation.
280:, it can be shown that the one-dimensional single-electron Schrödinger equation gives two qualitatively different types of solutions.
4670:
El
Boudouti, E. H.; Djafari-Rouhani, B.; Akjouj, A.; Dobrzynski, L. (2009). "Acoustic waves in solid and fluid layered materials".
1836:{\displaystyle {\begin{aligned}E&={\frac {\hbar ^{2}}{2m}}\left({\frac {\pi }{a}}+\kappa \right)^{2}\pm |V|\left\end{aligned}}}
3620:
In contrast to the nearly free electron model used to describe the
Shockley states, the Tamm states are suitable to describe also
3282:
is the effective mass of the electron. The matching conditions at the crystal surface, i.e. at z=0, have to be satisfied for each
1537:
3716:
A naturally simple but fundamental question is how many surface states are in a band gap in a one-dimensional crystal of length
1019:
just inside the crystal and an increased negative charge density just outside the surface, leading to the formation of a dipole
3693:
374:
208:
4705:
El
Hassouani, Y.; El Boudouti, E.H.; Djafari-Rouhani, B. (2013). "One-Dimensional Phononic Crystals". In Deymier, P.A. (ed.).
3780:
then for nearly a century, as shown, for example, in. However, a recent new investigation gives an entirely different answer.
758:
here. The solution in the crystal is then a linear combination of an incoming wave and a wave reflected from the surface. For
4722:
3880:
2124:, the bulk solution has to be fitted to an exponentially decaying solution, which is compatible with the constant potential
439:
of the lattice. The
Shockley states are then found as solutions to the one-dimensional single electron Schrödinger equation
1057:
4748:
3996:
4494:
Ren, Shang Yuan; Chang, Yia-Chung (2007). "Theory of confinement effects in finite one-dimensional phononic crystals".
4453:
2570:{\displaystyle {\begin{aligned}\Psi _{i}(z\leq 0)&=Fe^{qz}\left\pm \exp \left\right]e^{\mp i\delta }\end{aligned}}}
2106:{\displaystyle {\begin{aligned}\Psi _{i}&=Ce^{i\kappa z}\left(e^{i\pi z/a}+\lefte^{-i\pi z/a}\right)\end{aligned}}}
1230:{\displaystyle {\begin{alignedat}{2}V(z)&=V\left\\&=2V\cos \left({\frac {2\pi z}{a}}\right),\\\end{alignedat}}}
4531:"Two types of modes in finite size one-dimensional coaxial photonic crystals: General rules and experimental evidence"
3941:
An ideal simple three-dimensional finite crystal may have vertex-like, edge-like, surface-like, and bulk-like states.
3598:
are often called Tamm states. In the tight binding approach, the electronic wave functions are usually expressed as a
2865:
1858:
is given by 2V. The electronic wave functions deep inside the crystal, attributed to the different bands are given by
4753:
4191:
4051:
4005:
3975:
3688:
An experimental technique to measure the dispersion of surface states is angle resolved photoemission spectroscopy (
3578:. At energies where a surface and a bulk state are degenerate, the surface and the bulk state can mix, forming a
4259:
Proceedings of the Royal
Society of London. Series A, Containing Papers of a Mathematical and Physical Character
95:, eigenstates of the single-electron Schrödinger equation with a perfectly periodic potential, a crystal, are
3705:
3675:
surface states cannot easily be characterized in terms of their chemical, physical or structural properties.
3268:{\displaystyle {\begin{aligned}E_{s}=E_{0}+{\frac {\hbar ^{2}{\textbf {k}}_{||}^{2}}{2m^{*}}},\end{aligned}}}
17:
3539:
3470:
50:. In the weakened potential at the surface, new electronic states can be formed, so called surface states.
4629:
4530:
3396:
3323:
3285:
4743:
4340:
Ren, Shang Yuan (2002). "Two Types of
Electronic States in One-dimensional Crystals of Finite length".
3625:
3503:
3434:
1020:
347:
355:
43:
3897:
Truncation of the lattice's periodic potential may or may not lead to a surface state in a band gap.
4208:
2363:{\displaystyle {\begin{aligned}i\sin(2\delta )&=-i{\frac {\hbar ^{2}\pi q}{maV}}\end{aligned}}}
1246:
1023:. The dipole perturbs the potential at the surface leading, for example, to a change of the metal
1607:
are found by substitution into the Schrödinger equation. This leads to the following eigenvalues
382:
3664:
Interfaces between two materials, such as a semiconductor-oxide or semiconductor-metal interface
343:
1488:
1328:
3810:
1294:
3617:. The energy levels of such states are expected to significantly shift from the bulk values.
742:
is the normalization factor. The solution must be obtained independently for the two domains
395:
3130:
385:. Qualitatively, Tamm states resemble localized atomic or molecular orbitals at the surface.
4679:
4644:
4592:
4545:
4503:
4468:
4401:
4359:
4266:
4220:
4140:
4076:
3840:
3790:
3369:
3903:
8:
4707:
Acoustic
Metamaterials and Phononic Crystals, Springer Series in Solid-State Sciences 173
402:
305:
have to belong to one of the allowed energy bands. The second type of solution exists in
260:
A simplified model of the crystal potential in one dimension can be sketched as shown in
4683:
4648:
4605:
4596:
4580:
4549:
4507:
4472:
4405:
4363:
4270:
4224:
4144:
4080:
3719:
2252:
It can be shown that the matching conditions can be fulfilled for every possible energy
4610:
4375:
4349:
4284:
4164:
4130:
3860:
3762:
3742:
1527:
406:
38:
4718:
4614:
4581:"Surface and confined acoustic waves in finite size 1D solid-fluid phononic crystals"
4561:
4288:
4209:"Spin-polarized quantum confinement in nanostructures: Scanning tunneling microscopy"
4187:
4156:
4047:
4001:
3971:
3884:
319:
4379:
4168:
2932:
parallel to the surface and a function representing a one-dimensional surface state
1534:). Since the solutions of interest are close to the Brillouin zone boundary, we set
4710:
4687:
4652:
4600:
4553:
4511:
4476:
4367:
4322:
4274:
4228:
4148:
4084:
3636:
Surface states originating from clean and well ordered surfaces are usually called
3621:
378:
331:
34:
4691:
4102:
I. Tamm (1932). "On the possible bound states of electrons on a crystal surface".
4067:
W. Shockley (1939). "On the
Surface States Associated with a Periodic Potential".
3358:
a single, but generally different energy level for the surface state is obtained.
