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ɋɈȼɊȿɆȿɇɇɕȿ ɉɊɈȻɅȿɆɕ ɎɂɁɂɄɂ ɌȼȿɊȾɈȽɈ ɌȿɅȺ

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ɝɞɟ  ɞɟɣɫɬɜɢɟ (3.7) ɜ ɨɬɫɭɬɫɬɜɢɟ ɜɨɡɛɭɠɞɟɧɢɹ. ɉɨɥɭ-

ɱɢɜɲɟɟɫɹ ɜɵɪɚɠɟɧɢɟ (3.9) ɨɩɢɫɵɜɚɟɬ ɜɡɚɢɦɨɞɟɣɫɬɜɢɟ ɨɞɧɨ-

ɤɨɦɩɨɧɟɧɬɧɨɣ ɩɥɚɡɦɵ ɡɚɪɹɠɟɧɧɵɯ ɱɚɫɬɢɰ (ɫ ɡɚɪɹɞɨɦ e*=m)

ɫ ɬɨɱɟɱɧɵɦ ɡɚɪɹɞɨɦ, ɧɚɯɨɞɹɳɢɦɫɹ ɜ ɬɨɱɤɟ z

)

. ɉɥɚɡɦɚ ɛɭɞɟɬ

ɷɤɪɚɧɢɪɨɜɚɬɶ ɷɬɨɬ ɡɚɪɹɞ, ɫɨɡɞɚɜɚɹ ɪɚɜɧɵɣ ɩɨ ɜɟɥɢɱɢɧɟ ɢ

ɩɪɨɬɢɜɨɩɨɥɨɠɧɵɣ ɩɨ ɡɧɚɤɭ ɡɚɪɹɞ ɜɨɤɪɭɝ z

. ȼɢɞɧɨ ɢɡ (3.9),

ɱɬɨ ɜ ɬɨɱɤɟ z

ɦɵ ɢɦɟɟɦ ɟɞɢɧɢɱɧɵɣ ɡɚɪɹɞ (ɚɧɟm). Ɉɞɧɚɤɨ

ɫɥɟɞɭɟɬ ɜɫɩɨɦɧɢɬɶ, ɱɬɨ ɪɟɚɥɶɧɨ ɞɟɣɫɬɜɢɟ (3.7), (3.9) ɨɩɢɫɵ-

ɜɚɟɬ ɧɟ ɝɢɩɨɬɟɬɢɱɟɫɤɢɟ ɱɚɫɬɢɰɵ ɫ ɡɚɪɹɞɨɦ e*=m, ɚ ɷɥɟɤɬɪɨ-

ɧɵ ɫ ɡɚɪɹɞɨɦ e, ɢ ɬɨɥɶɤɨ ɩɨ ɚɧɚɥɨɝɢɢ ɦɵ ɫɨɩɨɫɬɚɜɥɹɟɦ ɪɟ-

ɚɥɶɧɭɸ ɢ “ɩɥɚɡɦɟɧɧɭɸ” ɫɢɫɬɟɦɵ. ɋɨɨɬɜɟɬɫɬɜɟɧɧɨ ɪɚɫɫɦɚɬ-

ɪɢɜɚɟɦɨɟ ɜɨɡɛɭɠɞɟɧɢɟ ɨɬɧɨɫɢɬɟɥɶɧɨ ɪɟɚɥɶɧɵɯ ɷɥɟɤɬɪɨɧɨɜ

ɛɭɞɟɬ ɢɦɟɬɶ ɞɪɨɛɧɵɣ ɡɚɪɹɞ e*=e/m

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, Ʌɚɮɥɢɧ ɩɪɟɞɫɤɚɡɚɥ ɷɥɟɦɟɧɬɚɪɧɵɟ ɜɨɡ-

ɛɭɠɞɟɧɢɹ ɫ ɞɪɨɛɧɵɦ ɡɚɪɹɞɨɦ! ɋɥɟɞɭɟɬ ɡɚɦɟɬɢɬɶ, ɱɬɨ ɛɭɤ-

ɜɚɥɶɧɨ ɡɚ ɝɨɞ ɞɨ ɷɬɨɝɨ, ȼɢɥɤɫɟɤ (Wilczek, 1982) ɪɚɫɫɦɨɬɪɟɥ

ɬɟɨɪɟɬɢɱɟɫɤɢ ɤɜɚɧɬɨɜɭɸ ɫɬɚɬɢɫɬɢɤɭ ɱɚɫɬɢɰ ɜ ɞɜɭɦɟɪɧɨɦ

ɩɪɨɫɬɪɚɧɫɬɜɟ. Ɉɧ ɩɨɤɚɡɚɥ, ɱɬɨ ɤɪɨɦɟ ɛɨɡɟ- ɢ ɮɟɪɦɢ- ɜɚɪɢɚɧ-

ɬɚ ɫɭɳɟɫɬɜɭɟɬ ɜ ɞɜɭɦɟɪɢɢ ɢ ɞɪɨɛɧɚɹ ɫɬɚɬɢɫɬɢɤɚ, ɤɨɝɞɚ ɜɨɥ-

ɧɨɜɚɹ ɮɭɧɤɰɢɹ ɩɪɢɨɛɪɟɬɚɟɬ ɤɨɦɩɥɟɤɫɧɵɣ ɮɚɡɨɜɵɣ ɦɧɨɠɢ-

ɬɟɥɶ ɩɪɢ ɩɟɪɟɫɬɚɧɨɜɤɟ ɞɜɭɯ ɱɚɫɬɢɰ (ɢɯ ɧɚɡɵɜɚɸɬ ɧɟ ɛɨɡɨɧɚ-

ɦɢ ɢɥɢ ɮɟɪɦɢɨɧɚɦɢ, ɚ ɷɧɢɨɧɚɦɢ (anyon)).Ɍɚɤɚɹ ɷɧɢɨɧɧɚɹ

ɫɬɚɬɢɫɬɢɤɚ ɡɚɧɢɦɚɟɬ ɩɪɨɦɟɠɭɬɨɱɧɨɟ ɩɨɥɨɠɟɧɢɟ ɦɟɠɞɭ ɛɨ-

ɡɟ- ɢ ɮɟɪɦɢ- ɫɬɚɬɢɫɬɢɤɚɦɢ, ɤɨɬɨɪɵɟ ɩɪɨɫɬɨ ɨɛɥɚɞɚɸɬ ɫɨɨɬ-

ɜɟɬɫɬɜɟɧɧɨ ɩɪɟɞɟɥɶɧɵɦɢ ɮɚɡɚɦɢ 0 ɢ

. ɋɚɦɨɟ ɢɧɬɟɪɟɫɧɨɟ,

ɱɬɨ ɜ ɪɚɛɨɬɟ ȼɢɥɤɫɟɤɚ ɱɢɫɬɨ ɝɢɩɨɬɟɬɢɱɟɫɤɢ ɪɚɫɫɦɚɬɪɢɜɚ-

ɥɢɫɶ ɫɨɫɬɚɜɧɵɟ ɤɜɚɡɢɱɚɫɬɢɰɵ, ɩɪɟɞɫɬɚɜɥɹɸɳɢɟ ɫɨɛɨɣ ɡɚɪɹ-

ɠɟɧɧɵɟ ɱɚɫɬɢɰɵ, ɜɪɚɳɚɸɳɢɟɫɹ ɜɨɤɪɭɝ ɬɪɭɛɨɤ ɦɚɝɧɢɬɧɨɝɨ

ɩɨɬɨɤɚ (ɫɦ. ɬɚɤɠɟ ɡɚɞɚɱɭ (3.6)). Ⱥ ɜ ɫɥɭɱɚɟ ɞɪɨɛɧɨɝɨ ɤɜɚɧɬɨ-

ɜɨɝɨ ɏɨɥɥɚ ɭɠɟ ɜɩɨɥɧɟ ɪɟɚɥɶɧɵɟ ɜɨɡɛɭɠɞɟɧɢɹ, ɤɚɤ ɩɨɤɚɡɵ-

ɜɚɟɬ Ʌɚɮɥɢɧ, ɢɦɟɸɬ ɞɪɨɛɧɵɣ ɡɚɪɹɞ e/m, ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɩɨ

ɬɟɪɦɢɧɨɥɨɝɢɢ ȼɢɥɤɫɟɤɚ ɢɦɟɸɬ 1/m- ɞɪɨɛɧɭɸ ɫɬɚɬɢɫɬɢɤɭ, ɢ

ɞɟɣɫɬɜɭɟɬ ɬɚɤ, ɤɚɤ ɟɫɥɢ ɛɵ ɨɧɢ ɛɵɥɢ ɛɨɡɨɧɨɦ (ɥɢɛɨ ɮɟɪ-

ɦɢɨɧɨɦ  ɚ ɜɟɪɧɟɟ  ɷɧɢɨɧɨɦ), ɫɰɟɩɥɟɧɧɵɣ ɫ ɦɚɝɧɢɬɧɵɦ

ɩɨɬɨɤɨɦ )=)

/m.

