Кашурников В.А., Маймистов А.И. Современные проблемы физики твердого тела. Часть1

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ɋɈȼɊȿɆȿɇɇɕȿ ɉɊɈȻɅȿɆɕ ɎɂɁɂɄɂ ɌȼȿɊȾɈȽɈ ɌȿɅȺ

_________________________________________________________________________________________________________________________________________________________________________________________________

ɩɨɥɟ ~1Ɍɥ ɥɚɪɦɨɪɨɜɚ ɱɚɫɬɨɬɚ

~ 10

c. Ɍɢɩɢɱɧɨɟ ɜɪɟɦɹ

ɪɚɫɫɟɹɧɢɹ ɧɚ ɩɪɢɦɟɫɹɯ

W

~ 10

-12

-10

-14

c. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɞɚɠɟ

ɜ ɛɨɥɶɲɢɯ ɩɨɥɹɯ (> ~ 1 Tɥ) ɪɟɚɥɢɡɭɟɬɫɹ ɪɟɠɢɦ “ɫɥɚɛɨɝɨ

ɩɨɥɹ” ɜ ɫɦɵɫɥɟ ɪɚɧɟɟ ɫɤɚɡɚɧɧɨɝɨ:

<<1 . Ɉɞɧɚɤɨ ɷɬɨ ɧɟ

ɦɟɲɚɟɬ ɧɚɛɥɸɞɚɬɶ ɷɮɮɟɤɬ ɏɨɥɥɚ, ɩɨɫɤɨɥɶɤɭ ɜɫɟ ɪɚɜɧɨ

ɦɨɠɧɨ ɫ ɜɵɫɨɤɨɣ ɬɨɱɧɨɫɬɶɸ ɢɡɦɟɪɹɬɶ ɩɨɩɟɪɟɱɧɭɸ ɗ.Ⱦ.ɋ.

ɏɨɥɥɚ, ɜɵɪɚɠɚɸɳɭɸ ɦɟɪɭ ɨɬɤɥɨɧɟɧɢɹ ɷɥɟɤɬɪɨɧɨɜ ɩɪɨɜɨ-

ɞɢɦɨɫɬɢ ɜ ɩɨɩɟɪɟɱɧɨɦ ɦɚɝɧɢɬɧɨɦɭ ɩɨɥɸ ɧɚɩɪɚɜɥɟɧɢɢ.

ȼ ɪɟɚɥɶɧɨɦ ɬɜɟɪɞɨɦ ɬɟɥɟ ɷɥɟɤɬɪɨɧ ɩɪɨɜɨɞɢɦɨɫɬɢ ɧɟ ɹɜ-

ɥɹɟɬɫɹ ɫɜɨɛɨɞɧɨɣ ɱɚɫɬɢɰɟɣ, ɢ ɟɝɨ ɦɚɫɫɚ ɜ ɨɛɳɟɦ ɫɥɭɱɚɟ ɨɬ-

ɥɢɱɧɚ ɨɬ ɦɚɫɫɵ ɫɜɨɛɨɞɧɨɝɨ ɷɥɟɤɬɪɨɧɚ m (ɩɨ ɨɩɪɟɞɟɥɟɧɢɸ

ɬɟɧɡɨɪ ɷɮɮɟɤɬɢɜɧɵɯ ɦɚɫɫ m

DE

ɞɥɹ ɷɥɟɤɬɪɨɧɚ ɫ ɷɧɟɪɝɢɟɣ E(k)

ɢɦɟɟɬ ɜɢɞ m



1 !

] d

E(k)/dk

= x,y,z), ɨɞɧɚɤɨ ɜ

ɤɜɚɡɢɤɥɚɫɫɢɱɟɫɤɨɦ ɩɪɢɛɥɢɠɟɧɢɢ ɩɪɢ ɭɫɥɨɜɢɢ ɦɟɞɥɟɧɧɨɫɬɢ

ɢɡɦɟɧɟɧɢɹ ɜɧɟɲɧɢɯ ɩɨɥɟɣ ɢ ɨɬɫɭɬɫɬɜɢɢ ɷɥɟɤɬɪɢɱɟɫɤɨɝɨ ɢ

ɦɚɝɧɢɬɧɨɝɨ ɩɪɨɛɨɹ ɫɨɯɪɚɧɹɟɬɫɹ ɫɨɨɬɧɨɲɟɧɢɟ (1.1) (ɬɨɥɶɤɨ

ɭɠɟ ɧɟ ɞɥɹ ɪɟɚɥɶɧɨɝɨ ɢɦɩɭɥɶɫɚ, ɚ ɞɥɹ ɤɜɚɡɢɢɦɩɭɥɶɫɚ k ɛɥɨ-

ɯɨɜɫɤɨɝɨ ɷɥɟɤɬɪɨɧɚ  ɷɥɟɤɬɪɨɧɚ ɜ ɡɨɧɧɨɣ ɫɯɟɦɟ  ɷɬɢ ɫɨɨɬ-

ɧɨɲɟɧɢɹ ɦɨɠɧɨ ɞɨɤɚɡɚɬɶ  ɫɦ. ɡɚɞɚɱɭ ɜ ɤɨɧɰɟ ɩɚɪɚɝɪɚɮɚ).