802:
622:
3965:
3658:
Surfaces with defects, where the translational symmetry of the surface is broken.
3574:
only and are therefore localized at the surface, similar to the picture given in
4714:
4424:
Electronic States in Crystals of Finite Size: Quantum Confinement of Bloch Waves
4398:
Electronic States in Crystals of Finite Size: Quantum Confinement of Bloch Waves
4326:
3944:
A surface state is always in a band gap is only valid for one-dimensional cases.
578:{\displaystyle {\begin{aligned}\left\Psi (z)&=&E\Psi (z),\end{aligned}}}
4656:
4557:
4515:
4232:
4152:
3879:
This state is either a band-edge state or a surface state in the band gap(see,
3606:
3497:
1851:
1016:
4579:
El Boudouti, E. H.; El Hassouani, Y.; Djafari-Rouhani, B.; Aynaou, H. (2007).
4024:
K. Oura; V.G. Lifshifts; A.A. Saranin; A. V. Zotov; M. Katayama (2003). "11".
2260:
at the surface. A qualitative plot of the wave function is shown in figure 2.
1239:
whereas at the surface the potential is modeled as a step function of height V
762:>0 the solution will be required to decrease exponentially into the vacuum
4737:
4181:
4160:
3614:
3595:
1855:
1475:{\displaystyle {\begin{aligned}\Psi (z)&=Ae^{ikz}+Be^{iz}.\end{aligned}}}
1285:
1024:
751:
428:
370:
362:
298:
4578:
4023:
435:. Within the crystal the potential is assumed periodic with the periodicity
4628:
El Hassouani, Y.; El Boudouti, E. H.; Djafari-Rouhani, B.; Rais, R (2008).
4565:
4371:
4279:
4254:
4088:
2858:
1284:. At the Brillouin zone boundaries, Bragg reflection occurs resulting in a
1011:
The wave function for a state at a metal surface is qualitatively shown in
4669:
4627:
4354:
3566:
3565:
is extending into the three-dimensional Brillouin zone of the Bulk. Bulk
1289:
302:
96:
4454:"Localized modes in a one-dimensional diatomic chain of coupled spheres"
4121:
Hasan, M. Z.; Kane, C. L. (2010). "Colloquium: Topological insulators".
3837:
always has one and only one state whose energy and properties depend on
3787:
The theory found that a one-dimensional finite crystal with two ends at
2242:{\displaystyle {\begin{aligned}\Psi _{0}&=D\exp \left\end{aligned}}}
3583:
2253:
755:
315:
92:
4704:
4480:
3937:
Further investigations extended to multi-dimensional cases found that
3646:
3638:
3431:
parallel to the surface, while a bulk state is characterized by both
335:
3571:
1035:
306:
4255:"Notes on some electronic properties of conductors and insulators"
4135:
2826:{\displaystyle 0\leq q\leq q_{max}={\frac {maV}{\hbar ^{2}\pi }}}
2373:
one obtains solutions with a decaying amplitude into the crystal
3963:
2841:
2755:
E is real for large negative z, as required. Also in the range
2257:
1280:, where the dispersion relation will be parabolic, as shown in
58:
47:
3586:, while retaining an enhanced amplitude close to the surface.
3689:
3361:
294:
3683:
3582:. Such a state can propagate deep into the bulk, similar to
342:
Historically, surface states that arise as solutions to the
80:
69:
2846:
1588:{\displaystyle k_{\perp }={\bigl (}\pi /a{\bigr )}+\kappa }
988:
711:
4709:. Vol. 173. Berlin, Springer-Verlag. pp. 45–93.
4182:
Frederick Seitz; Henry Ehrenreich; David Turnbull (1996).
3594:
Surface states that are calculated in the framework of a
361:
Surface states that are calculated in the framework of a
241:{\displaystyle u_{n{\boldsymbol {k}}}({\boldsymbol {r}})}
248:
is a function with the same periodicity as the crystal,
53:
4439:
The Spectral Theory of Periodic Differential Equations
3906:
3863:
3843:
3813:
3793:
3765:
3745:
3722:
3704:
The surface state dispersion can be measured using a
3542:
3506:
3473:
3437:
3399:
3372:
3326:
3288:
3170:
3133:
2941:
2868:
2761:
2589:
2382:
2284:
2140:
1867:
1616:
1540:
1491:
1371:
1331:
1297:
1249:
1055:
771:
597:
448:
264:. In the crystal, the potential has the periodicity,
211:
107:
4452:
Hladky-Henniona, Anne-Christine; Allan, Guy (2005).
4451:
3959:
3957:
293:
The first type of solution can be obtained for both
4200:
4041:
1030:
3921:
3900:An ideal one-dimensional crystal of finite length
3869:
3849:
3828:
3799:
3771:
3751:
3731:
3609:is responsible for the chemical bonding, e.g. the
3557:
3528:
3488:
3459:
3423:
3385:
3350:
3312:
3267:
3153:
3116:
2924:
2825:
2744:
2569:
2362:
2241:
2105:
1835:
1587:
1514:
1474:
1354:
1317:
1272:
1229:
1000:
723:
577:
325:
240:
194:
3954:
2120:is a normalization constant. Near the surface at
4735:
4316:
3699:
3127:The energy of this state is increased by a term
2925:{\displaystyle {\textbf {k}}_{||}=(k_{x},k_{y})}
369:. In the tight binding approach, the electronic
4028:. Springer-Verlag, Berlin Heidelberg New York.
27:Electronic states at the surface of materials
2842:Surface states of a three-dimensional crystal
1599:is a small quantity. The arbitrary constants
1574:
1556:
389:
4528:
4430:
4066:
3964:Sidney G. Davison; Maria Steslicka (1992).
3678:
3667:Interfaces between solid and liquid phases.
3631:
3366:A surface state is described by the energy
417:
301:. In semiconductors though, the associated
4445:
4417:
4415:
4391:
4389:
3362:True surface states and surface resonances
4604:
4353:
4278:
4134:
4120:
3993:
3684:Angle resolved photoemission spectroscopy
350:for clean and ideal surfaces, are called
4493:
4114:
3989:
3987:
2845:
1034:
79:
68:
57:
4698:
4621:
4572:
4436:
4412:
4386:
4333:
4310:
4101:
4060:
231:
221:
178:
168:
153:
145:
121:
14:
4736:
4663:
4522:
4317:Davison, S. D.; Stęślicka, M. (1992).