ɄȼȺɇɌɈȼɕȿ ɗɎɎȿɄɌɕ ɏɈɅɅȺ

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ɂɦɟɧɧɨ ɨɫɧɨɜɵɜɚɹɫɶ ɧɚ ɬɚɤɨɦ ɩɪɟɞɫɬɚɜɥɟɧɢɢ, ɏɨɥɞɟɣɧ

(Haldane, 1983), ɢ Ƚɚɥɶɩɟɪɢɧ (Halperin, 1984) ɩɪɟɞɫɤɚɡɚɥɢ

ɡɧɚɱɟɧɢɟ ɞɪɨɛɟɣ ɜ ɢɟɪɚɪɯɢɢ ɞɪɨɛɧɵɯ ɤɜɚɧɬɨɜɵɯ ɫɨɫɬɨɹɧɢɣ.

Ɍɚɤ, ɛɵɥɨ ɩɪɟɞɫɤɚɡɚɧɨ, ɱɬɨ FQHE-ɷɮɮɟɤɬ ɦɨɠɟɬ ɪɟɚɥɢɡɨ-

ɜɚɬɶɫɹ ɩɪɢ , p  ɰɟɥɨɟ, m  ɧɟɱɟɬɧɨɟ (ɚ ɬɚɤɠɟ ɩɪɢ

ɢɡ-ɡɚ ɷɥɟɤɬɪɨɧ-ɞɵɪɨɱɧɨɣ ɫɢɦɦɟɬɪɢɢ). ɉɪɢɜɟ-

ɞɟɦ ɬɚɤɠɟ ɛɨɥɟɟ ɨɛɳɟɟ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɮɚɤɬɨɪɚ ɡɚɩɨɥɧɟɧɢɹ

“ɫɚɬɟɥɥɢɬɧɵɯ”, ɢɥɢ “ɞɨɱɟɪɧɢɯ” ɫɨɫɬɨɹɧɢɣ:

mp / P

mp /1  P

.6,4,2;1;...5,3,1;

....

...



Ɉɞɧɨɡɧɚɱɧɨɣ ɢɧɮɨɪɦɚɰɢɢ ɨɛ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɨɦ ɩɪɨ-

ɹɜɥɟɧɢɢ ɞɪɨɛɧɨɝɨ ɡɚɪɹɞɚ ɧɨɫɢɬɟɥɟɣ ɞɨ ɩɨɫɥɟɞɧɟɝɨ ɜɪɟɦɟɧɢ

ɧɟ ɛɵɥɨ. Ɍɚɤ, ɩɟɪɜɵɟ ɭɤɚɡɚɧɢɹ ɧɚ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɨ ɧɚɛɥɸ-

ɞɚɟɦɵɣ ɜɨɡɛɭɠɞɟɧɢɣ ɜ FQHE, ɢɦɟɸɳɢɯ ɞɪɨɛɧɵɣ ɡɚɪɹɞ

ɩɨɹɜɢɥɨɫɶ ɟɳɟ ɜ 1993 ɝɨɞɭ (ɫɦ., ɧɚɩɪɢɦɟɪ, Goldman V.J. et

al, 1993). Ɉɞɧɚɤɨ ɞɚɧɧɵɟ ɛɵɥɢ ɩɪɨɬɢɜɨɪɟɱɢɜɵ, ɢɦɟɥɢ

ɛɨɥɶɲɭɸ ɩɨɝɪɟɲɧɨɫɬɶ, ɢ ɞɨɩɭɫɤɚɥɢ ɚɥɶɬɟɪɧɚɬɢɜɧɨɟ ɨɛɴɹɫ-

ɧɟɧɢɟ.

ȼ 1997 ɝɨɞɭ ɫɢɬɭɚɰɢɹ ɢɡɦɟɧɢɥɚɫɶ.

ȼ ɫɟɧɬɹɛɪɟ ɮɪɚɧɰɭɡ-

ɫɤɢɦ ɢɫɫɥɟɞɨɜɚɬɟɥɹɦ ɭɞɚɥɨɫɶ ɩɪɟɞɫɬɚɜɢɬɶ ɭɛɟɞɢɬɟɥɶɧɵɟ

ɞɨɤɚɡɚɬɟɥɶɫɬɜɚ ɧɚɛɥɸɞɟɧɢɹ ɤɜɚɡɢɱɚɫɬɢɱɧɵɯ ɜɨɡɛɭɠɞɟɧɢɣ ɫ

ɡɚɪɹɞɨɦ ɜ ɬɪɢ ɪɚɡɚ ɦɟɧɶɲɟɦ ɡɚɪɹɞɚ ɷɥɟɤɬɪɨɧɚ

(L.Saminadayar, D.C.Glatti, Y.Lin and B.Etienne, 1997) ɫɨɨɬ-

ɜɟɬɫɬɜɟɧɧɨ ɩɪɢ ɮɚɤɬɨɪɟ ɡɚɩɨɥɧɟɧɢɹ 1/3. ȼ ɫɬɚɧɞɚɪɬɧɨɣ

ɯɨɥɥɨɜɫɤɨɣ ɝɟɨɦɟɬɪɢɢ ɨɧɢ ɢɫɫɥɟɞɨɜɚɥɢ ɝɟɬɟɪɨɫɬɪɭɤɬɭɪɵ

GaAs/Ga(Al)As ɫ ɞɜɭɦɟɪɧɨɣ ɷɥɟɤɬɪɨɧɧɨɣ ɩɥɨɬɧɨɫɬɶɸ ~ 10

ɋɈȼɊȿɆȿɇɇɕȿ ɉɊɈȻɅȿɆɕ ɎɂɁɂɄɂ ɌȼȿɊȾɈȽɈ ɌȿɅȺ

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ɫɦ

-2

ɜ ɫɢɥɶɧɨɦ ɩɨɩɟɪɟɱɧɨɦ ɦɚɝɧɢɬɧɨɦ ɩɨɥɟ ~ 15 Ɍɥ ɢ

ɫɜɟɪɯɧɢɡɤɨɣ ɬɟɦɩɟɪɚɬɭɪɟ (0.13 - 0.15K ). ɂɫɩɨɥɶɡɭɹ ɷɥɟɤ-

ɬɪɨɧɧɭɸ ɥɢɬɨɝɪɚɮɢɸ, ɭɱɟɧɵɦ ɭɞɚɥɨɫɶ ɢɡɝɨɬɨɜɢɬɶ ɬɭɧɧɟɥɶ-

ɧɵɣ ɯɨɥɥɨɜɫɤɢɣ ɤɨɧɬɚɤɬ ɲɢɪɢɧɨɣ 275 ɧɦ. ɉɨ ɞɨɫɬɢɠɟɧɢɢ

ɥɚɮɥɢɧɨɜɫɤɨɝɨ ɫɨɫɬɨɹɧɢɹ ɩɪɢ ɡɚɩɨɥɧɟɧɢɢ 1/3, ɤɨɬɨɪɨɟ ɬɟɫ-

ɬɢɪɨɜɚɥɨɫɶ ɩɨ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɣ ɫɬɭɩɟɧɶɤɟ ɧɚ ɯɨɥɥɨɜɫɤɨɣ

