3741:`s transparency window and the most extensively employed wavelength for telecommunications. By fine tuning material parameters such as quantum well depth, biaxial strain and silicon content in the well, it is also possible to tailor the optical band gap of the Ge/SiGe quantum well system to modulate at 1310 nm, which also corresponds to a transparency window for optical fibers. Electro-optic modulation by QCSE using Ge/SiGe quantum wells has been demonstrated up to 23 GHz with energies per bit as low as 108 fJ. and integrated in a waveguide configuration on a SiGe waveguide
1582:
3587:
2923:
2673:
527:
1173:
1941:
1881:
824:
2679:
2429:
4146:
Rouifed, Mohamed Said; Chaisakul, Papichaya; Marris-Morini, Delphine; Frigerio, Jacopo; Isella, Giovanni; Chrastina, Daniel; Edmond, Samson; Roux, Xavier Le; Coudevylle, Jean-René; Vivien, Laurent (18 September 2012). "Quantum-confined Stark effect at 13 μm in Ge/Si_035Ge_065 quantum-well structure".
2004:
towards lower energies of the optical transitions, the DC electric field also induces a decrease in magnitude of the absorption coefficient, as it decreases the overlapping integrals of relating valence and conduction band wave functions. Given the approximations made so far and the absence of any
59:
set of energy subbands. Only a discrete set of frequencies of light may be absorbed or emitted by the system. When an external electric field is applied, the electron states shift to lower energies, while the hole states shift to higher energies. This reduces the permitted light absorption or
87:
The shift in absorption lines can be calculated by comparing the energy levels in unbiased and biased quantum wells. It is a simpler task to find the energy levels in the unbiased system, due to its symmetry. If the external electric field is small, it can be treated as a perturbation to the
3618:. An electric field can be induced perpendicularly to the quantum wells by applying an external, reverse bias to the PIN diode, causing QCSE. This mechanism can be employed to modulate wavelengths below the band gap of the unbiased system and within the reach of the QCSE induced redshift.
1577:{\displaystyle E_{1}^{(2)}=\sum _{k\neq 1}{\frac {|\langle k^{(0)}|eFz|1^{(0)}\rangle |^{2}}{E_{1}^{(0)}-E_{k}^{(0)}}}\approx {\frac {|\langle 2^{(0)}|eFz|1^{(0)}\rangle |^{2}}{E_{1}^{(0)}-E_{2}^{(0)}}}=-24\left({\frac {2}{3\pi }}\right)^{6}{\frac {e^{2}F^{2}m_{e}^{*}L^{4}}{\hbar ^{2}}}}
2316:
3423:
then excitons cease to exist in the bulk material. This somewhat limits the applicability of Franz-Keldysh for modulation purposes, as the redshift induced by the applied electric field is countered by shift towards higher energies due to the absence of exciton generations.
667:
427:
1862:
4240:
Chaisakul, Papichaya; Marris-Morini, Delphine; Frigerio, Jacopo; Chrastina, Daniel; Rouifed, Mohamed-Said; Cecchi, Stefano; Crozat, Paul; Isella, Giovanni; Vivien, Laurent (11 May 2014). "Integrated germanium optical interconnects on silicon substrates".
3431:, strong excitonic effects will be present no matter the magnitude of the applied electric field. Furthermore, quantum wells behave as two dimensional systems, which strongly enhance excitonic effects with respect to bulk material. In fact, solving the
209:
1587:
for electron, where the additional approximation of neglecting the perturbation terms due to the bound states with k even and > 2 has been introduced. By comparison, the perturbation terms from odd-k states are zero due to symmetry.
4039:
Kuo, Yu-Hsuan; Lee, Yong Kyu; Ge, Yangsi; Ren, Shen; Roth, Jonathan E.; Kamins, Theodore I.; Miller, David A. B.; Harris, James S. (October 2005). "Strong quantum-confined Stark effect in germanium quantum-well structures on silicon".
3550:
2918:{\displaystyle \phi _{v,1}=\phi _{v,1}^{0}+\phi _{v,1}^{'}={\frac {1}{A}}\left(\cos \left({\frac {\pi z}{L}}\right)+\left({\frac {2}{3\pi }}\right)^{4}{\frac {2m_{h}^{*}eFL^{3}}{\hbar ^{2}}}\sin \left({\frac {\pi z}{L}}\right)\right)}
2668:{\displaystyle \phi _{c,1}=\phi _{c,1}^{0}+\phi _{c,1}^{'}={\frac {1}{A}}\left(\cos \left({\frac {\pi z}{L}}\right)-\left({\frac {2}{3\pi }}\right)^{4}{\frac {2m_{e}^{*}eFL^{3}}{\hbar ^{2}}}\sin \left({\frac {\pi z}{L}}\right)\right)}
3186:
60:
emission frequencies. Additionally, the external electric field shifts electrons and holes to opposite sides of the well, decreasing the overlap integral, which in turn reduces the recombination efficiency (i.e. fluorescence
4190:
Chaisakul, Papichaya; Marris-Morini, Delphine; Rouifed, Mohamed-Saïd; Isella, Giovanni; Chrastina, Daniel; Frigerio, Jacopo; Le Roux, Xavier; Edmond, Samson; Coudevylle, Jean-René; Vivien, Laurent (26 January 2012).
924:
1161:
3065:
3418:
819:{\displaystyle \phi _{n}(z)={\sqrt {\frac {2}{L}}}\times {\begin{cases}\cos \left({\frac {n\pi z}{L}}\right)&n\,{\text{odd}}\\\sin \left({\frac {n\pi z}{L}}\right)&n\,{\text{even}}\end{cases}}.}
2157:
3337:. Due to their opposite electric charges, the electron and the hole constituting the exciton will be pulled apart under the influence of the external electric field. If the field is strong enough
2173:
295:
2996:
1727:
1735:
1005:
2088:
3087:
has to be taken into account. Excitons are quasiparticles consisting of a bound state of an electron-hole pair, whose binding energy in a bulk material can be modelled as that of an
3323:
3271:
659:
598:
481:
1872:
The approximations made so far are quite crude, nonetheless the energy shift does show experimentally a square law dependence from the applied electric field, as predicted.
3649:
on top of a silicon substrate, provided the presence of some buffer layer in between the two. This is a decisive advantage as it allows Ge/SiGe QCSE to be integrated with
3083:
The description of quantum-confined Stark effect given by second order perturbation theory is extremely simple and intuitive. However to correctly depict QCSE the role of
1051:
563:
521:
1653:
1621:
3926:
Weiner, Joseph S.; Miller, David A. B.; Chemla, Daniel S. (30 March 1987). "Quadratic electro-optic effect due to the quantum-confined Stark effect in quantum wells".
3683:
1994:
1968:
1934:
1908:
106:
3735:
3712:
3216:
954:
287:
260:
3240:
64:) of the system. The spatial separation between the electrons and holes is limited by the presence of the potential barriers around the quantum well, meaning that
2398:
2372:
2346:
3576:
3445:
2421:
3427:
This problem does not exist in QCSE, as electrons and holes are confined in the quantum wells. As long as the quantum well depth is comparable to the excitonic
2946:
618:
450:
233:
79:
of the material, the QCSE may shift the energy to values lower than the gap. This was evidenced recently in the study of quantum discs embedded in a nanowire.