4252:
4246:
4037:
4035:
4019:
4017:
3711:
3694:ultraviolet photoelectron spectroscopy
375:linear combinations of atomic orbitals
4585:Journal of Physics: Conference Series
4441:. Edinburgh, Scottish Academic Press.
4306:. New York, McGraw-Hill. p. 323.
4301:
4295:
4175:
3984:
3881:Particle in a one-dimensional lattice
3600:linear combination of atomic orbitals
3558:{\displaystyle \mathbf {k} _{\perp }}
3496:wave numbers. In the two-dimensional
3489:{\displaystyle \mathbf {k} _{\perp }}
54:Origin at condensed matter interfaces
2857:The results for surface states of a
2580:The energy eigenvalues are given by
4421:
4395:
4339:
4206:
4186:. Academic Press. pp. 80–150.
4095:
4042:Feng Duan; Jin Guojin (2005). "7".
4032:
4014:
3997:Introduction to Solid State Physics
3403:
3330:
3292:
3217:
3090:
3066:
3031:
3005:
2961:
2872:
24:
4426:(2 ed.). Singapore, Springer.
3500:of the surface, for each value of
3424:{\displaystyle {\textbf {k}}_{||}}
3351:{\displaystyle {\textbf {k}}_{||}}
3313:{\displaystyle {\textbf {k}}_{||}}
2947:
2388:
2146:
1873:
1376:
776:
556:
532:
412:
348:nearly free electron approximation
136:
113:
25:
4765:
4044:Condensed Matter Physics:Volume 1
3931:only one surface state at one end
3929:with two ends can have, at most,
3529:{\displaystyle \mathbf {k} _{||}}
3460:{\displaystyle \mathbf {k} _{||}}
952:
3545:
3509:
3476:
3440:
1031:Surface states in semiconductors
309:of semiconductors as well as in
4487:
974:
884:
697:
661:
427:of the vacuum in the form of a
326:Shockley states and Tamm states
4319:Basic Theory of Surface States
3967:Basic Theory of Surface States
3589:
3520:
3515:
3451:
3446:
3415:
3410:
3342:
3337:
3304:
3299:
3229:
3224:
3145:
3140:
3102:
3097:
3078:
3073:
3049:
3043:
3038:
3025:
3017:
3012:
2994:
2988:
2966:
2956:
2919:
2893:
2884:
2879:
2409:
2397:
2307:
2298:
2222:
2203:
2037:
2029:
1984:
1976:
1799:
1791:
1746:
1738:
1698:
1690:
1457:
1454:
1437:
1428:
1385:
1379:
1069:
1063:
942:
923:
785:
779:
646:
631:
611:
605:
565:
559:
541:
535:
524:
518:
235:
227:
182:
174:
46:from the bulk material to the
13:
1:
4692:10.1016/j.surfrep.2009.07.005
4606:10.1088/1742-6596/92/1/012113
4207:Oka, H.; et al. (2014).
3948:
3759:is the potential period, and
3706:scanning tunneling microscope
3700:Scanning tunneling microscopy
1273:{\displaystyle k=\pm \pi /a}
588:with the periodic potential
7:
4715:10.1007/978-3-642-31232-8_3
4529:El Boudouti, E. H. (2007).
4327:10.1007/978-3-642-31232-8_3
4321:. Oxford, Clarendon Press.
4304:The Modern Theory of Solids
3626:wide-bandgap semiconductors
10:
4770:
4749:Electronic band structures
4657:10.1103/PhysRevB.78.174306
4558:10.1103/PhysRevE.76.026607
4516:10.1103/PhysRevB.75.212301
4461:Journal of Applied Physics
4233:10.1103/RevModPhys.86.1127
4153:10.1103/revmodphys.82.3045
4000:. Wiley. pp. 80–150.
1288:consisting of a wave with
393:
390:Topological surface states
334:and the Russian physicist
1854:, where the width of the
1515:{\displaystyle G=2\pi /a}
1355:{\displaystyle k=-\pi /a}
373:are usually expressed as
356:narrow gap semiconductors
44:electronic band structure
4754:Semiconductor structures
4437:Eastham, M.S.P. (1973).
4422:Ren, Shang Yuan (2017).
4396:Ren, Shang Yuan (2006).
3829:{\displaystyle Na+\tau }
3679:Experimental observation
3661:Surfaces with adsorbates
3632:Extrinsic surface states
3320:separately and for each
1318:{\displaystyle k=\pi /a}
418:Surface states in metals
346:in the framework of the
4672:Surface Science Reports
2263:If imaginary values of
383:wide gap semiconductors
276:Given the potential in
4400:. New York, Springer.
4372:10.1006/aphy.2002.6298
4280:10.1098/rspa.1933.0103
4089:10.1103/PhysRev.56.317
3923:
3871:
3851:
3830:
3801:
3773:
3753:
3733:
3559:
3530:
3490:
3461:
3425:
3387:
3352:
3314:
3269:
3155:
3154:{\displaystyle E_{||}}
3118:
2926:
2859:monatomic linear chain
2854:
2827:
2746:
2571:
2364:
2243:
2107:
1837:
1589:
1516:
1476:
1356:
1319:
1274:
1231:
1043:
1002:
725:
579:
252:is the band index and
242:
196:
88:
77:
66:
4253:Fowler, R.H. (1933).
3924:
3872:
3852:
3850:{\displaystyle \tau }
3831:
3802:
3800:{\displaystyle \tau }
3774:
3754:
3734:
3560:
3531:
3491:
3462:
3426:
3388:
3386:{\displaystyle E_{s}}
3353:
3315:
3270:
3156:
3119:
2927:
2849:
2828:
2747:
2572:
2365:
2267:are considered, i.e.
2244:
2108:
1838:
1590:
1517:
1477:
1357:
1320:
1275:
1232:
1038:
1003:
726:
580:
396:topological insulator
243:
197:
83:
72:
61:
4104:Phys. Z. Sowjetunion
4046:. World Scientific.
3922:{\displaystyle L=Na}
3904:
3861:
3841:
3811:
3791:
3763:
3743:
3720:
3692:) or angle resolved
3572:forbidden energy gap
3540:
3504:
3471:
3435:
3397:
3393:and its wave vector
3370:
3324:
3286:
3168:
3131:
2939:
2866:
2759:
2587:
2380:
2282:
2138:
1865:
1850:at the edges of the
1614:
1538:
1489:
1369:
1329:
1295:
1247:
1053:
769:
595:
446:
344:Schrödinger equation
307:forbidden energy gap
209:
105:
4684:2009SurSR..64..471E
4649:2008PhRvB..78q4306E
4643:(1): 174306（1–23）.