Ɉɛɨɛɳɟɧɢɟ ɬɟɧɡɨɪɚ ɦɚɝɧɢɬɨɫɨɩɪɨɬɢɜɥɟɧɢɹ ɧɚ ɫɥɭɱɚɣ ɡɨɧ-

ɧɨɝɨ ɷɥɟɤɬɪɨɧɚ ɭɠɟ ɧɟ ɬɚɤ ɬɪɢɜɢɚɥɶɧɨ. Ɍɚɤ, ɪɚɫɫɦɨɬɪɢɦ

ɫɥɟɞɭɸɳɭɸ ɦɨɞɟɥɶɧɭɸ ɡɚɞɚɱɭ. ɉɭɫɬɶ ɢɦɟɟɬɫɹ ɞɜɭɯɡɨɧɧɚɹ

ɫɢɫɬɟɦɚ, ɬ.ɟ. ɷɥɟɤɬɪɨɧɵ ɜ ɤɚɠɞɨɣ ɫɢɫɬɟɦɟ ɯɚɪɚɤɬɟɪɢɡɭɸɬɫɹ

ɫɜɨɢɦ ɬɟɧɡɨɪɨɦ ɦɚɝɧɢɬɨɫɨɩɪɨɬɢɜɥɟɧɢɹ.





; i=1,2. (1.4)

ɋɨɝɥɚɫɧɨ ɫɨɨɬɧɨɲɟɧɢɹɦ

iiii

mne /1

ɢ R

=1/n

c ɦɵ

ɬɟɦ ɫɚɦɵɦ ɡɚɞɚɟɦ ɤɨɧɰɟɧɬɪɚɰɢɸ, ɡɚɪɹɞ ɢ ɷɮɮɟɤɬɢɜɧɭɸ

ɦɚɫɫɭ ɧɨɫɢɬɟɥɟɣ ɡɚɪɹɞɚ ɨɬɞɟɥɶɧɨ ɩɨ ɡɨɧɚɦ (ɩɨɥɨɠɢɦ ɞɥɹ

ɩɪɨɫɬɨɬɵ ɜɪɟɦɹ ɪɚɫɫɟɹɧɢɹ ɨɞɢɧɚɤɨɜɵɦ ɞɥɹ ɨɛɟɢɯ ɡɨɧ). ɉɨ-

ɥɭɱɢɦ ɦɚɬɪɢɰɭ ɦɚɝɧɢɬɨɫɨɩɪɨɬɢɜɥɟɧɢɹ ɫɢɫɬɟɦɵ, ɫɭɦɦɢɪɭɹ

ɄȼȺɇɌɈȼɕȿ ɗɎɎȿɄɌɕ ɏɈɅɅȺ

_________________________________________________________________________________________________________________________________________________________________________________________________

ɬɨɤɢ ɩɨ ɡɨɧɚɦ (ɩɨɞɪɨɛɧɟɟ ɫɦ. ɡɚɞɚɱɢ ɜ ɤɨɧɰɟ ɩɚɪɚɝɪɚɮɚ). ȼ

ɢɬɨɝɟ ɜ ɩɪɟɞɟɥɟ ɫɢɥɶɧɨɝɨ ɢ ɫɥɚɛɨɝɨ ɩɨɥɹ ɧɚɯɨɞɢɦ:

ɚ) (1.5)

2121

)/()(

);/(

:)1(0

UUUU

UUUUU





o

RRR

ɛ)

)/(

);/()(

:)1(

2121

RRRRR

RRRR





!!fo

UUU

(1.6)

ȼɨ ɜɬɨɪɨɦ ɫɥɭɱɚɟ, ɜ ɫɢɥɶɧɵɯ ɩɨɥɹɯ, ɦɨɠɧɨ ɩɪɟɨɛɪɚɡɨɜɚɬɶ

ɩɨɫɬɨɹɧɧɭɸ ɏɨɥɥɚ:

RRRRR R R

nec nec



12 1 2 1 2

11 2 2

11 1

/( ) /( / / )

/( )

(1.7)