3097:
4303:
3333:
towards lower energies. If an electric field is applied to a bulk semiconductor, a further redshift in the absorption spectrum is observed due to
2164:
89:
3847:
3273:
is the relative electric permittivity. The exciton binding energy has to be included in the energy balance of photon absorption processes:
957:
835:
1062:
3970:
3004:
68:
are able to exist in the system even under the influence of an electric field. The quantum-confined Stark effect is used in QCSE
3343:
2096:
3848:"Nanometer Scale Spectral Imaging of Quantum Emitters in Nanowires and Its Correlation to Their Atomically Resolved Structure"
3075:, which says that transition probability depends on the above overlapping integral, optical transition strength is weakened.
2311:{\displaystyle \phi _{n}^{'}=\sum _{k\neq n}{\frac {\langle \phi _{n}|H'|\phi _{k}\rangle }{E_{n}-E_{k}}}|\phi _{k}\rangle }
1868:
The downward shift in the confined energy level discussed in the above equation is referred to as the Franz-Keldysh effect.
75:
Even if
Quantum Objects (Wells, Dots or Discs, for instance) emit and absorb light generally with higher energies than the
3657:
422:{\displaystyle \psi (\mathbf {r} )=\phi _{n}(z){\frac {1}{\sqrt {A}}}e^{i(k_{x}\cdot {x}+k_{y}\cdot {y})}u(\mathbf {r} ).}
3594:
Quantum-confined Stark effect's most promising application lies in its ability to perform optical modulation in the near
4325:
1857:{\displaystyle \Delta E\approx -24\left({\frac {2}{3\pi }}\right)^{6}{\frac {e^{2}F^{2}m_{tot}^{*}L^{4}}{\hbar ^{2}}}.}
2951:
530:
On the left: wave functions corresponding to the n=1 and n=3 levels in a quantum well with no applied electric field (
1658:
4330:
4304:
https://web.archive.org/web/20100728030241/http://www.rle.mit.edu/sclaser/6.973%20lecture%20notes/Lecture%2013c.pdf
4098:"Design of Ge–SiGe Quantum-Confined Stark Effect Electroabsorption Heterostructures for CMOS Compatible Photonics"
3986:
Miller, David A.B. (2017). "Attojoule
Optoelectronics for Low-Energy Information Processing and Communications".
3626:
974:
3773:
Miller, D. (1984). "Band-Edge
Electroabsorption in Quantum Well Structures: The Quantum-Confined Stark Effect".
3750:
3334:
2008:
3590:
Animated image showing change in absorption spectrum of GaAs/AlGaAs quantum wells by externally applied voltage
661:. Under this simplified model, analytical expressions for the bound state wavefunctions exist, with the form
3611:
3432:
3279:
3072:
262:
is the height of the potential barriers. The bound states in the well lie at a set of discrete energies,
3249:
631:
568:
289:
and the associated wavefunctions can be written using the envelope function approximation as follows:
4250:
4204:
4156:
4112:
4049:
4005:
3935:
3869:
3782:
3603:
2163:
To calculate how this integral is modified by the quantum-confined Stark effect we once again employ
455:
713:
130:
32:
533:
490:
4128:
4073:
4021:
3995:
3893:
3859:
3828:
3436:
484:
1626:
1594:
204:{\displaystyle V(z)={\begin{cases}0;&|z|<L/2\\V_{0};&{\mbox{otherwise}}\end{cases}}}
4222:
4172:
4065:
3966:
3885:
3642:
3599:
625:
69:
36:
3811:
Miller, David A.B. (2009). "Device
Requirements for Optical Interconnects to Silicon Chips".
3668:
1973:
1947:
1913:
1887:
1016:
4266:
4258:
4212:
4164:
4120:
4057:
4013:
3943:
3877:
3820:
3790:
3646:
3622:
3219:
4097:
3915:. pp. Section 5.8: MODULATION OF EXCITONIC TRANSITIONS: QUANTUM CONFINED STARK EFFECT.
3717:
3694:
3545:{\displaystyle E_{X,n}={\frac {\mu }{m_{e}\varepsilon _{r}^{2}}}{\frac {R_{H}}{n^{2}-1/2}}}
3194:
1729:, the energy shift of the first optical transition induced by QCSE can be approximated to:
932:
265:
238:
3225:
2377:
2351:
2325:
4254:
4208:
4160:
4116:
4053:
4009:
3939:
3873:
3786:
3558:
2403:
452:
is the cross-sectional area of the system, perpendicular to the quantization direction,
3665:
3181:{\displaystyle E_{X,n}={\frac {\mu }{m_{e}\varepsilon _{r}^{2}}}{\frac {R_{H}}{n^{2}}}}
2931:
603:
435:
218:
28:
1591:
Similar calculations can be applied to holes by replacing the electron effective mass
4319:
3738:
61:
52:
48:
4132:
4025:
3897:
3832:
4077:
3661:
3615:
3586:
3243:
526:
40:
19:
3794:
3691:. QCSE in Ge/SiGe quantum wells can therefore be used to modulate light at 1.55
3428:
2948:
has been introduced as a normalization constant. For any applied electric field
72:, which allow optical communications signals to be switched on and off rapidly.
3824:
4096:
Lever, L; Ikonić, Z; Valavanis, A; Cooper, J D; Kelsall, R W (November 2010).
3088:
4124:
4017:
3687:, with a direct bandgap of 0.8 eV, which corresponds to a wavelength of 1550
4262:
3688:
3645:. Differently from III/V semiconductors, Ge/SiGe quantum well stacks can be
3638:
3637:
quantum wells, QCSE started to generate interest after its demonstration in
3607:
56:
4226:
4176:
4069:
3889:
1056:
The first order correction to the energy levels is zero due to symmetry.
4298:
Quantum Theory of the
Optical and Electronic Properties of Semiconductors
4217:
4192:
4168:
3595:
3330:
2001:
76:
44:
4271:
4061:
3664:. However it also has a relative minimum in the conduction band at the
3614:
region contains multiple quantum wells and acts as a waveguide for the
3084:
565:). On the right: the perturbative effect of the applied electric field
65:
3881:
1940:
1880:
919:{\displaystyle E_{n}={\frac {\hbar ^{2}n^{2}\pi ^{2}}{2m^{*}L^{2}}},}
3947:
1884:
Experimental demonstration of quantum-confined Stark effect in Ge/Si
4193:"23 GHz Ge/SiGe multiple quantum well electro-absorption modulator"
4000:
3864:
3585:
3555:
which is four times as high as the three dimensional case for the
3439:
in a two dimensional system yields an excitonic binding energy of
1156:{\displaystyle E_{n}^{(1)}=\langle n^{(0)}|eFz|n^{(0)}\rangle =0}
3650:
624:
If the quantum well is very deep, it can be approximated by the
2348:
energy level and consider only the perturbation from the level
3714:, which is crucial for silicon photonics applications as 1.55
968:
Supposing the electric field is biased along the z direction,
88:
unbiased system and its approximate effect can be found using
2005:
applied electric field along z, the overlapping integral for
3060:{\displaystyle \langle \phi _{c,1}|\phi _{v,1}\rangle <1}
3413:{\displaystyle -e{\vec {F}}\cdot {\vec {r_{X}}}>|E_{X}|}
1944:
Numerical simulation of the absorption coefficient of Ge/Si
809:
197:
3606:. A QCSE based electro-absorption modulator consists of a
2152:{\displaystyle \langle \phi _{c,n}|\phi _{v,n}\rangle =1}
2928:
for the conduction and valence band respectively, where
487:
for the energy band edge in the bulk semiconductor and
2167:. The first order correction for the wave function is
523:
is a slowly varying envelope function for the system.