4597:2007JPhCS..92a2113E
4550:2007PhRvE..76b6607E
4508:2007PhRvB..75u2301R
4502:(21): 212301(1-4).
4473:2005JAP....98e4909H
4467:(5): 054909 (1-7).
4406:2006escf.book.....R
4364:2002AnPhy.301...22R
4271:1933RSPSA.141...56F
4225:2014RvMP...86.1127O
4184:Solid State Physics
4145:2010RvMP...82.3045H
4081:1939PhRv...56..317S
3970:. Clarendon Press.
3712:A recent new theory
3596:tight-binding model
3536:therefore a rod of
3239:
738:is an integer, and
363:tight-binding model
4544:(2): 026607(1-9).
4302:Seitz, F. (1940).
3994:C. Kittel (1996).
3919:
3877:for each band gap.
3867:
3847:
3826:
3797:
3769:
3749:
3732:{\displaystyle Na}
3729:
3555:
3526:
3486:
3457:
3421:
3383:
3348:
3310:
3265:
3263:
3214:
3151:
3114:
3112:
2922:
2855:
2823:
2742:
2740:
2567:
2565:
2360:
2358:
2239:
2237:
2103:
2101:
1846:demonstrating the
1833:
1831:
1585:
1528:reciprocal lattice
1512:
1472:
1470:
1352:
1315:
1270:
1227:
1225:
1044:
998:
996:
986:
721:
719:
709:
575:
573:
407:geometric topology
238:
192:
190:
89:
78:
67:
4744:Materials science
4724:978-3-642-31231-1
4637:Physical Review B
4538:Physical Review E
4496:Physical Review B
4481:10.1063/1.2034082
4342:Annals of Physics
3933:in each band gap.
3885:Particle in a box
3870:{\displaystyle N}
3772:{\displaystyle N}
3752:{\displaystyle a}
3622:transition metals
3580:surface resonance
3405:
3332:
3294:
3256:
3219:
3092:
3068:
3033:
3007:
2963:
2874:
2821:
2736:
2724:
2644:
2624:
2521:
2467:
2354:
2225:
2201:
2060:
2042:
1989:
1822:
1804:
1751:
1667:
1651:
1214:
1160:
1117:
971:
955:
945:
881:
694:
658:
510:
481:
405:is calculated in
379:transition metals
365:are often called
320:exponential decay
35:electronic states
16:(Redirected from
4761:
4729:
4728:
4702:
4696:
4695:
4667:
4661:
4660:
4634:
4625:
4619:
4618:
4608:
4576:
4570:
4569:
4535:
4526:
4520:
4519:
4491:
4485:
4484:
4458:
4449:
4443:
4442:
4434:
4428:
4427:
4419:
4410:
4409:
4393:
4384:
4383:
4357:
4355:cond-mat/0204211
4337:
4331:
4330:
4314:
4308:
4307:
4299:
4293:
4292:
4282:
4250:
4244:
4243:
4241:
4239:
4204:
4198:
4197:
4179:
4173:
4172:
4138:
4129:(4): 3045–3067.
4118:
4112:
4111:
4099:
4093:
4092:
4064:
4058:
4057:
4039:
4030:
4029:
4021:
4012:
4011:
3991:
3982:
3981:
3961:
3928:
3926:
3925:
3920:
3876:
3874:
3873:
3868:
3856:
3854:
3853:
3848:
3835:
3833:
3832:
3827:
3806:
3804:
3803:
3798:
3778:
3776:
3775:
3770:
3758:
3756:
3755:
3750:
3738:
3736:
3735:
3730:
3605:If a particular
3564:
3562:
3561:
3556:
3554:
3553:
3548:
3535:
3533:
3532:
3527:
3525:
3524:
3523:
3518:
3512:
3495:
3493:
3492:
3487:
3485:
3484:
3479:
3466:
3464:
3463:
3458:
3456:
3455:
3454:
3449:
3443:
3430:
3428:
3427:
3422:
3420:
3419:
3418:
3413:
3407:
3406:
3392:
3390:
3389:
3384:
3382:
3381:
3357:
3355:
3354:
3349:
3347:
3346:
3345:
3340:
3334:
3333:
3319:
3317:
3316:
3311:
3309:
3308:
3307:
3302:
3296:
3295:
3274:
3272:
3271:
3266:
3264:
3257:
3255:
3254:
3253:
3240:
3238:
3233:
3232:
3227:
3221:
3220:
3213:
3212:
3202:
3197:
3196:
3184:
3183:
3161:so that we have
3160:
3158:
3157:
3152:
3150:
3149:
3148:
3143:
3123:
3121:
3120:
3115:
3113:
3109:
3108:
3107:
3106:
3105:
3100:
3094:
3093:
3083:
3082:
3081:
3076:
3070:
3069:
3048:
3047:
3046:
3041:
3035:
3034:
3024:
3023:
3022:
3021:
3020:
3015:
3009:
3008:
2987:
2986:
2965:
2964:
2955:
2954:
2931:
2929:
2928:
2923:
2918:
2917:
2905:
2904:
2889:
2888:
2887:
2882:
2876:
2875:
2832:
2830:
2829:
2824:
2822:
2820:
2816:
2815:
2805:
2794:
2789:
2788:
2751:
2749:
2748:
2743:
2741:
2737:
2735:
2734:
2729:
2725:
2723:
2712:
2705:
2704:
2694:
2681:
2673:
2669:
2668:
2667:
2655:
2654:
2649:
2645:
2637:
2625:
2623:
2615:
2614:
2605:
2576:
2574:
2573:
2568:
2566:
2562:
2561:
2546:
2542:
2541:
2537:
2536:
2532:
2522:
2514:
2487:
2483:
2482:
2478:
2468:
2460:
2434:
2433:
2396:
2395:
2369:
2367:
2366:
2361:
2359:
2355:
2353:
2342:
2335:
2334:
2324:
2275:and one defines
2248:
2246:
2245:
2240:
2238:
2234:
2230:
2226:
2215:
2214:
2202:
2200:
2199:
2190:
2182:
2180:
2154:
2153:
2112:
2110:
2109:
2104:
2102:
2098:
2094:
2093:
2092:
2088:
2066:
2062:
2061:
2053:
2052:
2047:
2043:
2041:
2040:
2032:
2020:
2013:
2012:
2002:
1995:
1990:
1988:
1987:
1979:
1967:
1960:
1959:
1949:
1936:
1935:
1931:
1907:
1906:
1881:
1880:
1842:
1840:
1839:
1834:
1832:
1828:
1824:
1823:
1815:
1814:
1809:
1805:
1803:
1802:
1794:
1782:
1775:
1774:
1764:
1757:
1752:
1750:
1749:
1741:
1729:
1722:
1721:
1711:
1701:
1693:
1685:
1684:
1679:
1675:
1668:
1660:
1652:
1650:
1642:
1641:
1632:
1594:
1592:
1591:
1586:
1578:
1577:
1568:
1560:
1559:
1550:
1549:
1521:
1519:
1518:
1513:
1508:
1481:
1479:
1478:
1473:
1471:
1464:
1463:
1450:
1413:
1412:
1361:
1359:
1358:
1353:
1348:
1325:and wave vector
1324:
1322:
1321:
1316:
1311:
1279:
1277:
1276:
1271:
1266:
1236:
1234:
1233:
1228:
1226:
1219:
1215:
1210:
1199:
1175:
1171:
1167:
1166:
1162:
1161:
1156:
1145:
1123:
1119:
1118:
1113:
1102:
1007:
1005:
1004:
999:
997:
990:
987:
973:
972:
969:
961:
957:
956:
948:
946:
935:
934:
916:
883:
882:
879:
871:
870:
855:
854:
839:
838:
820:
819:
730:
728:
727:
722:
720:
713:
710:
696:
695:
692:
684:
683:
660:
659:
656:
584:
582:
581:
576:
574:
531:
527:
511:
509:
508:
507:
494:
493:
484:
482:
480:
472:
471:
462:
332:William Shockley
247:
245:
244:
239:
234:
226:
225:
224:
201:
199:
198:
193:
191:
181:
173:
172:
171:
158:
157:
156:
148:
139:
126:
125:
124:
21:
4769:
4768:
4764:
4763:
4762:
4760:
4759:
4758:
4734:
4733:
4732:
4725:
4703:
4699:
4668:
4664:
4632:
4626:
4622:
4577:
4573:
4533:
4527:
4523:
4492:
4488:
4456:
4450:
4446:
4435:
4431:
4420:
4413:
4394:
4387:
4338:
4334:
4315:
4311:
4300:
4296:
4251:
4247:
4237:
4235:
4205:
4201:
4194:
4180:
4176:
4119:
4115:
4100:
4096:
4065:
4061:
4054:
4040:
4033:
4026:Surface Science
4022:
4015:
4008:
3992:
3985:
3978:
3962:
3955:
3951:
3905:
3902:
3901:
3862:
3859:
3858:
3842:
3839:
3838:
3812:
3809:
3808:
3792:
3789:
3788:
3764:
3761:
3760:
3744:
3741:
3740:
3721:
3718:
3717:
3714:
3702:
3686:
3681:
3634:
3592:
3549:
3544:
3543:
3541:
3538:
3537:
3519:
3514:
3513:
3508:
3507:
3505:
3502:
3501:
3480:
3475:
3474:
3472:
3469:
3468:
3450:
3445:
3444:
3439:
3438:
3436:
3433:
3432:
3414:
3409:
3408:
3402:
3401:
3400:
3398:
3395:
3394:
3377:
3373:
3371:
3368:
3367:
3364:
3341:
3336:
3335:
3329:
3328:
3327:
3325:
3322:
3321:
3303:
3298:
3297:
3291:
3290:
3289:
3287:
3284:
3283:
3262:
3261:
3249:
3245:
3241:
3234:
3228:
3223:
3222:
3216:
3215:
3208:
3204:
3203:
3201:
3192:
3188:
3179:
3175:
3171:
3169:
3166:
3165:
3144:
3139:
3138:
3134:
3132:
3129:
3128:
3111:
3110:
3101:
3096:
3095:
3089:
3088:
3087:
3077:
3072:
3071:
3065:
3064:
3063:
3056:
3052:
3042:
3037:
3036:
3030:
3029:
3028:
3016:
3011:
3010:
3004:
3003:
3002:
3001:
2997:
2982:
2978:
2976:
2969:
2960:
2959:
2950:
2946:
2942:
2940:
2937:
2936:
2913:
2909:
2900:
2896:
2883:
2878:
2877:
2871:
2870:
2869:
2867:
2864:
2863:
2844:
2811:
2807:
2806:
2795:
2793:
2778:
2774:
2760:
2757:
2756:
2739:
2738:
2730:
2713:
2700:
2696:
2695:
2693:
2689:
2688:
2680:
2663:
2659:
2650:
2636:
2632:
2631:
2630:
2626:
2616:
2610:
2606:
2604:
2597:
2590:
2588:
2585:
2584:
2564:
2563:
2551:
2547:
2513:
2512:
2508:
2501:
2497:
2459:
2458:
2454:
2450:
2446:
2439:
2435:
2426:
2422:
2412:
2391:
2387:
2383:
2381:
2378:
2377:
2357:
2356:
2343:
2330:
2326:
2325:
2323:
2310:
2285:
2283:
2280:
2279:
2236:
2235:
2210:
2206:
2195:
2191:
2183:
2181:
2179:
2175:
2171:
2155:
2149:
2145:
2141:
2139:
2136:
2135:
2129:
2100:
2099:
2084:
2071:
2067:
2048:
2036:
2028:
2021:
2008:
2004:
2003:
2001:
1997:
1996:
1994:
1983:
1975:
1968:
1955:
1951:
1950:
1948:
1944:
1940:
1927:
1917:
1913:
1912:
1908:
1896:
1892:
1882:
1876:
1872:
1868:
1866:
1863:
1862:
1830:
1829:
1810:
1798:
1790:
1783:
1770:
1766:
1765:
1763:
1759:
1758:
1756:
1745:
1737:
1730:
1717:
1713:
1712:
1710:
1706:
1702:
1697:
1689:
1680:
1659:
1658:
1654:
1653:
1643:
1637:
1633:
1631:
1624:
1617:
1615:
1612:
1611:
1573:
1572:
1564:
1555:
1554:
1545:
1541:
1539:
1536:
1535:
1504:
1490:
1487:
1486:
1469:
1468:
1446:
1424:
1420:
1402:
1398:
1388:
1372:
1370:
1367:
1366:
1344:
1330:
1327:
1326:
1307:
1296:
1293:
1292:
1262:
1248:
1245:
1244:
1242:
1224:
1223:
1200:
1198:
1194:
1173:
1172:
1146:
1144:
1137:
1133:
1103:
1101:
1097:
1093:
1086:
1082:
1072:
1056:
1054:
1051:
1050:
1033:
995:
994:
985:
984:
968:
967:
965:
947:
930:
926:
915:
911:
907:
895:
894:
878:
877:
875:
860:
856:
850:
846:
825:
821:
812:
808:
801:
797:
795:
788:
772:
770:
767:
766:
718:
717:
708:
707:
691:
690:
688:
679:
675:
672:
671:
655:
654:
652:
621:
617:
598:
596:
593:
592:
572:
571:
551:
544:
503:
499:
495:
489:
485:
483:
473:
467:
463:
461:
457:
453:
449:
447:
444:
443:
426:
420:
415:
413:Shockley states
398:
392:
352:Shockley states
328:
322:into the bulk.