Ɍɚɤ, ɟɫɥɢ ɦɵ ɢɦɟɟɦ ɫɨɛɫɬɜɟɧɧɵɣ ɩɨɥɭɩɪɨɜɨɞɧɢɤ, ɚɞɜɟɡɨ-

ɧɵ  ɷɬɨ ɡɨɧɚ ɩɪɨɜɨɞɢɦɨɫɬɢ ɢ ɜɚɥɟɧɬɧɚɹ ɡɨɧɚ, ɬɨ

. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɜ ɞɨɫɬɚɬɨɱɧɨ ɫɢɥɶɧɵɯ

ɩɨɥɹɯ, ɤɨɝɞɚ ɡɨɧɵ ɞɚɸɬ ɚɞɞɢɬɢɜɧɵɣ ɜɤɥɚɞ ɜ ɩɪɨɜɨɞɢɦɨɫɬɶ,

ɦɨɠɧɨ ɢɡɦɟɪɹɬɶ ɫɭɦɦɚɪɧɭɸ ɤɨɧɰɟɧɬɪɚɰɢɸ ɧɨɫɢɬɟɥɟɣ ɡɚɪɹ-

ɞɚ, ɧɟɫɦɨɬɪɹ ɧɚ ɫɥɨɠɧɭɸ ɷɧɟɪɝɟɬɢɱɟɫɤɭɸ ɫɬɪɭɤɬɭɪɭ. Ɍɨ

ɟɫɬɶ, ɧɟɨɛɯɨɞɢɦɨ, ɱɬɨɛɵ ɜ ɫɢɥɶɧɵɯ ɩɨɥɹɯ ɜɤɥɚɞ ɪɚɡɥɢɱɧɵɯ

ɡɨɧ ɪɚɫɰɟɩɥɹɥɫɹ, ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɤɨɷɮɮɢɰɢɟɧ-

ɬɚ ɏɨɥɥɚ ɨɫɬɚɟɬɫɹ ɬɚɤɢɦ ɠɟ ɭɧɢɜɟɪɫɚɥɶɧɵɦ, ɩɨɡɜɨɥɹɹ ɩɨɥɭ-

ɱɚɬɶ ɢɧɮɨɪɦɚɰɢɸ ɨ ɫɭɦɦɚɪɧɨɣ ɤɨɧɰɟɧɬɪɚɰɢɢ ɧɨɫɢɬɟɥɟɣ ɢ

ɡɧɚɤɟ ɡɚɪɹɞɚ. Ʉɪɨɦɟ ɬɨɝɨ, ɥɢɧɟɣɧɚɹ ɡɚɜɢɫɢɦɨɫɬɶ ɩɨɩɟɪɟɱɧɨɣ

ɤɨɦɩɨɧɟɧɬɵ

Recnn

1/| | ( )

(ɯɨɥɥɨɜɫɤɨɝɨ ɫɨɩɪɨɬɢɜɥɟɧɢɹ) ɨɬ ɦɚɝɧɢɬ-

ɧɨɝɨ ɩɨɥɹ ɜ ɲɢɪɨɤɨɦ ɢɧɬɟɪɜɚɥɟ ɬɟɦɩɟɪɚɬɭɪ ɢ ɩɨɥɟɣ ɞɥɹ

ɦɧɨɝɢɯ ɨɞɧɨɡɨɧɧɵɯ ɜɟɳɟɫɬɜ ɩɨɡɜɨɥɹɟɬ ɤɨɧɫɬɪɭɢɪɨɜɚɬɶ ɞɚɬ-

ɱɢɤɢ ɏɨɥɥɚ  ɢɡɦɟɪɢɬɟɥɢ ɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ, ɨɛɥɚɞɚɸɳɢɟ

ɜɵɫɨɤɨɣ ɱɭɜɫɬɜɢɬɟɥɶɧɨɫɬɶɸ ɢ ɦɚɥɵɦɢ ɪɚɡɦɟɪɚɦɢ. Ʉɚɤ ɜɢɞ-

ɧɨ ɢɡ ɚɧɚɥɢɡɚ, ɫɚɦɨ ɭɫɥɨɜɢɟ “ɫɢɥɶɧɨɝɨ ɩɨɥɹ” ɫɨɨɬɜɟɬɫɬɜɭɟɬ

ɋɈȼɊȿɆȿɇɇɕȿ ɉɊɈȻɅȿɆɕ ɎɂɁɂɄɂ ɌȼȿɊȾɈȽɈ ɌȿɅȺ

_________________________________________________________________________________________________________________________________________________________________________________________________