ɩɪɨɜɨɞɢɦɨɫɬɢ, ɢɡɦɟɪɹɥɫɹ ɬɭɧɧɟɥɶɧɵɣ ɬɨɤ ɱɟɪɟɡ ɤɨɧɬɚɤɬ ɢ

ɟɝɨ ɮɥɭɤɬɭɚɰɢɢ (ɲɭɦ). ɋɨɝɥɚɫɧɨ ɫɨɨɬɧɨɲɟɧɢɹɦ ɒɨɬɤɢ,

ɨɬɧɨɲɟɧɢɟ

ɚɦɩɥɢɬɭɞɵ ɤɨɪɨɬɤɨɜɨɥɧɨɜɨɝɨ ɲɭɦɚ ɬɭɧɧɟɥɶɧɨɝɨ

ɬɨɤɚ ɤ ɜɟɥɢɱɢɧɟ ɫɚɦɨɝɨ ɬɨɤɚ ɧɚɩɪɹɦɭɸ ɡɚɜɢɫɢɬ ɨɬ ɡɚɪɹɞɚ

ɬɭɧɧɟɥɢɪɭɸɳɢɯ ɤɜɚɡɢɱɚɫɬɢɰ. ɗɤɫɩɟɪɢɦɟɧɬɚɬɨɪɵ ɧɚɛɥɸɞɚ-

ɥɢ ɡɚɜɢɫɢɦɨɫɬɶ ɷɬɨɣ ɮɢɡɢɱɟɫɤɨɣ ɜɟɥɢɱɢɧɵ ɨɬ ɬɨɤɚ ɱɟɪɟɡ

ɤɨɧɬɚɤɬ, ɢ ɩɨ ɧɚɤɥɨɧɭ ɤɪɢɜɨɣ ɨɩɪɟɞɟɥɢɥɢ ɡɚɪɹɞ ɜɨɡɛɭɠɞɟ-

ɧɢɣ e*=e/3, ɱɬɨ ɩɨɥɧɨɫɬɶɸ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɬɟɨɪɟɬɢɱɟɫɤɢɦ

ɩɪɟɞɫɤɚɡɚɧɢɹɦ Ʌɚɮɥɢɧɚ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɨɩɢɫɚɧɧɚɹ ɜɵɲɟ

ɪɚɛɨɬɚ ɹɜɥɹɟɬɫɹ ɩɟɪɜɵɦ ɭɛɟɞɢɬɟɥɶɧɵɦ ɞɨɤɚɡɚɬɟɥɶɫɬɜɨɦ

ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɨɝɨ

ɧɚɛɥɸɞɟɧɢɹ ɜɨɡɛɭɠɞɟɧɢɣ ɫ ɞɪɨɛɧɵɦ

ɡɚɪɹɞɨɦ ɜ ɮɢɡɢɤɟ ɤɨɧɞɟɧɫɢɪɨɜɚɧɧɵɯ ɫɪɟɞ. ɋɨɨɛɳɟɧɢɹ ɨ

ɧɚɛɥɸɞɟɧɢɢ ɷɥɟɦɟɧɬɚɪɧɵɯ ɜɨɡɛɭɠɞɟɧɢɣ, ɯɚɪɚɤɬɟɪɢɡɭɸ-

ɳɢɯɫɹ ɞɪɨɛɧɵɦ ɡɚɪɹɞɨɦ, ɩɨɞɬɜɟɪɠɞɚɸɬɫɹ ɢ ɞɪɭɝɢɦɢ ɢɫɫɥɟ-

ɞɨɜɚɬɟɥɹɦɢ (B. Daviss, 1998, G. P. Collins, 1997).

Ʉɜɚɡɢɱɚɫɬɢɰɵ, “ɤɜɚɡɢɞɵɪɤɚ” ɢ “ɤɜɚɡɢɷɥɟɤɬɪɨɧ”, ɥɨɤɚɥɢ-

ɡɨɜɚɧɵ ɜ ɨɛɥɚɫɬɢ ɩɪɨɫɬɪɚɧɫɬɜɚ ɪɚɡɦɟɪɨɦ ɩɨɪɹɞɤɚ ɦɚɝɧɢɬ-

ɧɨɣ ɞɥɢɧɵ. Ʉɨɧɟɱɧɵɣ ɪɚɡɦɟɪ ɩɨɡɜɨɥɹɟɬ ɨɰɟɧɢɬɶ ɷɧɟɪɝɢɸ,

ɧɟɨɛɯɨɞɢɦɭɸ ɞɥɹ ɫɨɡɞɚɧɢɹ ɬɚɤɢɯ ɤɜɚɡɢɱɚɫɬɢɰ. ɉɨɥɶɡɭɹɫɶ

ɚɧɚɥɨɝɢɟɣ ɫ

ɩɥɚɡɦɨɣ, ɢ ɢɫɯɨɞɹ ɢɡ ɷɧɟɪɝɢɢ ɷɤɪɚɧɢɪɨɜɚɧɢɹ

ɩɥɚɡɦɨɣ ɟɞɢɧɢɱɧɨɝɨ ɡɚɪɹɞɚ, Ʌɚɮɥɢɧ ɨɰɟɧɢɥ ɷɧɟɪɝɟɬɢɱɟ-

ɫɤɭɸ ɳɟɥɶ ɜ ɫɩɟɤɬɪɟ, ɤɨɬɨɪɚɹ ɫɬɚɧɞɚɪɬɧɨ ɜɵɫɱɢɬɵɜɚɟɬɫɹ ɢɡ

ɫɥɟɞɭɸɳɟɣ ɪɚɡɧɨɫɬɢ:

)(2)()(

EEE <<< '



Ɉɰɟɧɤɢ ɞɚɸɬ

~ e

H

~ 4-5 K ɩɪɢ ɩɨɥɟ ~ 10 Ɍɥ, m=3, 5,

~ 10 (ɞɢɷɥɟɤɬɪɢɱɟɫɤɚɹ ɤɨɧɫɬɚɧɬɚ) ɢ ɩɨɞɬɜɟɪɠɞɚɸɬɫɹ ɬɨɱ-

ɧɵɦɢ ɱɢɫɥɟɧɧɵɦɢ ɪɚɫɱɟɬɚɦɢ.

ɄȼȺɇɌɈȼɕȿ ɗɎɎȿɄɌɕ ɏɈɅɅȺ

_________________________________________________________________________________________________________________________________________________________________________________________________

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɨɫɧɨɜɧɨɟ ɫɨɫɬɨɹɧɢɟ ɠɢɞɤɨɫɬɢ Ʌɚɮɥɢɧɚ

ɨɬɞɟɥɟɧɨ ɨɬ ɜɨɡɛɭɠɞɟɧɧɨɝɨ ɫɨɫɬɨɹɧɢɹ ɤɨɧɟɱɧɨɣ ɷɧɟɪɝɟɬɢ-

ɱɟɫɤɨɣ ɳɟɥɶɸ. ɋɨɨɬɜɟɬɫɬɜɟɧɧɨ ɤɨɧɟɱɧɚɹ ɳɟɥɶ ɝɨɜɨɪɢɬ ɨ

ɬɨɦ, ɱɬɨ ɷɬɚ ɠɢɞɤɨɫɬɶ ɧɟɫɠɢɦɚɟɦɚ, ɢ ɨɰɟɧɤɢ ɷɧɟɪɝɟɬɢɱɟ-

ɫɤɨɣ ɳɟɥɢ

ɜɩɨɥɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɬ ɷɤɫɩɟɪɢɦɟɧɬɭ (ɧɚɩɨɦ-

ɧɢɦ, ɱɬɨ ɧɚɛɥɸɞɚɟɬɫɹ ɩɨɜɟɞɟɧɢɟ

~ exp[-

/T]). Ʌɚɮɥɢ-

ɧɨɜɫɤɨɟ ɫɨɫɬɨɹɧɢɟ ɨɛɥɚɞɚɟɬ ɫɩɟɰɢɮɢɱɟɫɤɢɦ ɞɚɥɶɧɢɦ ɩɨ-

ɪɹɞɤɨɦ, ɱɟɦ-ɬɨ ɧɚɩɨɦɢɧɚɸɳɢɦ ɫɜɟɪɯɩɪɨɜɨɞɹɳɟɟ ɫɨɫɬɨɹ-

ɧɢɟ. ȼɟɥɢɱɢɧɚ ɷɧɟɪɝɟɬɢɱɟɫɤɨɣ ɳɟɥɢ

ɢɝɪɚɟɬ ɩɪɚɤɬɢɱɟɫɤɢ

ɬɭ ɠɟ ɪɨɥɶ ɜ ɬɟɨɪɢɢ ɥɚɮɥɢɧɨɜɫɤɨɣ ɠɢɞɤɨɫɬɢ, ɤɚɤ ɢ ɦɨɞɭɥɶ

ɩɚɪɚɦɟɬɪɚ ɩɨɪɹɞɤɚ ɜ ɫɜɟɪɯɩɪɨɜɨɞɧɢɤɚɯ. ɇɚɥɢɱɢɟ ɤɨɧɟɱɧɨɣ

ɷɧɟɪɝɟɬɢɱɟɫɤɨɣ ɳɟɥɢ ɩɪɢɜɨɞɢɬ ɤ ɩɨɱɬɢ ɢɞɟɚɥɶɧɨɣ ɩɪɨɜɨ-

ɞɢɦɨɫɬɢ ɷɥɟɤɬɪɨɧɨɜ ɜ ɥɚɮɥɢɧɨɜɫɤɨɦ ɫɨɫɬɨɹɧɢɢ (ɬɚɤ ɱɬɨ

ɫɨɩɪɨɬɢɜɥɟɧɢɟ, ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɷɤɫɩɟɪɢɦɟɧɬɨɦ, ɷɤɫɩɨɧɟɧ-