188:
3720:
3697:
3671:
3561:
3448:
3346:
3282:
3252:
3228:
3197:
3100:
3007:
2954:
2934:
2682:
2432:
2406:
2380:
2354:
2328:
2176:
2099:
2011:
1976:
1950:
1916:
1890:
1738:
1661:
1629:
1597:
1176:
1065:
1019:
977:
935:
838:
670:
634:
606:
600:
modifies the wave functions and decreases the energy
571:
536:
493:
458:
438:
298:
268:
241:
221:
109:
43:(QW). In the absence of an external electric field,
3913:
100:The potential for a quantum well may be written as
3729:
3706:
3677:
3570:
3544:
3412:
3329:Exciton generation therefore redshift the optical
3317:
3265:
3234:
3210:
3180:
3059:
2990:
2940:
2917:
2667:
2415:
2392:
2366:
2340:
2310:
2151:
2082:
1988:
1962:
1928:
1902:
1856:
1721:
1647:
1615:
1576:
1167:The second order correction is, for instance n=1,
1155:
1045:
999:
948:
918:
818:
653:
612:
592:
557:
515:
475:
444:
421:
281:
254:
227:
203:
2991:{\displaystyle {\vec {F}}\cdot {\hat {z}}\neq 0}
3598:spectral range, which is of great interest for
1722:{\displaystyle m_{tot}^{*}=m_{e}^{*}+m_{h}^{*}}
8:
3048:
3008:
2305:
2259:
2218:
2140:
2100:
1409:
1355:
1279:
1225:
1144:
1090:
3806:
3804:
1000:{\displaystyle \mathbf {F} =F\mathbf {z} ,}
3653:technology and silicon photonics systems.
2083:{\displaystyle n_{valence}=n_{conduction}}
4270:
4216:
3999:
3863:
3768:
3766:
3719:
3696:
3670:
3560:
3531:
3519:
3508:
3502:
3493:
3488:
3478:
3468:
3453:
3447:
3405:
3399:
3390:
3375:
3369:
3368:
3354:
3353:
3345:
3309:
3296:
3281:
3257:
3251:
3227:
3202:
3196:
3170:
3160:
3154:
3145:
3140:
3130:
3120:
3105:
3099:
3036:
3027:
3015:
3006:
2971:
2970:
2956:
2955:
2953:
2933:
2891:
2873:
2862:
2846:
2841:
2831:
2825:
2806:
2779:
2754:
2741:
2730:
2717:
2706:
2687:
2681:
2641:
2623:
2612:
2596:
2591:
2581:
2575:
2556:
2529:
2504:
2491:
2480:
2467:
2456:
2437:
2431:
2405:
2379:
2353:
2327:
2299:
2290:
2281:
2268:
2253:
2244:
2231:
2225:
2215:
2203:
2186:
2181:
2175:
2128:
2119:
2107:
2098:
2047:
2016:
2010:
1980:
1975:
1954:
1949:
1920:
1915:
1894:
1889:
1843:
1832:
1822:
1811:
1801:
1791:
1784:
1778:
1759:
1737:
1713:
1708:
1695:
1690:
1677:
1666:
1660:
1639:
1634:
1628:
1607:
1602:
1596:
1566:
1555:
1545:
1540:
1530:
1520:
1513:
1507:
1488:
1459:
1454:
1435:
1430:
1418:
1413:
1397:
1388:
1374:
1362:
1350:
1347:
1329:
1324:
1305:
1300:
1288:
1283:
1267:
1258:
1244:
1232:
1220:
1217:
1205:
1186:
1181:
1175:
1132:
1123:
1109:
1097:
1075:
1070:
1064:
1018:
989:
978:
976:
960:of an electron in a given semiconductor.
940:
934:
904:
894:
879:
869:
859:
852:
843:
837:
801:
800:
773:
754:
753:
726:
708:
693:
675:
669:
639:
633:
605:
573:
572:
570:
538:
537:
535:
498:
492:
465:
457:
437:
408:
392:
383:
371:
362:
351:
335:
320:
305:
297:
273:
267:
246:
240:
220:
187:
176:
160:
149:
141:
125:
108:
1939:
1879:
525:
51:within the quantum well may only occupy
4091:
4089:
4087:
3963:Physics of Photonics Devices, Chapter 3
3762:
2870:
2620:
1840:
1655:. Introducing the total effective mass
1563:
856:
3660:semiconductor, with a bandgap of 0.66
27:) describes the effect of an external
829:The energies of the bound states are
7:
3318:{\displaystyle h\nu >E_{g}-E_{X}}
2165:time independent perturbation theory
2374:(notice that the perturbation from
1010:the perturbing Hamiltonian term is
3672:
1739:
648:
14:
3266:{\displaystyle \varepsilon _{r}}
990:
979:
654:{\displaystyle V_{0}\to \infty }
593:{\displaystyle {\vec {F}}\neq 0}
466:
409:
306:
4105:Journal of Lightwave Technology
3988:Journal of Lightwave Technology
3621:Although first demonstrated in
476:{\displaystyle u(\mathbf {r} )}
3406:
3391:
3381:
3359:
3246:of the electron-hole pair and
3028:
2976:
2961:
2291:
2245:
2232:
2120:
1466:
1460:
1442:
1436:
1414:
1404:
1398:
1389:
1375:
1369:
1363:
1351:
1336:
1330:
1312:
1306:
1284:
1274:
1268:
1259:
1245:
1239:
1233:
1221:
1193:
1187:
1139:
1133:
1124:
1110:
1104:
1098:
1082:
1076:
687:
681:
645:
578:
543:
510:
504:
470:
462:
413:
405:
397:
355:
332:
326:
310:
302:
150:
142:
119:
113:
1:
2423:due to symmetry). We obtain
1623:with the hole effective mass
235:is the width of the well and
4310:Physics of Photonics Devices
4291:Optical properties of solids
558:{\displaystyle {\vec {F}}=0}
516:{\displaystyle \phi _{n}(z)}
3795:10.1103/PhysRevLett.53.2173
4347:
3961:Chuang, Shun Lien (2009).
3825:10.1109/JPROC.2009.2014298
2322:Once again we look at the
4300:, World Scientific, 2004.
4293:, Oxford, New York, 2001.
1648:{\displaystyle m_{h}^{*}}
1616:{\displaystyle m_{e}^{*}}
4125:10.1109/JLT.2010.2081345
4018:10.1109/JLT.2017.2647779
4263:10.1038/NPHOTON.2014.73
3928:Applied Physics Letters
3846:Zagonel, L. F. (2011).