ɫɥɭɱɚɸ ɚɞɞɢɬɢɜɧɨɝɨ ɜɤɥɚɞɚ ɤɚɠɞɨɣ ɡɨɧɵ ɜ ɯɨɥɥɨɜɫɤɢɣ ɬɨɤ

ɢ ɩɪɟɧɟɛɪɟɠɢɦɨ ɦɚɥɨɣ ɢɧɬɟɪɮɟɪɟɧɰɢɢ ɢɯ ɦɟɠɞɭ ɫɨɛɨɣ. Ʉɚɤ

ɩɪɚɜɢɥɨ, ɩɟɪɟɧɨɪɦɢɪɨɜɤɚ ɷɮɮɟɤɬɢɜɧɨɣ ɦɚɫɫɵ (ɚ ɨɧɚ ɡɞɟɫɶ 

ɰɢɤɥɨɬɪɨɧɧɚɹ, ɜ ɤɨɬɨɪɭɸ ɞɚɟɬ ɜɤɥɚɞ ɬɨɥɶɤɨ ɱɚɫɬɶ ɬɟɧɡɨɪɚ

ɷɮɮɟɤɬɢɜɧɨɣ ɦɚɫɫɵ, ɩɨɩɟɪɟɱɧɚɹ ɦɚɝɧɢɬɧɨɦɭ ɩɨɥɸ - m

, m

) ɢ ɜɪɟɦɟɧɢ ɪɚɫɫɟɹɧɢɹ ɩɪɢɜɨɞɢɬ ɤ ɬɨɦɭ, ɱɬɨ

ɪɟɠɢɦ “ɫɢɥɶɧɨɝɨ” ɩɨɥɹ ɜɩɨɥɧɟ ɞɨɫɬɢɠɢɦ ɞɥɹ ɪɟɚɥɶɧɵɯ ɩɨ-

ɥɟɣ ~ 1 Ɍɥ ɢ ɜ ɪɟɚɥɶɧɵɯ ɬɜɟɪɞɵɯ ɬɟɥɚɯ.

Ɂɚɦɟɬɢɦ, ɱɬɨ ɛɨɥɟɟ ɫɥɨɠɧɵɟ ɩɨɞɯɨɞɵ ɤ ɨɩɢɫɚɧɢɸ ɷɮ-

ɮɟɤɬɚ ɏɨɥɥɚ ɭɱɢɬɵɜɚɸɬ ɷɧɟɪɝɟɬɢɱɟɫɤɭɸ ɡɚɜɢɫɢɦɨɫɬɶ ɜɪɟ-

ɦɟɧɢ ɪɟɥɚɤɫɚɰɢɢ, ɱɬɨ ɬɚɤɠɟ ɩɪɢɜɨɞɢɬ ɤ ɩɨɥɟɜɨɣ ɡɚɜɢɫɢɦɨ-

ɫɬɢ ɦɚɝɧɢɬɨɫɨɩɪɨɬɢɜɥɟɧɢɹ ɢ ɗ.Ⱦ.ɋ. ɏɨɥɥɚ ɞɚɠɟ ɜ ɨɞɧɨɡɨɧ-

ɧɨɦ ɫɥɭɱɚɟ.

ȼ ɡɚɤɥɸɱɟɧɢɟ ɤɪɚɬɤɨɝɨ ɨɩɢɫɚɧɢɹ ɨɛɵɱɧɨɝɨ (ɬɪɚɞɢɰɢɨɧ-

ɧɨɝɨ) ɷɮɮɟɤɬɚ ɏɨɥɥɚ ɨɬɦɟɬɢɦ, ɱɬɨ ɩɨɩɟɪɟɱɧɵɣ ɪɚɡɦɟɪ ɨɛ-

ɪɚɡɰɚ, ɜ ɤɨɬɨɪɨɦ ɧɚɛɥɸɞɚɟɬɫɹ ɷɬɚ ɯɨɥɥɨɜɫɤɚɹ ɗ.Ⱦ.ɋ., ɧɟ

ɤɪɢɬɢɱɟɧ ɞɥɹ ɷɮɮɟɤɬɚ, ɥɢɲɶ ɛɵ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɨɤɚ ɩɨ ɫɟ-