ɰɢɚɥɶɧɨ ɫɬɪɟɦɢɬɫɹ ɤ ɧɭɥɸ, ɤɚɤ ɢ ɜ ɰɟɥɨɦ ɤɜɚɧɬɨɜɨɦ ɷɮɮɟɤ-

ɬɟ ɏɨɥɥɚ).

Ȼɨɥɟɟ ɬɨɝɨ, ɫɨɛɫɬɜɟɧɧɨ ɫɬɭɩɟɧɶɤɢ ɞɪɨɛɧɨɝɨ ɤɜɚɧ-

ɬɨɜɨɝɨ ɷɮɮɟɤɬɚ ɏɨɥɥɚ ɛɨɥɟɟ ɱɟɬɤɨ ɨɬɫɥɟɠɢɜɚɸɬɫɹ ɢɦɟɧɧɨ

ɩɨ ɪɟɡɤɨɦɭ ɭɦɟɧɶɲɟɧɢɸ ɜɟɥɢɱɢɧɵ

(ɪɢɫ.3.1).

ɉɪɢɱɢɧɚ ɧɟɫɠɢɦɚɟɦɨɫɬɢ ɜ ɞɚɧɧɨɦ ɤɨɧɤɪɟɬɧɨɦ ɫɥɭɱɚɟ

ɦɨɠɧɨ ɩɪɨɢɥɥɸɫɬɪɢɪɨɜɚɬɶ ɫɥɟɞɭɸɳɢɦ: ɥɸɛɨɟ ɫɠɚɬɢɟ ɢɥɢ

ɪɚɫɲɢɪɟɧɢɟ ɪɚɜɧɨɫɢɥɶɧɨ ɢɧɠɟɤɬɢɪɨɜɚɧɢɸ ɤɜɚɡɢɱɚɫɬɢɰ.

ɉɪɢ ɷɬɨɦ ɢɡɦɟɧɟɧɢɟ ɷɧɟɪɝɢɢ ɤɜɚɞɪɚɬɢɱɧɨ ɩɨ ɩɪɢɪɚɳɟɧɢɸ

ɜɟɤɬɨɪɧɨɝɨ ɩɨɬɟɧɰɢɚɥɚ

E ~(

, ɬ.ɟ. ɧɟɨɛɯɨɞɢɦɚ ɡɧɚɱɢ-

ɬɟɥɶɧɚɹ ɮɥɭɤɬɭɚɰɢɹ ɩɨɥɹ ɞɥɹ ɜɨɡɛɭɠɞɟɧɢɹ ɫɢɫɬɟɦɵ ɩɭɬɟɦ

ɪɚɫɲɢɪɟɧɢɹ ɢɥɢ ɫɠɚɬɢɹ. ɇɟɫɠɢɦɚɟɦɨɫɬɶ ɜɵɡɵɜɚɟɬ ɫɩɟɰɢ-

ɮɢɱɟɫɤɢɟ ɩɪɨɞɨɥɶɧɵɟ ɤɨɥɥɟɤɬɢɜɧɵɟ ɜɨɡɛɭɠɞɟɧɢɹ, ɜ ɨɬɫɭɬ-

ɫɬɜɢɟ ɡɜɭɤɨɜɨɣ ɜɨɥɧɵ ɪɚɫɬɹɠɟɧɢɹ-ɫɠɚɬɢɹ, ɢ ɢɦɟɸɳɢɟ ɷɧɟɪ-

ɝɢɸ ~

ɜ ɞɥɢɧɧɨɜɨɥɧɨɜɨɦ ɩɪɟɞɟɥɟ. Ɉɧɢ ɢ ɪɟɚɥɢɡɭɸɬ ɩɪɨ-

ɜɨɞɢɦɨɫɬɶ ɛɟɡ ɪɟɡɢɫɬɢɜɧɵɯ ɩɨɬɟɪɶ ɩɪɢ ɧɭɥɟɜɨɣ ɬɟɦɩɟɪɚɬɭ-

ɪɟ. ɋɢɬɭɚɰɢɹ ɬɚɤɚɹ ɠɟ, ɤɚɤ ɢ ɜ ɫɥɭɱɚɟ ɩɨɥɧɨɫɬɶɸ ɡɚɩɨɥɧɟɧ-

ɧɨɣ ɡɨɧɵ Ʌɚɧɞɚɭ (m=1) ɞɥɹ IQHE, ɤɨɝɞɚ ɩɨɯɨɠɢɟ ɤɨɥɥɟɤ-

ɬɢɜɧɵɟ ɜɨɡɛɭɠɞɟɧɢɹ ɢɦɟɸɬ ɦɟɫɬɨ ɩɪɢ ɷɧɟɪɝɢɢ

ȿɳɟ ɨɞɧɨ ɩɨɥɟɡɧɨɟ ɫɪɚɜɧɟɧɢɟ ɩɨ ɩɨɜɨɞɭ ɨɬɤɥɢɤɚ ɫɢɫɬɟ-

ɦɵ ɜ ɥɚɮɥɢɧɨɜɫɤɨɦ ɫɨɫɬɨɹɧɢɢ ɧɚ ɫɢɥɵ ɫɠɚɬɢɹ. Ɉɧ ɬɚɤɨɣ

ɋɈȼɊȿɆȿɇɇɕȿ ɉɊɈȻɅȿɆɕ ɎɂɁɂɄɂ ɌȼȿɊȾɈȽɈ ɌȿɅȺ

_________________________________________________________________________________________________________________________________________________________________________________________________