3813:Proceedings of the IEEE
3678:{\displaystyle \Gamma }
1989:{\displaystyle _{0.82}}
1963:{\displaystyle _{0.18}}
1929:{\displaystyle _{0.82}}
1903:{\displaystyle _{0.18}}
1046:{\displaystyle H'=eFz.}
83:Theoretical description
3731:
3708:
3679:
3591:
3572:
3546:
3414:
3319:
3267:
3236:
3212:
3182:
3061:
2992:
2942:
2919:
2669:
2417:
2394:
2368:
2342:
2312:
2153:
2090:transitions will be:
2084:
1997:
1990:
1964:
1937:
1930:
1904:
1876:Absorption coefficient
1858:
1723:
1649:
1617:
1578:
1157:
1047:
1001:
950:
920:
820:
655:
621:
620:of the n=1 transition.
614:
594:
559:
517:
477:
446:
423:
283:
256:
229:
205:
3732:
3730:{\displaystyle \mu m}
3709:
3707:{\displaystyle \mu m}
3680:
3604:optical interconnects
3589:
3573:
3547:
3415:
3320:
3268:
3237:
3213:
3211:{\displaystyle R_{H}}
3183:
3062:
2993:
2943:
2920:
2670:
2418:
2395:
2369:
2343:
2313:
2154:
2085:
1991:
1965:
1943:
1931:
1905:
1883:
1859:
1724:
1650:
1618:
1579:
1158:
1048:
1002:
951:
949:{\displaystyle m^{*}}
921:
821:
656:
615:
595:
560:
529:
518:
478:
447:
424:
284:
282:{\displaystyle E_{n}}
257:
255:{\displaystyle V_{0}}
230:
206:
4218:10.1364/OE.20.003219
4169:10.1364/OL.37.003960
3751:Franz–Keldysh effect
3718:
3695:
3669:
3610:structure where the
3602:and down-scaling of
3559:
3446:
3433:Schrödinger equation
3344:
3335:Franz–Keldysh effect
3280:
3250:
3235:{\displaystyle \mu }
3226:
3195:
3098:
3005:
2952:
2932:
2680:
2430:
2404:
2378:
2352:
2326:
2174:
2097:
2009:
2000:Additionally to the
1974:
1948:
1914:
1888:
1736:
1659:
1627:
1595:
1174:
1063:
1017:
975:
933:
836:
668:
632:
604:
569:
534:
491:
456:
436:
432:In this expression,
296:
266:
239:
219:
107:
4326:Quantum electronics
4255:2014NaPho...8..482C
4209:2012OExpr..20.3219C
4161:2012OptL...37.3960R
4117:2010JLwT...28.3273L
4062:10.1038/nature04204
4054:2005Natur.437.1334K
4048:(7063): 1334–1336.
4010:2017JLwT...35..346M
3940:1987ApPhL..50..842W
3874:2011NanoL..11..568Z
3787:1984PhRvL..53.2173M
3498:
3150:
3073:Fermi's golden rule
3071:Thus, according to
2851:
2750:
2722:
2601:
2500:
2472:
2393:{\displaystyle n=3}
2367:{\displaystyle n=2}
2341:{\displaystyle n=1}
2195:
1827:
1718:
1700:
1682:
1644:
1612:
1550:
1470:
1446:
1340:
1316:
1197:
1086:
90:perturbation theory
33:absorption spectrum
4308:Shun Lien Chuang,
3727:
3704:
3675:
3592:
3582:Optical modulation
3571:{\displaystyle 1s}
3568:
3542:
3484:
3410:
3315:
3263:
3232:
3208:
3178:
3136:
3057:
2988:
2938:
2915:
2837:
2726:
2702:
2665:
2587:
2476:
2452:
2416:{\displaystyle =0}
2413:
2390:
2364:
2338:
2308:
2214:
2177:
2149:
2080:
1998:
1986:
1960:
1938:
1926:
1900:
1854:
1807:
1719:
1704:
1686:
1662:
1645:
1630:
1613:
1598:
1574:
1536:
1450:
1426:
1320:
1296:
1216:
1177:
1153:
1066:
1043:
997:
946:
916:
816:
808:
651:
622:
610:
590:
555:
513:
473:
442:
419:
279:
252:
225:
201:
196:
192:
70:optical modulators
4331:Quantum mechanics
3882:10.1021/nl103549t
3781:(22): 2173–2176.
3647:epitaxially grown
3600:silicon photonics
3540:
3500:
3437:Coulomb potential
3384:
3362:
3176:
3152:
2979:
2964:
2941:{\displaystyle A}
2904:
2879:
2819:
2792:
2762:
2654:
2629:
2569:
2542:
2512:
2288:
2199:
1849:
1772:
1572:
1501:
1472:
1342:
1201:
911:
804:
789:
757:
742:
703:
702:
626:particle in a box
613:{\displaystyle E}
581:
546:
445:{\displaystyle A}
345:
344:
228:{\displaystyle L}
191:
37:emission spectrum
18:quantum-confined
4338:
4277:
4276:
4274:
4243:Nature Photonics
4237:
4231:
4230:
4220:
4187:
4181:
4180:
4143:
4137:
4136:
4102:
4093:
4082:
4081:
4036:
4030:
4029:
4003:
3983:
3977:
3976:
3958:
3952:
3951:
3923:
3917:
3916:
3911:Singh, Jasprit.
3908:
3902:
3901:
3867:
3843:
3837:
3836:
3819:(7): 1166–1185.