ɱɟɧɢɸ ɛɵɥɨ ɨɞɧɨɪɨɞɧɵɦ. ȿɫɥɢ ɠɟ ɦɵ ɯɨɬɢɦ ɪɟɚɥɢɡɨɜɚɬɶ

ɢɫɬɢɧɧɨ ɞɜɭɦɟɪɧɭɸ ɝɟɨɦɟɬɪɢɸ ɷɤɫɩɟɪɢɦɟɧɬɚ (ɪɢɫ.1.1), ɫɥɟ-

ɞɭɟɬ ɩɪɢɦɟɧɹɬɶ ɮɨɥɶɝɢ ɢ ɩɥɟɧɤɢ ɫ ɬɚɤɢɦ ɩɨɩɟɪɟɱɧɵɦ ɪɚɡ-

ɦɟɪɨɦ, ɱɬɨɛɵ ɨɧ ɛɵɥ ɩɨɪɹɞɤɚ ɢ ɦɟɧɶɲɟ ɯɚɪɚɤɬɟɪɧɨɝɨ ɪɚ-

ɞɢɭɫɚ ɜɪɚɳɟɧɢɹ ɷɥɟɤɬɪɨɧɚ r

ɜ ɦɚɝɧɢɬɧɨɦ ɩɨɥɟ (ɥɚɪɦɨɪɨɜɚ

ɪɚɞɢɭɫɚ). Ɇɨɠɧɨ ɨɰɟɧɢɬɶ, ɱɬɨ ɩɪɢ ɩɨɥɟ H ~ 1-10Ɍɥ ɢɦɟɟɦ

~ 10

-3

- 10

-4

ɫɦ ( ~ 10

mnV

/)3(

3/12

ɫɦ/ɫ 

ɫɤɨɪɨɫɬɶ Ɏɟɪɦɢ, ɯɚɪɚɤɬɟɪɧɚɹ ɫɤɨɪɨɫɬɶ ɷɥɟɤɬɪɨɧɨɜ). Ɍɚɤɨɣ

ɩɨɩɟɪɟɱɧɵɣ ɪɚɡɦɟɪ ɨɛɪɚɡɰɚ ɜɩɨɥɧɟ ɞɨɫɬɢɠɢɦ. Ɉɞɧɚɤɨ ɩɪɢ

ɛɨɥɟɟ ɫɢɥɶɧɵɯ ɩɨɥɹɯ ɬɚɤɚɹ ɩɨɥɭɤɥɚɫɫɢɱɟɫɤɚɹ ɨɰɟɧɤɚ ɫɬɚɧɨ-

ɜɢɬɫɹ ɧɟ ɩɪɚɜɢɥɶɧɨɣ. Ȼɨɥɟɟ ɫɬɪɨɝɨɟ ɤɜɚɧɬɨɜɨɦɟɯɚɧɢɱɟɫɤɨɟ

ɨɩɢɫɚɧɢɟ ɞɚɟɬ ɞɪɭɝɨɣ ɩɚɪɚɦɟɬɪ ɞɥɹ ɨɛɥɚɫɬɢ ɥɨɤɚɥɢɡɚɰɢɢ

ɷɥɟɤɬɪɨɧɚ ɜ ɦɚɝɧɢɬɧɨɦ ɩɨɥɟ  l

(ɧɚɡɵɜɚɟɦɨɝɨ ɦɚɝɧɢɬɧɨɣ

ɞɥɢɧɨɣ). Ɍɨ ɟɫɬɶ, ɜɨɥɧɨɜɚɹ ɮɭɧɤɰɢɹ ɷɥɟɤɬɪɨɧɚ ɜ ɩɨɩɟɪɟɱ-

ɧɨɦ ɦɚɝɧɢɬɧɨɦɭ ɩɨɥɸ ɧɚɩɪɚɜɥɟɧɢɢ ɭɛɵɜɚɟɬ ɤɚɤ

, ɝɞɟ l

)2/exp(

lr

ɨɩɪɟɞɟɥɹɟɬɫɹ ɫɥɟɞɭɸɳɢɦ ɜɵɪɚɠɟɧɢ-

ɟɦ:

ɄȼȺɇɌɈȼɕȿ ɗɎɎȿɄɌɕ ɏɈɅɅȺ

_________________________________________________________________________________________________________________________________________________________________________________________________

! eH , (1.8)

ɱɬɨ ɞɥɹ ɩɨɥɹ H ~ 5

÷ 10 Ɍɥ ɩɪɢɜɨɞɢɬ ɤ l

~ 100 ɚɧɝɫɬɪɟɦ.