ɠɟ, ɤɚɤ ɧɚ ɩɪɢɥɨɠɟɧɧɨɟ ɦɚɝɧɢɬɧɨɟ ɩɨɥɟ ɜ ɫɜɟɪɯɩɪɨɜɨɞɧɢ-

ɤɚɯ ɜɬɨɪɨɝɨ ɪɨɞɚ. ɂɡɜɟɫɬɧɨ, ɱɬɨ ɫɜɟɪɯɩɪɨɜɨɞɧɢɤɢ ɜɬɨɪɨɝɨ

ɪɨɞɚ ɜ ɦɚɝɧɢɬɧɨɦ ɩɨɥɟ, ɦɟɧɶɲɟɦ ɧɟɤɨɬɨɪɨɝɨ ɩɟɪɜɨɝɨ ɤɪɢ-

ɬɢɱɟɫɤɨɝɨ H

, ɩɨɥɧɨɫɬɶɸ ɜɵɬɚɥɤɢɜɚɸɬ ɜɧɟɲɧɟɟ ɩɨɥɟ ɢɡ

ɫɜɨɟɝɨ ɨɛɴɟɦɚ (ɬɚɤ ɧɚɡɵɜɚɟɦɵɣ ɷɮɮɟɤɬ Ɇɟɣɫɫɧɟɪɚ). ɉɪɢ

ɩɨɥɟ, ɛɨɥɶɲɟɦ H

, ɫɢɬɭɚɰɢɹ ɧɟɫɤɨɥɶɤɨ ɢɧɚɹ - ɦɚɝɧɢɬɧɨɟ

ɩɨɥɟ ɩɪɨɧɢɤɚɟɬ ɜ ɫɜɟɪɯɩɪɨɜɨɞɧɢɤ, ɩɪɢɱɟɦ ɱɟɪɟɡ ɨɛɥɚɫɬɢ

ɧɨɪɦɚɥɶɧɨɣ ɮɚɡɵ, ɨɤɪɭɠɟɧɧɵɟ ɷɤɪɚɧɢɪɭɸɳɢɦɢ ɜɢɯɪɟɜɵɦɢ

ɬɨɤɚɦɢ, ɧɚɩɪɚɜɥɟɧɧɵɦɢ ɩɪɨɬɢɜ ɜɧɟɲɧɟɝɨ ɩɨɥɹ - ɬɚɤ ɧɚɡɵ-

ɜɚɟɦɵɟ ɜɢɯɪɢ Ⱥɛɪɢɤɨɫɨɜɚ - ɮɥɭɤɫɨɢɞɵ. ȼ ɫɥɭɱɚɟ ɥɚɮɥɢ-

ɧɨɜɫɤɨɝɨ ɫɨɫɬɨɹɧɢɹ ɫɢɬɭɚɰɢɹ ɩɨɞɨɛɧɚɹ: ɫɧɚɱɚɥɚ ɫɢɫɬɟɦɚ

ɝɟɧɟɪɢɪɭɟɬ ɯɨɥɥɨɜɫɤɢɟ ɬɨɤɢ ɛɟɡ ɫɠɚɬɢɹ, ɡɚɬɟɦ ɩɪɢ ɤɪɢɬɢɱɟ-

ɫɤɨɦ ɫɠɚɬɢɢ ɤɨɥɥɚɩɫɢɪɭɟɬ ɜ ɤɜɚɧɬɨɜɵɣ ɨɛɴɟɦ (ɦɚɤ-

ɪɨɫɤɨɩɢɱɟɫɤɚɹ ɤɚɩɥɹ). ɗɬɨ ɩɨɞɨɛɢɟ ɜɢɯɪɹ (ɤɚɤ ɜ ɫɜɟɪɯɩɪɨ-

ɜɨɞɧɢɤɚɯ ɜɬɨɪɨɝɨ ɪɨɞɚ), ɨɤɪɭɠɟɧɨ ɰɢɪɤɭɥɢɪɭɸɳɢɦ ɯɨɥɥɨɜ-

ɫɤɢɦ ɬɨɤɨɦ, ɧɚɩɪɚɜɥɟɧɧɵɦ ɩɪɨɬɢɜ ɫɠɚɬɢɹ.

lm S

ɋɨɛɫɬɜɟɧɧɨ ɞɥɹ ɨɛɴɹɫɧɟɧɢɹ ɫɬɭɩɟɧɟɤ FQHE ɫɥɟɞɭɟɬ

ɨɩɹɬɶ ɭɱɟɫɬɶ ɟɫɬɟɫɬɜɟɧɧɵɟ ɩɪɢɦɟɫɢ ɢ ɧɟɨɞɧɨɪɨɞɧɨɫɬɢ,

ɢɦɟɸɳɢɟ ɦɟɫɬɨ ɜ ɪɟɚɥɶɧɵɯ ɩɨɥɭɩɪɨɜɨɞɧɢɤɨɜɵɯ ɝɟɬɟɪɨɫɬ-

ɪɭɤɬɭɪɚɯ. Ɏɚɤɬɢɱɟɫɤɢ ɨɧɢ ɡɞɟɫɶ ɢɝɪɚɸɬ ɬɭ ɠɟ ɪɨɥɶ, ɤɚɤ ɢ ɜ

ɬɟɨɪɢɢ IQHE. “ɗɥɟɤɬɪɨɧɧɵɟ

” ɢ “ɞɵɪɨɱɧɵɟ” ɤɜɚɡɢɱɚɫɬɢɱ-

ɧɵɟ ɡɨɧɵ, ɪɚɡɞɟɥɟɧɧɵɟ ɜ ɱɢɫɬɨɦ, ɛɟɫɩɪɢɦɟɫɧɨɦ ɫɥɭɱɚɟ

ɷɧɟɪɝɟɬɢɱɟɫɤɨɣ ɳɟɥɶɸ 2

, ɭɲɢɪɟɧɵ ɜ ɤɨɧɬɢɧɭɭɦ ɫɨɫɬɨɹ-

ɳɢɣ ɢɡ ɞɜɭɯ ɡɨɧ ɜɨɡɛɭɠɞɟɧɧɵɯ ɫɨɫɬɨɹɧɢɣ, ɪɚɡɞɟɥɟɧɧɵɯ

ɡɨɧɨɣ ɥɨɤɚɥɢɡɨɜɚɧɧɵɯ ɫɨɫɬɨɹɧɢɣ, ɡɚɯɜɚɬɵɜɚɸɳɢɯ “ɤɪɵɥɶɹ”

ɡɨɧ. ɉɨɤɚ ɭɪɨɜɟɧɶ Ɏɟɪɦɢ ɜ ɨɛɥɚɫɬɢ ɥɨɤɚɥɢɡɚɰɢɢ, ɧɨɜɵɟ

ɱɚɫɬɢɰɵ ɩɪɢ ɟɝɨ ɜɚɪɢɚɰɢɹɯ ɡɚɯɜɚɬɵɜɚɸɬɫɹ ɩɪɢɦɟɫɧɵɦɢ

ɭɪɨɜɧɹɦɢ. ɉɪɢ ɷɬɨɦ ɯɨɥɥɨɜɫɤɚɹ ɩɪɨɜɨɞɢɦɨɫɬɶ ɪɟɚɥɢɡɭɟɬɫɹ

ɩɪɢ



m . Ʉɚɤ ɬɨɥɶɤɨ ɭɪɨɜɟɧɶ Ɏɟɪɦɢ ɩɪɨɯɨɞɢɬ ɱɟɪɟɡ

ɞɟɥɨɤɚɥɢɡɨɜɚɧɧɭɸ ɨɛɥɚɫɬɶ, ɩɪɨɢɫɯɨɞɢɬ ɩɟɪɟɯɨɞ ɞɨ ɫɥɟ-

ɞɭɸɳɟɝɨ ɫɨɫɬɨɹɧɢɹ Ʌɚɮɥɢɧɚ, ɢ ɜɨɡɪɚɫɬɚɸɬ ɪɟɡɢɫɬɢɜɧɵɟ

ɩɨɬɟɪɢ (

). Ɉɬɦɟɬɢɦ, ɱɬɨ ɩɪɢ ɭɜɟɥɢɱɟɧɢɢ m ( ɱɬɨ ɷɤɜɢɜɚ-

ɥɟɧɬɧɨ ɩɨ ɮɚɤɬɨɪɭ ɡɚɩɨɥɧɟɧɢɹ ɥɢɛɨ

->0, ɥɢɛɨ

->1 (1-

ɄȼȺɇɌɈȼɕȿ ɗɎɎȿɄɌɕ ɏɈɅɅȺ

_________________________________________________________________________________________________________________________________________________________________________________________________

p/m)) ɭɦɟɧɶɲɚɟɬɫɹ ɷɧɟɪɝɟɬɢɱɟɫɤɚɹ ɳɟɥɶ, ɢ FQHE ɩɪɨɹɜɥɹ-

ɟɬɫɹ ɜɫɟ ɫɥɚɛɟɟ.