3808:
3799:
3798:
3770:
3736:
3734:
3733:
3728:
3713:
3711:
3710:
3705:
3684:
3682:
3681:
3676:
3656:Germanium is an
3577:
3575:
3574:
3569:
3551:
3549:
3548:
3543:
3541:
3539:
3535:
3524:
3523:
3513:
3512:
3503:
3501:
3499:
3497:
3492:
3483:
3482:
3469:
3464:
3463:
3419:
3417:
3416:
3411:
3409:
3404:
3403:
3394:
3386:
3385:
3380:
3379:
3370:
3364:
3363:
3355:
3324:
3322:
3321:
3316:
3314:
3313:
3301:
3300:
3272:
3270:
3269:
3264:
3262:
3261:
3241:
3239:
3238:
3233:
3220:Rydberg constant
3217:
3215:
3214:
3209:
3207:
3206:
3187:
3185:
3184:
3179:
3177:
3175:
3174:
3165:
3164:
3155:
3153:
3151:
3149:
3144:
3135:
3134:
3121:
3116:
3115:
3066:
3064:
3063:
3058:
3047:
3046:
3031:
3026:
3025:
2997:
2995:
2994:
2989:
2981:
2980:
2972:
2966:
2965:
2957:
2947:
2945:
2944:
2939:
2924:
2922:
2921:
2916:
2914:
2910:
2909:
2905:
2900:
2892:
2880:
2878:
2877:
2868:
2867:
2866:
2850:
2845:
2832:
2830:
2829:
2824:
2820:
2818:
2807:
2797:
2793:
2788:
2780:
2763:
2755:
2749:
2748:
2740:
2721:
2716:
2698:
2697:
2674:
2672:
2671:
2666:
2664:
2660:
2659:
2655:
2650:
2642:
2630:
2628:
2627:
2618:
2617:
2616:
2600:
2595:
2582:
2580:
2579:
2574:
2570:
2568:
2557:
2547:
2543:
2538:
2530:
2513:
2505:
2499:
2498:
2490:
2471:
2466:
2448:
2447:
2422:
2420:
2419:
2414:
2399:
2397:
2396:
2391:
2373:
2371:
2370:
2365:
2347:
2345:
2344:
2339:
2317:
2315:
2314:
2309:
2304:
2303:
2294:
2289:
2287:
2286:
2285:
2273:
2272:
2262:
2258:
2257:
2248:
2243:
2235:
2230:
2229:
2216:
2213:
2194:
2193:
2185:
2158:
2156:
2155:
2150:
2139:
2138:
2123:
2118:
2117:
2089:
2087:
2086:
2081:
2079:
2078:
2039:
2038:
1995:
1993:
1992:
1987:
1985:
1984:
1969:
1967:
1966:
1961:
1959:
1958:
1935:
1933:
1932:
1927:
1925:
1924:
1909:
1907:
1906:
1901:
1899:
1898:
1863:
1861:
1860:
1855:
1850:
1848:
1847:
1838:
1837:
1836:
1826:
1821:
1806:
1805:
1796:
1795:
1785:
1783:
1782:
1777:
1773:
1771:
1760:
1728:
1726:
1725:
1720:
1717:
1712:
1699:
1694:
1681:
1676:
1654:
1652:
1651:
1646:
1643:
1638:
1622:
1620:
1619:
1614:
1611:
1606:
1583:
1581:
1580:
1575:
1573:
1571:
1570:
1561:
1560:
1559:
1549:
1544:
1535:
1534:
1525:
1524:
1514:
1512:
1511:
1506:
1502:
1500:
1489:
1473:
1471:
1469:
1458:
1445:
1434:
1424:
1423:
1422:
1417:
1408:
1407:
1392:
1378:
1373:
1372:
1354:
1348:
1343:
1341:
1339:
1328:
1315:
1304:
1294:
1293:
1292:
1287:
1278:
1277:
1262:
1248:
1243:
1242:
1224:
1218:
1215:
1196:
1185:
1162:
1160:
1159:
1154:
1143:
1142:
1127:
1113:
1108:
1107:
1085:
1074:
1052:
1050:
1049:
1044:
1027:
1006:
1004:
1003:
998:
993:
982:
955:
953:
952:
947:
945:
944:
925:
923:
922:
917:
912:
910:
909:
908:
899:
898:
885:
884:
883:
874:
873:
864:
863:
853:
848:
847:
825:
823:
822:
817:
812:
811:
805:
802:
794:
790:
785:
774:
758:
755:
747:
743:
738:
727:
704:
695:
694:
680:
679:
660:
658:
657:
652:
644:
643:
628:model, in which
619:
617:
616:
611:
599:
597:
596:
591:
583:
582:
574:
564:
562:
561:
556:
548:
547:
539:
522:
520:
519:
514:
503:
502:
482:
480:
479:
474:
469:
451:
449:
448:
443:
428:
426:
425:
420:
412:
401:
400:
396:
388:
387:
375:
367:
366:
346:
340:
336:
325:
324:
309:
288:
286:
285:
280:
278:
277:
261:
259:
258:
253:
251:
250:
234:
232:
231:
226:
210:
208:
207:
202:
200:
199:
193:
189:
181:
180:
164:
153:
145:
4346:
4345:
4341:
4340:
4339:
4337:
4336:
4335:
4316:
4315:
4286:
4284:General sources
4281:
4280:
4239:
4238:
4234:
4189:
4188:
4184:
4145:
4144:
4140:
4100:
4095:
4094:
4085:
4038:
4037:
4033:
3985:
3984:
3980:
3973:
3960:
3959:
3955:
3948:10.1063/1.98008
3934:(13): 842–844.
3925:
3924:
3920:
3910:
3909:
3905:
3845:
3844:
3840:
3810:
3809:
3802:
3775:Phys. Rev. Lett
3772:
3771:
3764:
3759:
3747:
3716:
3715:
3693:
3692:
3667:
3666:
3634:
3630:
3584:
3557:
3556:
3515:
3514:
3504:
3474:
3473:
3449:
3444:
3443:
3395:
3371:
3342:
3341:
3305:
3292:
3278:
3277:
3253:
3248:
3247:
3224:
3223:
3198:
3193:
3192:
3166:
3156:
3126:
3125:
3101:
3096:
3095:
3081:
3032:
3011:
3003:
3002:
2950:
2949:
2930:
2929:
2893:
2887:
2869:
2858:
2833:
2811:
2802:
2801:
2781:
2775:
2768:
2764:
2742:
2683:
2678:
2677:
2643:
2637:
2619:
2608:
2583:
2561:
2552:
2551:
2531:
2525:
2518:
2514:
2492:
2433:
2428:
2427:
2402:
2401:
2376:
2375:
2350:
2349:
2324:
2323:
2295:
2277:
2264:
2263:
2249:
2236:
2221:
2217:
2187:
2172:
2171:
2124:
2103:
2095:
2094:
2043:
2012:
2007:
2006:
1977:
1972:
1971:
1951:
1946:
1945:
1917:
1912:
1911:
1891:
1886:
1885:
1878:
1839:
1828:
1797:
1787:
1786:
1764:
1755:
1754:
1734:
1733:
1657:
1656:
1625:
1624:
1593:
1592:
1562:
1551:
1526:
1516:
1515:
1493:
1484:
1483:
1425:
1412:
1393:
1358:
1349:
1295:
1282:
1263:
1228:
1219:
1172:
1171:
1128:
1093:
1061:
1060:
1020:
1015:
1014:
973:
972:
966:
936:
931:
930:
900:
890:
886:
875:
865:
855:
854:
839:
834:
833:
807:
806:
795:
775:
769:
760:
759:
748:
728:
722:
709:
671:
666:
665:
635:
630:
629:
602:
601:
567:
566:
532:
531:
494:
489:
488:
454:
453:
434:
433:
379:
358:
347:
316:
294:
293:
269:
264:
263:
242:
237:
236:
217:
216:
195:
194:
185:
172:
169:
168:
139:
126:
105:
104:
98:
96:Unbiased system
85:
31:upon the light
12:
11:
5:
4344:
4342:
4334:
4333:
4328:
4318:
4317:
4314:
4313:
4312:, Wiley, 2009.
4306:
4301:
4296:Hartmut Haug,
4294:
4285:
4282:
4279:
4278:
4249:(6): 482–488.
4232:
4203:(3): 3219–24.
4197:Optics Express
4182:
4155:(19): 3960–2.
4149:Optics Letters
4138:
4083:
4031:
3994:(3): 346–396.
3978:
3972:978-0470293195
3971:
3953:
3918:
3903:
3858:(2): 568–573.