ɋɬɪɭɤɬɭɪɵ ɫ ɩɨɩɟɪɟɱɧɵɦ ɩɪɨɮɢɥɟɦ ɬɚɤɨɝɨ ɦɚɫɲɬɚɛɚ (ɧɚɧɨ-

ɫɬɪɭɤɬɭɪɵ) ɭɞɚɟɬɫɹ ɫɞɟɥɚɬɶ ɧɚ ɛɚɡɟ ɝɟɬɟɪɨɫɬɪɭɤɬɭɪ GaAs -

1-x

As, ɩɪɟɞɫɬɚɜɥɹɸɳɢɯ ɫɨɛɨɣ ɫɢɫɬɟɦɭ ɩɥɨɫɤɢɯ ɤɜɚɧ-

ɬɨɜɵɯ ɹɦ. Ɍɚɤɢɟ ɫɬɪɭɤɬɭɪɵ ɢɡɝɨɬɨɜɥɹɸɬ ɩɨɫɥɨɣɧɵɦ ɧɚɩɵ-

ɥɟɧɢɟɦ ɧɚ ɩɨɞɥɨɠɤɭ ɩɨɥɭɩɪɨɜɨɞɧɢɤɚ GaAs, ɱɟɪɟɞɭɹ ɟɝɨ ɫɨ

ɫɥɨɹɦɢ ɬɨɝɨ ɠɟ ɩɨɥɭɩɪɨɜɨɞɧɢɤɚ, ɞɨɩɢɪɨɜɚɧɧɨɝɨ ɚɥɸɦɢɧɢ-

ɟɦ. Ɍɪɟɯɜɚɥɟɧɬɧɵɣ Al, ɡɚɦɟɳɚɸɳɢɣ ɬɪɟɯɜɚɥɟɧɬɧɵɣ Ga, ɧɟ

ɦɟɧɹɟɬ ɤɨɧɰɟɧɬɪɚɰɢɸ ɷɥɟɤɬɪɨɧɨɜ, ɭɱɚɫɬɜɭɸɳɢɯ ɜ ɩɪɨɜɨ-

ɞɢɦɨɫɬɢ, ɨɞɧɚɤɨ ɢɦɟɟɬ ɧɟɫɤɨɥɶɤɨ ɢɧɨɣ ɢɨɧɧɵɣ ɪɚɞɢɭɫ ɢ

ɫɢɥɭ ɫɜɹɡɢ ɫ As, ɢ ɩɪɢɜɨɞɢɬ ɤ ɞɪɭɝɨɦɭ ɯɢɦɢɱɟɫɤɨɦɭ ɩɨɬɟɧ-

ɰɢɚɥɭ ɢ ɞɧɭ ɡɨɧɵ ɩɪɨɜɨɞɢɦɨɫɬɢ ɞɥɹ ɷɥɟɤɬɪɨɧɨɜ. ȼɪɟɡɭɥɶ-

ɬɚɬɟ ɞɨɩɢɪɨɜɚɧɧɵɟ ɚɥɸɦɢɧɢɟɦ ɨɛɥɚɫɬɢ ɫɬɚɧɨɜɹɬɫɹ ɷɧɟɪɝɟ-

ɬɢɱɟɫɤɢɦɢ ɛɚɪɶɟɪɚɦɢ ɞɥɹ ɷɥɟɤɬɪɨɧɧɨɝɨ ɝɚɡɚ, ɢ ɨɧ ɪɚɫɩɨɥɚ-

ɝɚɟɬɫɹ ɦɟɠɞɭ ɧɢɦɢ ɜ ɩɥɨɫɤɢɯ ɨɛɥɚɫɬɹɯ ɱɢɫɬɨɝɨ GaAs ɫɩɨ-

ɩɟɪɟɱɧɵɦ ɪɚɡɦɟɪɨɦ ɩɨɪɹɞɤɚ ɫɨɬɧɢ ɚɧɝɫɬɪɟɦ. Ɍɚɤɢɦ ɨɛɪɚ-

ɡɨɦ ɢ ɪɟɚɥɢɡɭɟɬɫɹ ɞɜɭɦɟɪɧɵɣ ɷɥɟɤɬɪɨɧɧɵɣ ɝɚɡ. Ɉɞɧɚɤɨ ɨɬ-

ɫɸɞɚ ɟɳɟ ɧɟ ɫɥɟɞɭɟɬ, ɱɬɨ ɞɥɹ ɷɥɟɤɬɪɨɧɧɨɝɨ ɝɚɡɚ ɨɬɫɭɬɫɬɜɭɟɬ

ɩɨɩɟɪɟɱɧɚɹ ɫɨɫɬɚɜɥɹɸɳɚɹ ɢɦɩɭɥɶɫɚ p

, ɢ ɷɥɟɤɬɪɨɧɵ ɞɜɭ-

ɦɟɪɧɵ ɜ ɮɢɡɢɱɟɫɤɨɦ ɫɦɵɫɥɟ. Ⱦɥɹ ɷɬɨɝɨ ɧɟɨɛɯɨɞɢɦɨ "ɜɤɥɸ-

ɱɢɬɶ" ɫɢɥɶɧɨɟ ɦɚɝɧɢɬɧɨɟ ɩɨɥɟ, ɱɬɨ ɢ ɩɪɢɜɨɞɢɬ, ɤɚɤ ɦɵ ɧɢɠɟ

ɩɨɤɚɠɟɦ, ɤ ɢɫɬɢɧɧɨɣ ɞɜɭɦɟɪɧɨɣ ɫɢɬɭɚɰɢɢ.

Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ ɬɨɝɨ ɮɚɤɬɚ, ɱɬɨ ɦɚɝɧɢɬɧɚɹ ɞɥɢɧɚ ɢ ɟɫɬɶ

ɢɫɤɨɦɵɣ ɯɚɪɚɤɬɟɪɧɵɣ ɦɚɫɲɬɚɛ ɜ ɩɪɨɞɨɥɶɧɨɦ ɩɨɥɸ ɧɚɩɪɚɜ-

ɥɟɧɢɢ ɜɵɜɟɞɟɦ ɢɡ ɫɥɟɞɭɸɳɢɯ ɫɨɨɛɪɚɠɟɧɢɣ. Ʉɚɤ ɢɡɜɟɫɬɧɨ (ɢ

ɩɨɡɞɧɟɟ ɦɵ ɷɬɨ ɨɛɫɭɞɢɦ), ɜ ɩɨɩɟɪɟɱɧɨɦ ɦɚɝɧɢɬɧɨɦ ɩɨɥɟ

ɞɜɢɠɟɧɢɟ ɷɥɟɤɬɪɨɧɚ ɩɪɨɜɨɞɢɦɨɫɬɢ ɮɢɧɢɬɧɨ. ɗɬɨ ɩɪɢɜɨɞɢɬ

ɤ ɞɨɩɨɥɧɢɬɟɥɶɧɨɦɭ ɤɜɚɧɬɨɜɚɧɢɸ ɟɝɨ ɷɧɟɪɝɢɢ ɩɨ ɬɚɤ ɧɚɡɵ-

ɜɚɟɦɵɦ ɭɪɨɜɧɹɦ Ʌɚɧɞɚɭ, ɬɚɤ ɱɬɨ ɟɝɨ ɷɧɟɪɝɢɹ ɤɚɤ ɮɭɧɤɰɢɹ

ɩɪɨɞɨɥɶɧɨɝɨ ɢɦɩɭɥɶɫɚ ɢ ɧɨɦɟɪɚ ɭɪɨɜɧɹ Ʌɚɧɞɚɭ ɢɦɟɟɬ ɜɢɞ:

ɋɈȼɊȿɆȿɇɇɕȿ ɉɊɈȻɅȿɆɕ ɎɂɁɂɄɂ ɌȼȿɊȾɈȽɈ ɌȿɅȺ

_________________________________________________________________________________________________________________________________________________________________________________________________

cZZj

jmkkE

!! )2/1(2/)(

 , j = 0, 1, 2,...