ɂɧɬɟɪɟɫɧɨ, ɱɬɨ ɷɥɟɤɬɪɨɧɧɵɣ ɝɚɡ ɜ ɭɫɥɨɜɢɹɯ ɞɪɨɛɧɨɝɨ

ɤɜɚɧɬɨɜɨɝɨ ɷɮɮɟɤɬɚ ɏɨɥɥɚ ɜ ɩɪɢɧɰɢɩɟ ɞɨɥɠɟɧ ɫɨɜɟɪɲɟɧɧɨ

ɩɨ ɢɧɨɦɭ ɪɟɚɝɢɪɨɜɚɬɶ ɧɚ ɢɡɦɟɧɟɧɢɟ ɤɨɧɰɟɧɬɪɚɰɢɢ ɩɪɢɦɟ-

ɫɟɣ, ɱɟɦ ɰɟɥɨɱɢɫɥɟɧɧɵɣ. ȿɫɥɢ ɜ ɰɟɥɨɱɢɫɥɟɧɧɨɦ ɷɮɮɟɤɬɟ

ɜɨɡɪɚɫɬɚɧɢɟ ɤɨɧɰɟɧɬɪɚɰɢɢ ɩɪɢɦɟɫɟɣ ɜ ɢɡɜɟɫɬɧɵɯ ɩɪɟɞɟɥɚɯ

ɭɲɢɪɹɟɬ ɩɥɚɬɨ ɧɚ ɩɨɥɟɜɨɣ ɡɚɜɢɫɢɦɨɫɬɢ ɯɨɥɥɨɜɫɤɨɝɨ

ɫɨɩɪɨ-

ɬɢɜɥɟɧɢɹ, ɬɨ ɜ ɞɪɨɛɧɨɦ ɷɮɮɟɤɬɟ ɭɜɟɥɢɱɟɧɢɟ ɤɨɧɰɟɧɬɪɚɰɢɢ

ɩɪɢɦɟɫɢ ɦɨɠɟɬ ɭɦɟɧɶɲɢɬɶ ɲɢɪɢɧɭ ɩɥɚɬɨ, ɬ.ɤ. ɤɨɧɟɱɧɚɹ

ɳɟɥɶ ɜ ɫɩɟɤɬɪɟ ɜɨɡɛɭɠɞɟɧɢɣ 2

ɧɟ ɛɭɞɟɬ ɡɚɦɚɡɵɜɚɬɶɫɹ ɩɪɢ-

ɦɟɫɹɦɢ ɬɨɥɶɤɨ ɩɪɢ ɭɫɥɨɜɢɢ, ɱɬɨ ɜɪɟɦɹ ɫɬɨɥɤɧɨɜɟɧɢɣ

W

. Ʉɚɤ ɬɨɥɶɤɨ ɩɪɢɦɟɫɟɣ ɦɧɨɝɨ, ɳɟɥɶ ɦɟɠɞɭ ɨɫɧɨɜɧɵɦ,

ɥɨɤɚɥɢɡɨɜɚɧɧɵɦ ɧɚ ɩɪɢɦɟɫɹɯ ɢ ɞɟɥɨɤɚɥɢɡɨɜɚɧɧɵɦɢ, ɤɜɚɡɢ-

ɱɚɫɬɢɱɧɵɦɢ ɫɨɫɬɨɹɧɢɹɦɢ ɭɦɟɧɶɲɚɟɬɫɹ. ɉɪɢ ɧɚɥɢɱɢɢ ɜɢɝ-

ɧɟɪɨɜɫɤɨɣ ɤɪɢɫɬɚɥɥɢɡɚɰɢɢ (ɫɦ. ɞɚɥɟɟ) ɜɛɥɢɡɢ ɩɥɚɬɨ ɦɨɠɧɨ

ɨɠɢɞɚɬɶ ɧɟɥɢɧɟɣɧɭɸ ɡɚɜɢɫɢɦɨɫɬɶ ɲɢɪɢɧɵ ɩɥɚɬɨ ɨɬ ɤɨɧ-

ɰɟɧɬɪɚɰɢɢ, ɬ.ɤ. ɥɨɤɚɥɢɡɚɰɢɹ ɷɥɟɤɬɪɨɧɨɜ ɡɚ ɫɱɟɬ ɤɭɥɨɧɨɜɫɤɨ-

ɝɨ “ɡɚɩɢɪɚɧɢɹ” ɦɨɠɟɬ ɪɚɡɪɭɲɚɬɶɫɹ ɩɪɢɦɟɫɹɦɢ.

'/!

Ɂɚɞɚɱɢ.

Ɂɚɞɚɱɚ 3.1 Ɋɚɫɫɱɢɬɚɬɶ ɜɨɥɧɨɜɵɟ ɮɭɧɤɰɢɢ ɢ ɫɩɟɤɬɪ ɭɟɞɢɧɟɧɧɨɝɨ

ɷɥɟɤɬɪɨɧɚ ɜ ɩɨɫɬɨɹɧɧɨɦ ɦɚɝɧɢɬɧɨɦ ɩɨɥɟ.

Ɋɟɲɟɧɢɟ

. ȼɵɛɢɪɚɟɦ ɰɢɥɢɧɞɪɢɱɟɫɤɢɟ ɤɨɨɪɞɢɧɚɬɵ

, z. ȼɟɤɬɨɪɧɵɣ

ɩɨɬɟɧɰɢɚɥ ɩɪɢ ɧɚɩɪɚɜɥɟɧɢɢ ɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ ɜɞɨɥɶ ɨɫɢ z ɩɪɟɞɫɬɚɜɢɦ ɜ

ɫɥɟɞɭɸɳɟɦ ɜɢɞɟ ɜ ɫɢɦɦɟɬɪɢɱɧɨɣ ɥɨɪɟɧɰɟɜɫɤɨɣ ɤɚɥɢɛɪɨɜɤɟ:

2/),(2/;0

HxyHAAA



Ɋɚɫɩɢɫɵɜɚɟɦ ɞɟɣɫɬɜɢɟ ɨɩɟɪɚɬɨɪɚ Ƚɚɦɢɥɶɬɨɧɚ ɧɚ ɜɨɥɧɨɜɭɸ ɮɭɧɤɰɢɸ ɜ

ɰɢɥɢɧɞɪɢɱɟɫɤɨɣ ɝɟɨɦɟɬɪɢɢ

ɋɈȼɊȿɆȿɇɇɕȿ ɉɊɈȻɅȿɆɕ ɎɂɁɂɄɂ ɌȼȿɊȾɈȽɈ ɌȿɅȺ

_________________________________________________________________________________________________________________________________________________________________________________________________

];

ˆˆ

[

*2*2*2

])/(*][2/1[

\\\\

\ \

AppA

AcepmH

);

(

&&&



*2*8

]

)(

[

\U

\

 \

cim

Ⱦɚɥɟɟ, ɢɳɟɦ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ ɒɪɟɞɢɧɝɟɪɚ ɜ ɜɢɞɟ:

]exp[)(),,(

imzikRz



ɉɨɞɫɬɚɜɥɹɹ ɷɬɨ ɜɵɪɚɠɟɧɢɟ ɜ ɭɪɚɜɧɟɧɢɟ , ɩɨɥɭɱɢɦ ɩɨɫɥɟ

ɚɥɝɟɛɪɚɢɱɟɫɤɢɯ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ ɞɥɹ ɪɚɞɢɚɥɶɧɨɣ ɱɚɫɬɢ ɫɥɟɞɭɸɳɟɟ ɭɪɚɜ-

ɧɟɧɢɟ:

./*2;2/12/

;2][/1][

222

kEmlceH

RRmRR



 wwww

EJUJ

UUU

Ⱦɚɥɟɟ ɩɪɨɜɨɞɢɦ ɩɟɪɟɧɨɪɦɢɪɨɜɤɭ ɪɚɞɢɚɥɶɧɨɣ ɩɟɪɟɦɟɧɧɨɣ ɧɚ ɦɚɝɧɢɬɧɭɸ

ɞɥɢɧɭ, ɢ ɜɜɨɞɢɦ ɛɟɡɪɚɡɦɟɪɧɵɣ ɩɚɪɚɦɟɬɪ , ɬɨɝɞɚ ɭɪɚɜɧɟɧɢɟ

(ɩɨɫɥɟ ɧɟɤɨɬɨɪɵɯ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ) ɩɪɢɧɢɦɚɟɬ ɜɢɞ:

JU[

.2/4/

,0]

[][][

RRR

JE O

[



[

O[ww[ww[

ɇɟɫɥɨɠɧɨ ɜɢɞɟɬɶ ɩɪɹɦɨ ɢɡ ɩɨɫɥɟɞɧɟɝɨ ɭɪɚɜɧɟɧɢɹ ɚɫɢɦɩɬɨɬɵ ɪɟɲɟɧɢɹ:



fo

.0;

;];2/exp[

~)(

2/||

[[

[

ɄȼȺɇɌɈȼɕȿ ɗɎɎȿɄɌɕ ɏɈɅɅȺ

_________________________________________________________________________________________________________________________________________________________________________________________________

ɂɳɟɦ ɪɟɲɟɧɢɟ ɜ ɜɢɞɟ:,ɝɞɟ ɧɚ ɮɭɧɤ-

ɰɢɸ W ɫɥɟɞɭɟɬ ɧɚɥɨɠɢɬɶ ɭɫɥɨɜɢɹ ɨɝɪɚɧɢɱɟɧɧɨɫɬɢ ɜ ɧɭɥɟ ɢ ɧɚ ɛɟɫɤɨɧɟɱ-

ɧɨɫɬɢ. ȼ ɢɬɨɝɟ ɩɨɫɥɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ ɩɨɥɭɱɚɟɦ ɨɤɨɧɱɚɬɟɥɶɧɨɟ ɭɪɚɜɧɟ-

ɧɢɟ:

2/||

]2/exp[)()(

[[[[



0]2/)1|(|[]1|[| O

[

[ WmWmW .

ɉɨɥɭɱɟɧɧɨɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ

 ɬɢɩɢɱɧɨɟ ɭɪɚɜɧɟɧɢɟ ɞɥɹ

ɜɵɪɨɠɞɟɧɧɨɣ ɝɢɩɟɪɝɟɨɦɟɬɪɢɱɟɫɤɨɣ ɮɭɧɤɰɢɢ (Ʉɭɦɦɟɪɚ):

).,1||],

1||

[()(

[



O [ m

Ɉɝɪɚɧɢɱɟɧɢɟ ɪɟɲɟɧɢɹ ɧɚ ɛɟɫɤɨɧɟɱɧɨɫɬɢ ɬɪɟɛɭɟɬ, ɱɬɨɛɵ ɜɵɩɨɥɧɹɥɨɫɶ

ɫɨɨɬɧɨɲɟɧɢɟ: n=

-(|m|+1)/2, n=0,1,2... , ɬɨɝɞɚ ɝɢɩɟɪɝɟɨɦɟɬɪɢɱɟɫɤɚɹ

ɮɭɧɤɰɢɹ ɩɪɟɜɪɚɳɚɟɬɫɹ ɜ ɤɨɧɟɱɧɵɣ ɩɨɥɢɧɨɦ. ɉɨɫɥɟɞɧɟɟ ɢ ɟɫɬɶ ɢɫɤɨɦɵɣ

ɫɩɟɤɬɪ, ɤɨɬɨɪɵɣ ɩɨɫɥɟ ɜɨɫɫɬɚɧɨɜɥɟɧɢɹ ɢɫɯɨɞɧɵɯ ɨɛɨɡɧɚɱɟɧɢɣ ɩɪɢɧɢ-

ɦɚɟɬ ɜɢɞ:

*2/)2/12/2/||()(

mkmmnkE

ZCZnm

!! 

Ɂɚɞɚɱɚ 3.2. ɉɨɥɭɱɢɬɶ ɜɨɥɧɨɜɵɟ ɮɭɧɤɰɢɢ ɜ ɩɟɪɜɨɣ ɡɨɧɟ Ʌɚɧɞɚɭ ɞɥɹ

ɷɥɟɤɬɪɨɧɚ ɜ ɩɥɨɫɤɨɣ ɝɟɨɦɟɬɪɢɢ.

Ɋɟɲɟɧɢɟ: ɍɱɢɬɵɜɚɹ ɬɨɱɧɨɟ ɪɟɲɟɧɢɟ ɜ ɜɢɞɟ ɜɵɪɨɠɞɟɧɧɨɣ ɝɢɩɟɪɝɟɨ-

ɦɟɬɪɢɱɟɫɤɨɣ ɮɭɧɤɰɢɢ (ɩɪɟɞɵɞɭɳɚɹ ɡɚɞɚɱɚ), ɞɥɹ ɩɟɪɜɨɣ ɡɨɧɵ Ʌɚɧɞɚɭ

(n=0) ɩɨɥɭɱɚɟɦ, ɱɬɨ ɮɭɧɤɰɢɹ

1),1||,0()(

[ [ mnFW ɬɨɱ-

ɧɨ, ɬ.ɟ. ɩɨɥɧɚɹ ɜɨɥɧɨɜɚɹ ɮɭɧɤɰɢɹ ɢɦɟɟɬ ɜɢɞ:

2/||

]2/exp[

]exp[)(),,(

imzik

imzikRz

[[M





ȼ ɩɥɨɫɤɨɣ ɝɟɨɦɟɬɪɢɢ k

=0, ɢ ɦɨɠɧɨ ɩɟɪɟɩɢɫɚɬɶ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ

(ɬɨɥɶɤɨ ɩɪɢ ɧɟɨɬɪɢɰɚɬɟɥɶɧɵɯ m ):

./)(;4/||exp

!22

)(

liyxzz

z  

ɋɈȼɊȿɆȿɇɇɕȿ ɉɊɈȻɅȿɆɕ ɎɂɁɂɄɂ ɌȼȿɊȾɈȽɈ ɌȿɅȺ

_________________________________________________________________________________________________________________________________________________________________________________________________

Ɉɤɨɧɱɚɬɟɥɶɧɨ, ɧɨɪɦɢɪɭɹ ɢɧɬɟɝɪɚɥ ɨɬ ɤɜɚɞɪɚɬɚ ɦɨɞɭɥɹ ɜɨɥɧɨɜɨɣ ɮɭɧɤ-

ɰɢɢ ɩɨ ɜɫɟɣ ɤɨɦɩɥɟɤɫɧɨɣ ɩɥɨɫɤɨɫɬɢ Z ɧɚ ɟɞɢɧɢɰɭ, ɢɦɟɟɦ:

4/||exp

!22

)(



Ɂɚɞɚɱɚ 3.3. ɉɨɤɚɡɚɬɶ, ɱɬɨ ɤɜɚɞɪɚɬ ɪɚɞɢɭɫɚ ɩɪɟɰɟɫɫɢɢ ɷɥɟɤɬɪɨɧɚ ɜ ɩɟɪ-

ɜɨɣ ɡɨɧɟ Ʌɚɧɞɚɭ, ɬ.ɟ. <x

>, ɞɥɹ m-ɝɨ ɨɪɛɢɬɚɥɶɧɨɝɨ ɦɨɦɟɧɬɚ ɪɚɜɟɧ

2(m+1)l

Ɂɚɞɚɱɚ 3.4. ɉɨɥɭɱɢɬɶ ɜɢɞ ɤɭɥɨɧɨɜɫɤɨɝɨ ɨɬɬɚɥɤɢɜɚɬɟɥɶɧɨɝɨ ɩɨɬɟɧɰɢɚɥɚ

ɞɥɹ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɡɚɪɹɞɨɜ e* ɜ ɱɢɫɬɨ ɞɜɭɦɟɪɧɨɣ ɫɢɬɭɚɰɢɢ.

Ɋɟɲɟɧɢɟ:

ɇɚ ɩɥɨɫɤɨɫɬɢ ɩɪɢɧɹɬɨ ɜɜɨɞɢɬɶ ɤɨɦɩɥɟɤɫɧɵɣ ɩɨɬɟɧɰɢɚɥ:

v(z)=

(x,y)-iA(x,y), z=x+iy . ɇɟɫɥɨɠɧɨ ɩɨɤɚɡɚɬɶ, ɱɬɨ

iEEdzzdv



 /)( .

ɉɭɫɬɶ ɞɜɭɦɟɪɧɵɣ ɡɚɪɹɞ e* ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɡɚɪɹɞ ɟɞɢɧɢɰɵ ɞɥɢɧɵ

ɛɟɫɤɨɧɟɱɧɨɣ ɡɚɪɹɠɟɧɧɨɣ ɩɪɹɦɨɣ ɧɢɬɢ (ɷɬɨ ɤɚɤ ɪɚɡ ɱɢɫɬɨ ɞɜɭɦɟɪɧɚɹ

ɫɢɬɭɚɰɢɹ), ɩɪɨɧɢɡɵɜɚɸɳɟɣ ɩɥɨɫɤɨɫɬɶ x,y. Ɍɨɝɞɚ ɧɚɩɪɹɠɟɧɧɨɫɬɶ ɩɨɥɹ ɜ

ɩɨɥɹɪɧɵɯ ɤɨɨɪɞɢɧɚɬɚɯ

UI

ɞɚɟɬɫɹ ɮɨɪɦɭɥɚɦɢ: E

=2e*/

, E

=0. Ⱦɢɮɮɟ-

ɪɟɧɰɢɪɭɹ ɢɯ, ɩɨɥɭɱɚɟɦ: v(z) = - 2e*

ln(z) = -2e*ln

U

-2ie*

. Ɍɨɝɞɚ ɞɥɹ

ɷɧɟɪɝɢɢ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɞɜɭɯ ɩɥɨɫɤɢɯ ɡɚɪɹɞɨɜ ɜ ɬɨɱɤɚɯ z

, z

(ɬɨɥɶɤɨ

ɞɟɣɫɬɜɢɬɟɥɶɧɚɹ ɱɚɫɬɶ) ɢɦɟɟɦ:

e* v(z

-z

) = - 2 e*

ln | z

- z

| .