230:
220:
216:
212:
210:
207:
206:
189:
188:
177:
167:
163:
159:
152:
144:
140:
135:
134:
127:
120:
116:
112:
108:
106:
103:
102:
93:Bloch's theorem
56:
28:
23:
22:
15:
12:
11:
5:
4767:
4757:
4756:
4751:
4746:
4731:
4730:
4723:
4697:
4678:(1): 471–594.
4662:
4620:
4571:
4521:
4486:
4444:
4429:
4411:
4385:
4332:
4309:
4294:
4265:(843): 56–71.
4245:
4213:Rev. Mod. Phys
4199:
4192:
4174:
4123:Rev. Mod. Phys
4113:
4094:
4075:(4): 317–323.
4059:
4052:
4031:
4013:
4006:
3983:
3976:
3952:
3950:
3947:
3946:
3945:
3942:
3935:
3934:
3918:
3915:
3912:
3909:
3898:
3895:
3866:
3846:
3825:
3822:
3819:
3816:
3796:
3768:
3748:
3728:
3725:
3713:
3710:
3701:
3698:
3685:
3682:
3680:
3677:
3669:
3668:
3665:
3662:
3659:
3633:
3630:
3615:dangling bonds
3591:
3588:
3552:
3547:
3522:
3517:
3511:
3498:Brillouin zone
3483:
3478:
3453:
3448:
3442:
3417:
3412:
3380:
3376:
3363:
3360:
3344:
3339:
3306:
3301:
3276:
3275:
3260:
3252:
3248:
3244:
3237:
3231:
3226:
3211:
3207:
3200:
3195:
3191:
3187:
3182:
3178:
3174:
3173:
3147:
3142:
3137:
3125:
3124:
3104:
3099:
3086:
3080:
3075:
3062:
3059:
3055:
3051:
3045:
3040:
3027:
3019:
3014:
3000:
2996:
2993:
2990:
2985:
2981:
2977:
2975:
2972:
2970:
2968:
2958:
2953:
2949:
2945:
2944:
2921:
2916:
2912:
2908:
2903:
2899:
2895:
2892:
2886:
2881:
2843:
2840:
2819:
2814:
2810:
2804:
2801:
2798:
2792:
2787:
2784:
2781:
2777:
2773:
2770:
2767:
2764:
2753:
2752:
2733:
2728:
2722:
2719:
2716:
2711:
2708:
2703:
2699:
2692:
2687:
2684:
2679:
2676:
2672:
2666:
2662:
2658:
2653:
2648:
2643:
2640:
2635:
2629:
2622:
2619:
2613:
2609:
2603:
2600:
2598:
2596:
2593:
2592:
2578:
2577:
2560:
2557:
2554:
2550:
2545:
2540:
2535:
2531:
2528:
2525:
2520:
2517:
2511:
2507:
2504:
2500:
2496:
2493:
2490:
2486:
2481:
2477:
2474:
2471:
2466:
2463:
2457:
2453:
2449:
2445:
2442:
2438:
2432:
2429:
2425:
2421:
2418:
2415:
2413:
2411:
2408:
2405:
2402:
2399:
2394:
2390:
2386:
2385:
2371:
2370:
2352:
2349:
2346:
2341:
2338:
2333:
2329:
2322:
2319:
2316:
2313:
2311:
2309:
2306:
2303:
2300:
2297:
2294:
2291:
2288:
2287:
2250:
2249:
2233:
2229:
2224:
2221:
2218:
2213:
2209:
2205:
2198:
2194:
2189:
2186:
2178:
2174:
2170:
2167:
2164:
2161:
2158:
2156:
2152:
2148:
2144:
2143:
2127:
2114:
2113:
2097:
2091:
2087:
2083:
2080:
2077:
2074:
2070:
2065:
2059:
2056:
2051:
2046:
2039:
2035:
2031:
2027:
2024:
2019:
2016:
2011:
2007:
2000:
1993:
1986:
1982:
1978:
1974:
1971:
1966:
1963:
1958:
1954:
1947:
1943:
1939:
1934:
1930:
1926:
1923:
1920:
1916:
1911:
1905:
1902:
1899:
1895:
1891:
1888:
1885:
1883:
1879:
1875:
1871:
1870:
1852:Brillouin zone
1848:band splitting
1844:
1843:
1827:
1821:
1818:
1813:
1808:
1801:
1797:
1793:
1789:
1786:
1781:
1778:
1773:
1769:
1762:
1755:
1748:
1744:
1740:
1736:
1733:
1728:
1725:
1720:
1716:
1709:
1705:
1700:
1696:
1692:
1688:
1683:
1678:
1674:
1671:
1666:
1663:
1657:
1649:
1646:
1640:
1636:
1630:
1627:
1625:
1623:
1620:
1619:
1584:
1581:
1576:
1571:
1567:
1563:
1558:
1553:
1548:
1544:
1524:lattice vector
1511:
1507:
1503:
1500:
1497:
1494:
1483:
1482:
1467:
1462:
1459:
1456:
1453:
1449:
1445:
1442:
1439:
1436:
1433:
1430:
1427:
1423:
1419:
1416:
1411:
1408:
1405:
1401:
1397:
1394:
1391:
1389:
1387:
1384:
1381:
1378:
1375:
1374:
1351:
1347:
1343:
1340:
1337:
1334:
1314:
1310:
1306:
1303:
1300:
1269:
1265:
1261:
1258:
1255:
1252:
1240:
1222:
1218:
1213:
1209:
1206:
1203:
1197:
1193:
1190:
1187:
1184:
1181:
1178:
1176:
1174:
1170:
1165:
1159:
1155:
1152:
1149:
1143:
1140:
1136:
1132:
1129:
1126:
1122:
1116:
1112:
1109:
1106:
1100:
1096:
1092:
1089:
1085:
1081:
1078:
1075:
1073:
1071:
1068:
1065:
1062:
1059:
1058:
1032:
1029:
1017:charge density
1009:
1008:
993:
989:
983:
980:
977:
966:
964:
960:
954:
951:
944:
941:
938:
933:
929:
925:
922:
919:
914:
910:
906:
903:
900:
897:
896:
893:
890:
887:
876:
874:
869:
866:
863:
859:
853:
849:
845:
842:
837:
834:
831:
828:
824:
818:
815:
811:
807:
804:
803:
800:
796:
794:
791:
789:
787:
784:
781:
778:
775:
774:
752:wave functions
732:
731:
716:
712:
706:
703:
700:
689:
687:
682:
678:
674:
673:
670:
667:
664:
653:
651:
648:
645:
642:
639:
636:
633:
630:
627:
624:
623:
620:
616:
613:
610:
607:
604:
601:
600:
586:
585:
570:
567:
564:
561:
558:
555:
552:
550:
547:
545:
543:
540:
537:
534:
530:
526:
523:
520:
517:
514:
506:
502:
498:
492:
488:
479:
476:
470:
466:
460:
456:
452:
451:
424:
419:
416:
414:
411:
391:
388:
387:
386:
371:wave functions
359:
327:
324:
318:leading to an
299:semiconductors
291:
290:
286:
237:
233:
229:
223:
219:
215:
203:
202:
187:
184:
180:
176:
170:
166:
162:
155:
151:
147:
143:
138:
133:
130:
128:
123:
119:
115:
111:
110:
55:
52:
31:Surface states
26:
9:
6:
4:
3:
2:
4766:
4755:
4752:
4750:
4747:
4745:
4742:
4741:
4739:
4726:
4720:
4716:
4712:
4708:
4701:
4693:
4689:
4685:
4681:
4677:
4673:
4666:
4658:
4654:
4650:
4646:
4642:
4638:
4631:
4624:
4616:
4612:
4607:
4602:
4598:
4594:
4590:
4586:
4582:
4575:
4567:
4563:
4559:
4555:
4551:
4547:
4543:
4539:
4532:
4525:
4517:
4513:
4509:
4505:
4501:
4497:
4490:
4482:
4478:
4474:
4470:
4466:
4462:
4455:
4448:
4440:
4433:
4425:
4418:
4416:
4407:
4403:
4399:
4392:
4390:
4381:
4377:
4373:
4369:
4365:
4361:
4356:
4351:
4347:
4343:
4336:
4328:
4324:
4320:
4313:
4305:
4298:
4290:
4286:
4281:
4276:
4272:
4268:
4264:
4260:
4256:
4249:
4234:
4230:
4226:
4222:
4218:
4214:
4210:
4203:
4195:
4193:0-12-607729-0
4189:
4185:
4178:
4170:
4166:
4162:
4158:
4154:
4150:
4146:
4142:
4137:
4132:
4128:
4124:
4117:
4109:
4105:
4098:
4090:
4086:
4082:
4078:
4074:
4070:
4063:
4055:
4053:981-256-070-X
4049:
4045:
4038:
4036:
4027:
4020:
4018:
4009:
4007:0-471-14286-7
4003:
3999:
3998:
3990:
3988:
3979:
3977:0-19-851990-7
3973:
3969:
3968:
3960:
3958:
3953:
3943:
3940:
3939:
3938:
3932:
3916:
3913:
3910:
3907:
3899:
3896:
3893:
3892:
3891:
3888:
3886:
3882:
3878:
3864:
3844:
3823:
3820:
3817:
3814:
3794:
3785:
3781:
3766:
3746:
3726:
3723:
3709:
3707:
3697:
3695:
3691:
3676:
3674:
3666:
3663:
3660:
3657:
3656:
3655:
3653:
3649:
3648:
3643:
3641:
3640:
3629:
3627:
3623:
3618:
3616:
3612:
3608:
3603:
3601:
3597:
3587:
3585:
3581:
3577:
3573:
3568:
3550:
3499:
3481:
3378:
3374:
3359:
3281:
3258:
3250:
3246:
3242:
3235:
3209:
3205:
3198:
3193:
3189:
3185:
3180:
3176:
3164:
3163:
3162:
3135:
3084:
3060:
3057:
3053:
2998:
2991:
2983:
2979:
2973:
2971:
2951:
2935:
2934:
2933:
2914:
2910:
2906:
2901:
2897:
2890:
2860:
2852:
2848:
2839:
2837:
2817:
2812:
2808:
2802:
2799:
2796:
2790:
2785:
2782:
2779:
2775:
2771:
2768:
2765:
2762:
2731:
2726:
2720:
2717:
2714:
2709:
2706:
2701:
2697:
2690:
2685:
2682:
2677:
2674:
2670:
2664:
2660:
2656:
2651:
2646:
2641:
2638:
2633:
2627:
2620:
2617:
2611:
2607:
2601:
2599:
2594:
2583:
2582:
2581:
2558:
2555:
2552:
2548:
2543:
2538:
2533:
2529:
2526:
2523:
2518:
2515:
2509:
2505:
2502:
2498:
2494:
2491:
2488:
2484:
2479:
2475:
2472:
2469:
2464:
2461:
2455:
2451:
2447:
2443:
2440:
2436:
2430:
2427:
2423:
2419:
2416:
2414:
2406:
2403:
2400:
2392:
2376:
2375:
2374:
2350:
2347:
2344:
2339:
2336:
2331:
2327:
2320:
2317:
2314:
2312:
2304:
2301:
2295:
2292:
2289:
2278:
2277:
2276:
2274:
2270:
2266:
2261:
2259:
2255:
2231:
2227:
2219:
2216:
2211:
2207:
2196:
2192:
2187:
2184:
2176:
2172:
2168:
2165:
2162:
2159:
2157:
2150:
2134:
2133:
2132:
2130:
2123:
2119:
2095:
2089:
2085:
2081:
2078:
2075:
2072:
2068:
2063:
2057:
2054:
2049:
2044:
2033:
2025:
2022:
2017:
2014:
2009:
2005:
1998:
1991:
1980:
1972:
1969:
1964:
1961:
1956:
1952:
1945:
1941:
1937:
1932:
1928:
1924:
1921:
1918:
1914:
1909:
1903:
1900:
1897:
1893:
1889:
1886:
1884:
1877:
1861:
1860:
1859:
1857:
1856:forbidden gap
1853:
1849:
1825:
1819:
1816:
1811:
1806:
1795:
1787:
1784:
1779:
1776:
1771:
1767:
1760:
1753:
1742:
1734:
1731:
1726:
1723:
1718:
1714:
1707:
1703:
1694:
1686:
1681:
1676:
1672:
1669:
1664:
1661:
1655:
1647:
1644:
1638:
1634:
1628:
1626:
1621:
1610:
1609:
1608:
1606:
1602:
1598:
1582:
1579:
1569:
1565:
1561:
1551:
1546:
1542:
1533:
1529:
1525:
1509:
1505:
1501:
1498:
1495:
1492:
1465:
1460:
1451:
1447:
1443:
1440:
1434:
1431:
1425:
1421:
1417:
1414:
1409:
1406:
1403:
1399:
1395:
1392:
1390:
1382:
1365:
1364:
1363:
1349:
1345:
1341:
1338:
1335:
1332:
1312:
1308:
1304:
1301:
1298:
1291:
1287:
1286:standing wave
1283:
1267:
1263:
1259:
1256:
1253:
1250:
1237:
1220:
1216:
1211:
1207:
1204:
1201:
1195:
1191:
1188:
1185:
1182:
1179:
1177:
1168:
1163:
1157:
1153:
1150:
1147:
1141:
1138:
1134:
1130:
1127:
1124:
1120:
1114:
1110:
1107:
1104:
1098:
1094:
1090:
1087:
1083:
1079:
1076:
1074:
1066:
1060:
1048:
1041:
1037:
1028:
1026:
1025:work function
1022:
1018:
1014:
991:
981:
978:
975:
962:
958:
949:
939:
936:
931:
927:
920:
917:
912:
908:
904:
901:
898:
891:
888:
885:
872:
867:
864:
861:
857:
851:
847:
843:
840:
835:
832:
829:
826:
822:
816:
813:
809:
805:
798:
792:
790:
782:
765:
764:
763:
761:
757:
753:
749:
745:
741:
737:
714:
704:
701:
698:
685:
680:
676:
668:
665:
662:
649:
643:
640:
637:
634:
628:
625:
618:
614:
608:
602:
591:
590:
589:
568:
562:
553:
548:
546:
538:
528:
521:
515:
512:
504:
500:
496:
490:
486:
477:
474:
468:
464:
458:
454:
442:
441:
440:
438:
434:
430:
429:step function
410:
408:
404:
397:
384:
380:
376:
372:
368:
364:
360:
357:
353:
349:
345:
341:
340:
339:
337:
333:
323:
321:
317:
312:
308:
304:
303:eigenenergies
300:
296:
287:
283:
282:
281:
279:
274:
271:
267:
263:
258:
255:
251:
217:
213:
185:
164:
160:
149:
141:
131:
129:
117:
101:
100:
99:
98:
94:
91:As stated by
86:
82:
75:
71:
64:
60:
51:
49:
45:
40:
37:found at the
36:
32:
19:
18:Surface state
4706:
4700:
4675:
4671:
4665:
4640:
4636:
4623:
4588:
4584:
4574:
4541:
4537:
4524:
4499:
4495:
4489:
4464:
4460:
4447:
4438:
4432:
4423:
4397:
4348:(1): 22–30.
4345:
4341:
4335:
4318:
4312:
4303:
4297:
4262:
4258:
4248:
4236:. Retrieved
4216:
4212:
4202:
4183:
4177:
4126:
4122:
4116:
4107:
4103:
4097:
4072:
4068:
4062:
4043:
4025:
3995:
3966:
3936:
3930:
3889:
3836:
3786:
3782:
3715:
3703:
3687:
3672:
3670:
3651:
3645:
3644:
3637:
3635:
3619:
3610:
3604:
3593:
3575:
3567:energy bands
3365:
3279:
3277:
3126:
2856:
2850:
2835:
2754:
2579:
2372:
2272:
2268:
2264:
2262:
2251:
2125:
2121:
2117:
2115:
1845:
1604:
1600:
1596:
1531:
1484:
1281:
1238:
1049:
1045:
1039:
1021:double layer
1012:
1010:
759:
747:
743:
739:
735:
733:
587:
436:
432:
421:
399:
366:
351:
329:
310:
292:
277:
275:
269:
265:
261:
259:
253:
249:
204:
90:
84:
73:
62:
30:
29:
4238:3 September
4219:(4): 1127.
3890:Therefore:
3671:Generally,
3590:Tamm states
3584:Bloch waves
1290:wave vector
756:Bloch waves
367:Tamm states
97:Bloch waves
4738:Categories
4591:(1): 1–4.
3949:References
2254:eigenvalue
746:<0 and
394:See also:
316:wavenumber
311:local gaps
4615:250673169
4289:122900909
4161:0034-6861
4136:1002.3895
4069:Phys. Rev
3845:τ
3824:τ
3795:τ
3696:(ARUPS).
3673:extrinsic
3652:extrinsic
3647:Extrinsic
3639:intrinsic
3551:⊥
3482:⊥
3251:∗
3206:ℏ
3085:⋅
3058:−
2980:ψ
2948:Ψ
2818:π
2809:ℏ
2772:≤
2766:≤
2707:π
2698:ℏ
2686:−
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2559:δ
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2006:ℏ
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557:Ψ
533:Ψ
465:ℏ
459:−
336:Igor Tamm
150:⋅
114:Ψ
4566:17930167
4380:14490431
4169:16066223
3857:but not
3576:figure 3
2851:Figure 5
2836:figure 3
1595:, where
1532:figure 4
1282:figure 4
1040:Figure 4
1013:figure 2
754:must be
433:figure 1
289:surface.
278:Figure 1
270:Figure 1
262:Figure 1
85:Figure 3
74:Figure 2
63:Figure 1
4680:Bibcode
4645:Bibcode
4593:Bibcode
4546:Bibcode
4504:Bibcode
4469:Bibcode
4402:Bibcode
4360:Bibcode
4267:Bibcode
4221:Bibcode
4141:Bibcode
4077:Bibcode
3607:orbital
1526:of the
285:vacuum.
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3278:where
2258:vacuum
2116:Where
748:z>0
734:where
295:metals
48:vacuum
4633:(PDF)
4611:S2CID
4534:(PDF)
4457:(PDF)
4376:S2CID
4350:arXiv
4285:S2CID
4165:S2CID
4131:arXiv
3690:ARPES
3654:are:
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2122:z = 0
1530:(see
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205:Here
4719:ISBN
4562:PMID
4240:2021
4188:ISBN
4157:ISSN
4048:ISBN
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