3838:
3800:
3761:
3760:
3758:
3755:
3754:
3753:
3746:
3743:
3726:
3723:
3703:
3700:
3674:
3632:
3628:
3616:carrier signal
3583:
3580:
3567:
3564:
3553:
3552:
3538:
3534:
3530:
3527:
3522:
3518:
3511:
3507:
3496:
3491:
3487:
3481:
3477:
3472:
3467:
3462:
3459:
3456:
3452:
3421:
3420:
3408:
3402:
3398:
3393:
3389:
3383:
3378:
3374:
3367:
3361:
3358:
3352:
3349:
3327:
3326:
3312:
3308:
3304:
3299:
3295:
3291:
3288:
3285:
3260:
3256:
3231:
3205:
3201:
3189:
3188:
3173:
3169:
3163:
3159:
3148:
3143:
3139:
3133:
3129:
3124:
3119:
3114:
3111:
3108:
3104:
3080:
3077:
3069:
3068:
3056:
3053:
3050:
3045:
3042:
3039:
3035:
3030:
3024:
3021:
3018:
3014:
3010:
2987:
2984:
2978:
2975:
2969:
2963:
2960:
2937:
2926:
2925:
2913:
2908:
2903:
2899:
2896:
2890:
2886:
2883:
2876:
2872:
2865:
2861:
2857:
2854:
2849:
2844:
2840:
2836:
2828:
2823:
2817:
2814:
2810:
2805:
2800:
2796:
2791:
2787:
2784:
2778:
2774:
2771:
2767:
2761:
2758:
2753:
2747:
2744:
2739:
2736:
2733:
2729:
2725:
2720:
2715:
2712:
2709:
2705:
2701:
2696:
2693:
2690:
2686:
2675:
2663:
2658:
2653:
2649:
2646:
2640:
2636:
2633:
2626:
2622:
2615:
2611:
2607:
2604:
2599:
2594:
2590:
2586:
2578:
2573:
2567:
2564:
2560:
2555:
2550:
2546:
2541:
2537:
2534:
2528:
2524:
2521:
2517:
2511:
2508:
2503:
2497:
2494:
2489:
2486:
2483:
2479:
2475:
2470:
2465:
2462:
2459:
2455:
2451:
2446:
2443:
2440:
2436:
2412:
2409:
2389:
2386:
2383:
2363:
2360:
2357:
2337:
2334:
2331:
2320:
2319:
2307:
2302:
2298:
2293:
2284:
2280:
2276:
2271:
2267:
2261:
2256:
2252:
2247:
2242:
2239:
2234:
2228:
2224:
2220:
2212:
2209:
2206:
2202:
2198:
2192:
2189:
2184:
2180:
2161:
2160:
2148:
2145:
2142:
2137:
2134:
2131:
2127:
2122:
2116:
2113:
2110:
2106:
2102:
2077:
2074:
2071:
2068:
2065:
2062:
2059:
2056:
2053:
2050:
2046:
2042:
2037:
2034:
2031:
2028:
2025:
2022:
2019:
2015:
1983:
1979:
1957:
1953:
1936:quantum wells.
1923:
1919:
1897:
1893:
1877:
1874:
1870:
1869:
1866:
1864:
1853:
1846:
1842:
1835:
1831:
1825:
1820:
1817:
1814:
1810:
1804:
1800:
1794:
1790:
1781:
1776:
1770:
1767:
1763:
1758:
1753:
1750:
1747:
1744:
1741:
1716:
1711:
1707:
1703:
1698:
1693:
1689:
1685:
1680:
1675:
1672:
1669:
1665:
1642:
1637:
1633:
1610:
1605:
1601:
1585:
1584:
1569:
1565:
1558:
1554:
1548:
1543:
1539:
1533:
1529:
1523:
1519:
1510:
1505:
1499:
1496:
1492:
1487:
1482:
1479:
1476:
1468:
1465:
1462:
1457:
1453:
1449:
1444:
1441:
1438:
1433:
1429:
1421:
1416:
1411:
1406:
1403:
1400:
1396:
1391:
1387:
1384:
1381:
1377:
1371:
1368:
1365:
1361:
1357:
1353:
1346:
1338:
1335:
1332:
1327:
1323:
1319:
1314:
1311:
1308:
1303:
1299:
1291:
1286:
1281:
1276:
1273:
1270:
1266:
1261:
1257:
1254:
1251:
1247:
1241:
1238:
1235:
1231:
1227:
1223:
1214:
1211:
1208:
1204:
1200:
1195:
1192:
1189:
1184:
1180:
1165:
1164:
1152:
1149:
1146:
1141:
1138:
1135:
1131:
1126:
1122:
1119:
1116:
1112:
1106:
1103:
1100:
1096:
1092:
1089:
1084:
1081:
1078:
1073:
1069:
1054:
1053:
1042:
1039:
1036:
1033:
1030:
1026:
1023:
1008:
1007:
996:
992:
988:
985:
981:
965:
962:
958:effective mass
943:
939:
927:
926:
915:
907:
903:
897:
893:
889:
882:
878:
872:
868:
862:
858:
851:
846:
842:
827:
826:
815:
810:
799:
796:
793:
788:
784:
781:
778:
772:
768:
765:
762:
761:
752:
749:
746:
741:
737:
734:
731:
725:
721:
718:
715:
714:
712:
707:
701:
698:
692:
689:
686:
683:
678:
674:
650:
647:
642:
638:
609:
589:
586:
580:
577:
554:
551:
545:
542:
512:
509:
506:
501:
497:
485:Bloch function
483:is a periodic
472:
468:
464:
461:
441:
430:
429:
418:
415:
411:
407:
404:
399:
395:
391:
386:
382:
378:
374:
370:
365:
361:
357:
354:
350:
343:
339:
334:
331:
328:
323:
319:
315:
312:
308:
304:
301:
276:
272:
249:
245:
224:
213:
212:
198:
186:
184:
179:
175:
171:
170:
167:
163:
159:
156:
152:
148:
144:
140:
138:
135:
132:
131:
129:
124:
121:
118:
115:
112:
97:
94:
84:
81:
29:electric field
13:
10:
9:
6:
4:
3:
2:
4343:
4332:
4329:
4327:
4324:
4323:
4321:
4311:
4307:
4305:
4302:
4299:
4295:
4292:
4288:
4287:
4283:
4273:
4268:
4264:
4260:
4256:
4252:
4248:
4244:
4236:
4233:
4228:
4224:
4219:
4214:
4210:
4206:
4202:
4198:
4194:
4186:
4183:
4178:
4174:
4170:
4166:
4162:
4158:
4154:
4150:
4142:
4139:
4134:
4130:
4126:
4122:
4118:
4114:
4110:
4106:
4099:
4092:
4090:
4088:
4084:
4079:
4075:
4071:
4067:
4063:
4059:
4055:
4051:
4047:
4043:
4035:
4032:
4027:
4023:
4019:
4015:
4011:
4007:
4002:
3997:
3993:
3989:
3982:
3979:
3974:
3968:
3964:
3957:
3954:
3949:
3945:
3941:
3937:
3933:
3929:
3922:
3919:
3914:
3907:
3904:
3899:
3895:
3891:
3887:
3883:
3879:
3875:
3871:
3866:
3861:
3857:
3853:
3849:
3842:
3839:
3834:
3830:
3826:
3822:
3818:
3814:
3807:
3805:
3801:
3796:
3792:
3788:
3784:
3780:
3776:
3769:
3767:
3763:
3756:
3752:
3749:
3748:
3744:
3742:
3740:
3739:optical fiber
3724:
3721:
3701:
3698:
3690:
3686:
3663:
3659:
3654:
3652:
3648:
3644:
3640:
3636:
3624:
3619:
3617:
3613:
3609:
3605:
3601:
3597:
3588:
3581:
3579:
3565:
3562:
3536:
3532:
3528:
3525:
3520:
3516:
3509:
3505:
3494:
3489:
3485:
3479:
3475:
3470:
3465:
3460:
3457:
3454:
3450:
3442:
3441:
3440:
3438:
3434:
3430:
3425:
3400:
3396:
3387:
3376:
3372:
3365:
3356:
3350:
3347:
3340:
3339:
3338:
3336:
3332:
3310:
3306:
3302:
3297:
3293:
3289:
3286:
3283:
3276:
3275:
3274:
3258:
3254:
3245:
3229:
3221:
3203:
3199:
3171:
3167:
3161:
3157:
3146:
3141:
3137:
3131:
3127:
3122:
3117:
3112:
3109:
3106:
3102:
3094:
3093:
3092:
3090:
3086:
3078:
3076:
3074:
3054:
3051:
3043:
3040:
3037:
3033:
3022:
3019:
3016:
3012:
3001:
3000:
2999:
2985:
2982:
2973:
2967:
2958:
2935:
2911:
2906:
2901:
2897:
2894:
2888:
2884:
2881:
2874:
2863:
2859:
2855:
2852:
2847:
2842:
2838:
2834:
2826:
2821:
2815:
2812:
2808:
2803:
2798:
2794:
2789:
2785:
2782:
2776:
2772:
2769:
2765:
2759:
2756:
2751:
2745:
2743:
2737:
2734:
2731:
2727:
2723:
2718:
2713:
2710:
2707:
2703:
2699:
2694:
2691:
2688:
2684:
2676:
2661:
2656:
2651:
2647:
2644:
2638:
2634:
2631:
2624:
2613:
2609:
2605:
2602:
2597:
2592:
2588:
2584:
2576:
2571:
2565:
2562:
2558:
2553:
2548:
2544:
2539:
2535:
2532:
2526:
2522:
2519:
2515:
2509:
2506:
2501:
2495:
2493:
2487:
2484:
2481:
2477:
2473:
2468:
2463:
2460:
2457:
2453:
2449:
2444:
2441:
2438:
2434:
2426:
2425:
2424:
2410:
2407:
2387:
2384:
2381:
2361:
2358:
2355:
2335:
2332:
2329:
2300:
2296:
2282:
2278:
2274:
2269:
2265:
2254:
2250:
2240:
2237:
2226:
2222:
2210:
2207:
2204:
2200:
2196:
2190:
2188:
2182:
2178:
2170:
2169:
2168:
2166:
2146:
2143:
2135:
2132:
2129:
2125:
2114:
2111:
2108:
2104:
2093:
2092:
2091:
2075:
2072:
2069:
2066:
2063:
2060:
2057:
2054:
2051:
2048:
2044:
2040:
2035:
2032:
2029:
2026:
2023:
2020:
2017:
2013:
2003:
1996:quantum wells
1981:
1978:
1955:
1952:
1942:
1921:
1918:
1895:
1892:
1882:
1875:
1873:
1867:
1865:
1851:
1844:
1833:
1829:
1823:
1818:
1815:
1812:
1808:
1802:
1798:
1792:
1788:
1779:
1774:
1768:
1765:
1761:
1756:
1751:
1748:
1745:
1742:
1732:
1731:
1730:
1714:
1709:
1705:
1701:
1696:
1691:
1687:
1683:
1678:
1673:
1670:
1667:
1663:
1640:
1635:
1631:
1608:
1603:
1599:
1589:
1567:
1556:
1552:
1546:
1541:
1537:
1531:
1527:
1521:
1517:
1508:
1503:
1497:
1494:
1490:
1485:
1480:
1477:
1474:
1463:
1455:
1451:
1447:
1439:
1431:
1427:
1419:
1401:
1394:
1385:
1382:
1379:
1366:
1359:
1344:
1333:
1325:
1321:
1317:
1309:
1301:
1297:
1289:
1271:
1264:
1255:
1252:
1249:
1236:
1229:
1212:
1209:
1206:
1202:
1198:
1190:
1182:
1178:
1170:
1169:
1168:
1150:
1147:
1136:
1129:
1120:
1117:
1114:
1101:
1094:
1087:
1079:
1071:
1067:
1059:
1058:
1057:
1040:
1037:
1034:
1031:
1028:
1024:
1021:
1013:
1012:
1011:
994:
986:
983:
971:
970:
969:
964:Biased system
963:
961:
959:
941:
937:
913:
905:
901:
895:
891:
887:
880:
876:
870:
866:
860:
849:
844:
840:
832:
831:
830:
813:
797:
791:
786:
782:
779:
776:
770:
766:
763:
750:
744:
739:
735:
732:
729:
723:
719:
716:
710:
705:
699:
696:
690:
684:
676:
672:
664:
663:
662:
640:
636:
627:
607:
587:
584:
575:
552:
549:
540:
528:
524:
507:
499:
495:
486:
459:
439:
416:
402:
393:
389:
384:
380:
376:
372:
368:
363:
359:
352:
348:
341:
337:
329:
321:
317:
313:
299:
292:
291:
290:
274:
270:
247:
243:
222:
182:
177:
173:
165:
161:
157:
154:
146:
136:
133:
127:
122:
116:
110:
103:
102:
101:
95:
93:
91:
82:
80:
78:
73:
71:
67:
63:
62:quantum yield
58:
54:
50:
46:
42:
38:
34:
30:
26:
22:
21:
4309:
4297:
4290:
4272:11311/849543
4246:
4242:
4235:
4200:
4196:
4185:
4152:
4148:
4141:
4111:(22): 3273.
4108:
4104:
4045:
4041:
4034:
3991:
3987:
3981:
3962:
3956:
3931:
3927:
3921:
3912:
3906:
3855:
3852:Nano Letters
3851:
3841:
3816:
3812:
3778:
3774:
3658:indirect gap
3655:
3620:
3593:
3554:
3426:
3422:
3328:
3244:reduced mass
3190:
3082:
3070:
2927:
2321:
2162:
1999:
1871:
1590:
1586:
1166:
1055:
1009:
967:
928:
828:
623:
431:
214:
99:
86:
74:
41:quantum well
24:
20:Stark effect
17:
15:
3429:Bohr radius
4320:Categories
4289:Mark Fox,
4001:1609.05510
3612:instrinsic
3578:solution.
3089:hydrogenic
2998:we obtain
3965:. Wiley.