Ɂɚɞɚɱɚ 3.5. ɉɨɤɚɡɚɬɶ, ɱɬɨ ɷɧɟɪɝɢɹ ɫɢɫɬɟɦɵ

)

, ɤɨɬɨɪɭɸ ɫɥɟɞɭɟɬ ɦɢɧɢ-

ɦɢɡɢɪɨɜɚɬɶ ɜ ɫɥɭɱɚɟ ɜɚɪɢɚɰɢɨɧɧɨɣ, ɩɪɨɛɧɨɣ ɜɨɥɧɨɜɨɣ ɮɭɧɤɰɢɢ

ɩɨɞɱɢɧɹɟɬɫɹ ɫɨɨɬɧɨɲɟɧɢɸ: ɩɪɢ .

]exp[||

)

)0( ofo T

Ɋɟɲɟɧɢɟ: ɫɥɟɞɭɟɬ ɜɫɩɨɦɧɢɬɶ, ɱɬɨ ɞɥɹ ɦɢɧɢɦɢɡɚɰɢɢ ɷɧɟɪɝɢɢ ɫɢɫɬɟ-

ɦɵ ɫ ɭɱɟɬɨɦ ɬɟɦɩɟɪɚɬɭɪɵ ɫɥɟɞɭɟɬ ɡɚɩɢɫɚɬɶ:

))! ) ]exp[]/1[

ɋ ɞɪɭɝɨɣ ɫɬɨɪɨɧɵ, ɞɥɹ ɜɚɪɢɚɰɢɨɧɧɨɣ ɮɭɧɤɰɢɢ ɩɪɢ ɬɟɦɩɟɪɚɬɭɪɟ, ɫɬɪɟ-

ɦɹɳɟɣɫɹ ɤ ɧɭɥɸ, ɧɟɨɛɯɨɞɢɦɨ ɦɢɧɢɦɢɡɢɪɨɜɚɬɶ ɜɵɪɚɠɟɧɢɟ:

)<<! )

]/1[

ɋɪɚɜɧɢɜɚɹ ɢɯ, ɞɨɤɚɡɵɜɚɟɦ ɢɫɤɨɦɨɟ ɫɨɨɬɧɨɲɟɧɢɟ.

ɄȼȺɇɌɈȼɕȿ ɗɎɎȿɄɌɕ ɏɈɅɅȺ

_________________________________________________________________________________________________________________________________________________________________________________________________

Ɂɚɞɚɱɚ 3.6. ɉɨɤɚɡɚɬɶ, ɱɬɨ ɩɨɫɥɟɞɧɢɣ ɱɥɟɧ ɜ ɤɥɚɫɫɢɱɟɫɤɨɦ ɞɟɣɫɬɜɢɢ

(1.3.7) ɟɫɬɶ ɜɡɚɢɦɨɞɟɣɫɬɜɢɟ ɱɚɫɬɢɰ ɡɚɪɹɞɨɦ e*=m ɫ ɨɞɧɨɪɨɞɧɵɦ ɧɟɣ-

ɬɪɚɥɢɡɭɸɳɢɦ ɮɨɧɨɦ ɫ ɩɨɜɟɪɯɧɨɫɬɧɨɣ ɩɥɨɬɧɨɫɬɶɸ

=1/2

Ɋɟɲɟɧɢɟ:

ɪɚɫɫɦɨɬɪɢɦ ɷɬɨ ɜɵɪɚɠɟɧɢɟ: 1)

zm .||

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(r) ɨɬ ɨɞɧɨɪɨɞɧɨɝɨ ɩɨɜɟɪɯ ɫ

ɫɬɶɸ ɡɚɪɹɞɚ

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ɟɫɬɶ

VSUVUSM

2)( rdr

. Ɍɨɝɞɚ ɷɧɟɪɝɢɹ

ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ

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rere

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VSM

ɋɪɚɜɧɢɜɚɹ ɩɨɫɥɟɞɧɟɟ ɫɨɨɬɧɨɲɟɧɢɟ ɫ 1), ɩɨɥɭɱɚɟɦ

, ɢ ɭɱɢɬɵɜɚɹ,

ɨɛɪɚɡɨɜɚɧɢɟ ɜɨɥɧɨɜɨɣ ɮɭɧɤɰɢɢ ɩɪɢ ɜɧɟɫɟɧɢɢ

=1/2

ɱɬɨ z ɨɛɟɡɪɚɡɦɟɪɟɧɨ ɧɚ ɦɚɝɧɢɬɧɭɸ ɞɥɢɧɭ, ɢɦɟɟɦ ɜ ɪɚɡɦɟɪɧɨɦ ɜɢɞɟ

ɨɤɨɧɱɚɬɟɥɶɧɨ:

=1/2

Ɂɚɞɚɱɚ 3.7. ɉɨɥɭɱɢɬɶ ɩɪɟ

ɜ ɫɢɫɬɟɦɭ ɡɚɪɹɠɟɧɧɵɯ ɱɚɫɬɢɰ ɬɪɭɛɨɤ ɦɚɝɧɢɬɧɨɝɨ ɩɨɬɨɤɚ ɫ ɞɪɨɛɧɵɦ

ɤɜɚɧɬɨɦ )=

)

(ɤɚɠɞɚɹ ɫɜɹɡɚɧɚ ɫ ɨɞɧɨɣ ɱɚɫɬɢɰɟɣ), ɜ ɞɜɭɦɟɪɧɨɣ ɝɟɨ-

ɦɟɬɪɢɢ (ɡɚɞɚɱɚ ɨɛ ɷɧɢɨɧɧɨɣ, ɞɪɨɛɧɨɣ ɫɬɚɬɢɫɬɢɤɟ).

Ɋɟɲɟɧɢɟ

: ȼ ɫɢɦɦɟɬɪɢɱɧɨɣ ɤɚɥɢɛɪɨɜɤɟ ɜɟɤɬɨɪɧɵɣ ɩɨɬɟɧɰɢɚɥ A

)

(r),

ɛɭɫɨ ɥɨɜɥɟɧɧɵɣ ɨɞɧɨɣ ɬɪɭɛɤɨɣ ɦɚɝɧɢɬɧɨɝɨ ɩɨɬɨɤɚ (ɬ.ɟ. ɛɟɫɤɨɧɟɱɧɵɦ

ɬɨɧɤɢɦ ɫɨɥɟɧɨɢɞɨɦ), ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ:

)



)

)(

irA

ɝɞɟ

I

- ɩɨɥɹɪɧɵɣ ɭɝɨɥ ɜ x,y - ɩɥɨɫɤɨɫɬɢ. ȼ ɫɢɫɬɟɦɟ ɦɧɨɝɢɯ ɱɚɫɬɢɰ ɧɚ

ɜɵɞɟɥɟɧɧɭɸ i-ɱɚɫɬɢɰɭ ɞɟɣɫɬɜɭɟɬ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɬɚɥɶɧɵɯ ɫɥɟɞɭɸɳɢɣ

ɩɨɬɟɧɰɢɚɥ:

)(

ijj

irA



)

¦¦

)

ɝɞɟ ɨɬɧɨɫɢɬɟɥɶɧɵɣ ɮɚɡɨɜɵɣ ɫɞɜɢɝ:

)()

jij

xxy(arctg

iij

y 

Ɋɚɫɫɦɨɬɪɢɦ Ƚɚɦɢɥɶɬɨɧɢɚɧ ɫɢɫɬɟɦɵ ɩɪɢ ɧɚɥɢɱɢɢ ɤɨɝɨ ɜɟɤɬɨɪɧɨɝɨɬɚ

ɩɨɬɟɧɰɢɚɥɚ:

Кашурников В.А., Маймистов А.И. Современные проблемы физики твердого тела. Часть1 - Целый и дробный квантовые эффекты Холла

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