3865:1209.0953
3757:Citations
3722:μ
3699:μ
3673:Γ
3526:−
3486:ε
3471:μ
3382:→
3366:⋅
3360:→
3348:−
3303:−
3287:ν
3255:ε
3230:μ
3138:ε
3123:μ
3049:⟩
3034:ϕ
3013:ϕ
3009:⟨
2983:≠
2977:^
2968:⋅
2962:→
2895:π
2885:
2871:ℏ
2848:∗
2816:π
2783:π
2773:
2728:ϕ
2704:ϕ
2685:ϕ
2645:π
2635:
2621:ℏ
2598:∗
2566:π
2549:−
2533:π
2523:
2478:ϕ
2454:ϕ
2435:ϕ
2400:would be
2306:⟩
2297:ϕ
2275:−
2260:⟩
2251:ϕ
2223:ϕ
2219:⟨
2208:≠
2201:∑
2179:ϕ
2141:⟩
2126:ϕ
2105:ϕ
2101:⟨
1841:ℏ
1824:∗
1769:π
1749:−
1746:≈
1740:Δ
1715:∗
1697:∗
1679:∗
1641:∗
1609:∗
1564:ℏ
1547:∗
1498:π
1478:−
1448:−
1410:⟩
1356:⟨
1345:≈
1318:−
1280:⟩
1226:⟨
1210:≠
1203:∑
1145:⟩
1091:⟨
942:∗
896:∗
877:π
857:ℏ
780:π
767:
733:π
720:
706:×
673:ϕ
649:∞
646:→
585:≠
579:→
544:→
496:ϕ
390:⋅
369:⋅
318:ϕ
300:ψ
190:otherwise
55:within a
45:electrons
4227:22330559
4177:23027245
4133:11011784
4070:16251959
4026:38932250
3898:18003378
3890:21182283
3833:15772363
3745:See also
3596:infrared
3331:band gap
3085:excitons
3079:Excitons
2746:′
2496:′
2241:′
2191:′
2002:redshift
1025:′
77:band gap
66:excitons
57:discrete
4251:Bibcode
4205:Bibcode
4157:Bibcode
4113:Bibcode
4078:4414993
4050:Bibcode
4006:Bibcode
3936:Bibcode
3870:Bibcode
3783:Bibcode
3737:is the
3242:is the
3218:is the
956:is the
4225:
4175:
4131:
4076:
4068:
4042:Nature
4024:
3969:
3896:
3888:
3831:
3435:for a
3191:where
929:where
215:where
53:states
4129:S2CID
4101:(PDF)
4074:S2CID
4022:S2CID
3996:arXiv
3894:S2CID
3860:arXiv
3829:S2CID
3685:point
3091:atom
49:holes
39:of a
4223:PMID
4173:PMID
4066:PMID
3967:ISBN
3886:PMID
3651:CMOS
3643:SiGe
3623:GaAs
3388:>
3290:>
3052:<
1982:0.82
1956:0.18
1922:0.82
1896:0.18
803:even
155:<
47:and
25:QCSE
16:The
4267:hdl
4259:doi
4213:doi
4165:doi
4121:doi
4058:doi
4046:437
4014:doi
3944:doi
3878:doi
3821:doi
3791:doi
3633:1-x
3608:PIN
2882:sin
2770:cos
2632:sin
2520:cos
764:sin
756:odd
717:cos
35:or
4322::
4265:.
4257:.
4245:.
4221:.
4211:.
4201:20
4199:.
4195:.
4171:.
4163:.
4153:37
4151:.
4127:.
4119:.
4109:28
4107:.
4103:.
4086:^
4072:.
4064:.
4056:.
4044:.
4020:.
4012:.
4004:.
3992:35
3990:.
3942:.
3932:50
3930:.
3892:.
3884:.
3876:.
3868:.
3856:11
3854:.
3850:.
3827:.
3817:97
3815:.
3803:^
3789:.
3779:53
3777:.
3765:^
3689:nm
3662:eV
3639:Ge
3635:As
3631:Ga
3627:Al
3222:,
1970:Ge
1910:Ge
1752:24
1481:24
92:.
4275:.
4269::
4261::
4253::
4247:8
4229:.
4215::
4207::
4179:.
4167::
4159::
4135:.
4123::
4115::
4080:.
4060::
4052::
4028:.
4016::
4008::
3998::
3975:.
3950:.
3946::
3938::
3900:.
3880::
3872::
3862::
3835:.
3823::
3797:.
3793::
3785::
3725:m
3702:m
3641:/
3629:x
3625:/
3566:s
3563:1
3537:2
3533:/
3529:1
3521:2
3517:n
3510:H
3506:R
3495:2
3490:r
3480:e
3476:m
3466:=
3461:n
3458:,
3455:X
3451:E
3407:|
3401:X
3397:E
3392:|
3377:X
3373:r
3357:F
3351:e
3325:.
3311:X
3307:E
3298:g
3294:E
3284:h
3259:r
3204:H
3200:R
3172:2
3168:n
3162:H
3158:R
3147:2
3142:r
3132:e
3128:m
3118:=
3113:n
3110:,
3107:X
3103:E
3067:.
3055:1
3044:1
3041:,
3038:v
3029:|
3023:1
3020:,
3017:c
2986:0
2974:z
2959:F
2936:A
2912:)
2907:)
2902:L
2898:z
2889:(
2875:2
2864:3
2860:L
2856:F
2853:e
2843:h
2839:m
2835:2
2827:4
2822:)
2813:3
2809:2
2804:(
2799:+
2795:)
2790:L
2786:z
2777:(
2766:(
2760:A
2757:1
2752:=
2738:1
2735:,
2732:v
2724:+
2719:0
2714:1
2711:,
2708:v
2700:=
2695:1
2692:,
2689:v
2662:)
2657:)
2652:L
2648:z
2639:(
2625:2
2614:3
2610:L
2606:F
2603:e
2593:e
2589:m
2585:2
2577:4
2572:)
2563:3
2559:2
2554:(
2545:)
2540:L
2536:z
2527:(
2516:(
2510:A
2507:1
2502:=
2488:1
2485:,
2482:c
2474:+
2469:0
2464:1
2461:,
2458:c
2450:=
2445:1
2442:,
2439:c
2411:0
2408:=
2388:3
2385:=
2382:n
2362:2
2359:=
2356:n
2336:1
2333:=
2330:n
2318:.
2301:k
2292:|
2283:k
2279:E
2270:n
2266:E
2255:k
2246:|
2238:H
2233:|
2227:n
2211:n
2205:k
2197:=
2183:n
2159:.
2147:1
2144:=
2136:n
2133:,
2130:v
2121:|
2115:n
2112:,
2109:c
2076:n
2073:o
2070:i
2067:t
2064:c
2061:u
2058:d
2055:n
2052:o
2049:c
2045:n
2041:=
2036:e
2033:c
2030:n
2027:e
2024:l
2021:a
2018:v
2014:n
1852:.
1845:2
1834:4
1830:L
1819:t
1816:o
1813:t
1809:m
1803:2
1799:F
1793:2
1789:e
1780:6
1775:)
1766:3
1762:2
1757:(
1743:E
1710:h
1706:m
1702:+
1692:e
1688:m
1684:=
1674:t
1671:o
1668:t
1664:m
1636:h
1632:m
1604:e
1600:m
1568:2
1557:4
1553:L
1542:e
1538:m
1532:2
1528:F
1522:2
1518:e
1509:6
1504:)
1495:3
1491:2
1486:(
1475:=
1467:)
1464:0
1461:(
1456:2
1452:E
1443:)
1440:0
1437:(
1432:1
1428:E
1420:2
1415:|
1405:)
1402:0
1399:(
1395:1
1390:|
1386:z
1383:F
1380:e
1376:|
1370:)
1367:0
1364:(
1360:2
1352:|
1337:)
1334:0
1331:(
1326:k
1322:E
1313:)
1310:0
1307:(
1302:1
1298:E
1290:2
1285:|
1275:)
1272:0
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1130:n
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134:0
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123:=
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111:V
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