ȼ ɝɟɨɦɟɬɪɢɢ ɩɟɪɢɨɞɢɱɟɫɤɨɝɨ ɬɪɟɯɦɟɪɧɨɝɨ “ɹɳɢɤɚ” L

x L

d ɩɪɨɞɨɥɶɧɵɣ ɢɦɩɭɥɶɫ ɬɚɤɠɟ ɜ ɫɜɨɸ ɨɱɟɪɟɞɶ ɤɜɚɧɬɭɟɬɫɹ: k

= [2

/d]n, n = 0,1,2.. ȿɫɥɢ ɩɨɩɟɪɟɱɧɵɣ ɪɚɡɦɟɪ ɫɬɪɭɤɬɭɪɵ d

ɦɚɥ, ɬɨ ɪɚɫɫɬɨɹɧɢɟ ɦɟɠɞɭ ɭɪɨɜɧɹɦɢ ɷɧɟɪɝɢɢ ɫ ɪɚɡɥɢɱɧɵɦ

ɢɦɩɭɥɶɫɨɦ k

(ɬɚɤ ɧɚɡɵɜɚɟɦɵɦɢ ɭɪɨɜɧɹɦɢ ɪɚɡɦɟɪɧɨɝɨ ɤɜɚɧ-

ɬɨɜɚɧɢɹ) ɜɟɥɢɤɨ. Ⱦɥɹ ɢɫɬɢɧɧɨɝɨ ɞɜɭɦɟɪɧɨɝɨ ɫɨɫɬɨɹɧɢɹ

ɫɢɫɬɟɦɵ ɧɟɨɛɯɨɞɢɦɨ, ɱɬɨɛɵ ɜɫɟ ɞɪɭɝɢɟ ɯɚɪɚɤɬɟɪɧɵɟ ɷɧɟɪ-

ɝɢɢ ɛɵɥɢ ɦɧɨɝɨ ɦɟɧɶɲɟ, ɱɟɦ ɪɚɫɫɬɨɹɧɢɟ ɦɟɠɞɭ ɭɪɨɜɧɹɦɢ

ɪɚɡɦɟɪɧɨɝɨ ɤɜɚɧɬɨɜɚɧɢɹ, ɬɨɝɞɚ ɮɚɤɬɢɱɟɫɤɢ ɜɫɟɝɞɚ ɛɭɞɟɬ

ɪɟɚɥɢɡɨɜɚɧ ɧɭɥɟɜɨɣ ɩɨɩɟɪɟɱɧɵɣ ɢɦɩɭɥɶɫ. Ɇɚɬɟɦɚɬɢɱɟɫɤɢ

ɫɥɟɞɭɟɬ ɡɚɩɢɫɚɬɶ ɭɫɥɨɜɢɟ:,ɨɬɤɭɞɚ ɧɟɦɟɞ-

ɥɟɧɧɨ ɫɥɟɞɭɟɬ ɢɫɤɨɦɨɟ ɫɨɨɬɧɨɲɟɧɢɟ : d < < l

!! !!2/

. Ɍɚɤɢɦ ɨɛɪɚ-

ɡɨɦ, ɟɫɥɢ ɦɵ ɜɵɛɢɪɚɟɦ ɤɜɚɧɬɨɜɭɸ ɹɦɭ ɫ ɬɚɤɢɦ ɩɨɩɟɪɟɱɧɵɦ

ɪɚɡɦɟɪɨɦ, ɬɨ ɷɥɟɤɬɪɨɧɧɵɣ ɝɚɡ ɜ ɧɟɣ ɛɭɞɟɬ ɜɫɟɝɞɚ ɧɚɯɨɞɢɬɶ-

ɫɹ ɧɚ ɧɭɥɟɜɨɦ ɭɪɨɜɧɟ ɩɨɩɟɪɟɱɧɨɝɨ ɪɚɡɦɟɪɧɨɝɨ ɤɜɚɧɬɨɜɚɧɢɹ,

ɬɨ ɟɫɬɶ ɮɢɡɢɱɟɫɤɢ ɛɭɞɟɬ ɪɟɚɥɢɡɨɜɚɧɚ ɞɜɭɦɟɪɧɚɹ ɫɢɬɭɚɰɢɹ.

Ɂɚɞɚɱɢ

Ɂɚɞɚɱɚ 1.1. Ⱦɨɤɚɡɚɬɶ, ɱɬɨ ɞɥɹ ɤɜɚɡɢɢɦɩɭɥɶɫɚ ɛɥɨɯɨɜɫɤɨɝɨ ɷɥɟɤɬɪɨɧɚ ɜɨ

ɜɧɟɲɧɟɦ ɷɥɟɤɬɪɢɱɟɫɤɨɦ ɩɨɥɟ ɫɩɪɚɜɟɞɥɢɜɨ ɭɪɚɜɧɟɧɢɹ ɞɜɢɠɟɧɢɹ:

Eedtkd

! / .

Ɋɟɲɟɧɢɟ:

ȼɚɠɧɨ ɨɬɦɟɬɢɬɶ, ɱɬɨ ɜ ɤɜɚɡɢɤɥɚɫɫɢɱɟɫɤɨɦ ɩɪɢɛɥɢɠɟɧɢɢ ɭɪɚɜɧɟɧɢɟ

ɞɜɢɠɟɧɢɹ ɷɥɟɤɬɪɨɧɚ ɫ ɤɜɚɡɢɢɦɩɭɥɶɫɨɦ

ɢɦɟɟɬ ɩɨɱɬɢ ɬɚɤɨɣ ɠɟ ɜɢɞ

ɤɚɤ ɢ ɜ ɬɟɨɪɢɢ Ⱦɪɭɞɟ, ɬɨɥɶɤɨ ɛɟɡ ɫɬɨɥɤɧɨɜɢɬɟɥɶɧɨɝɨ ɱɥɟɧɚ:



; n  ɧɨɦɟɪ ɡɨɧɵ.

[/ ] ()/1

wH w



ɝɪɭɩɩɨɜɚɹ ɫɤɨɪɨɫɬɶ. Ⱦɥɹ ɨɛɨɫɧɨɜɚɧɢɹ ɩɟɪɜɨɣ ɱɚɫɬɢ ɷɬɨɝɨ ɫɨɨɬɧɨɲɟɧɢɹ

ɢɫɯɨɞɢɦ ɢɡ ɬɨɝɨ, ɱɬɨ ɞɨɥɠɧɚ ɛɵɬɶ ɩɨɫɬɨɹɧɧɚ ɜɨ ɜɪɟɦɟɧɢ ɷɧɟɪɝɢɹ ɷɥɟɤ-

ɬɪɨɧɚ:

kert t() [()] []

const . Ɍɨɝɞɚ ɜɚɪɶɢɪɭɟɦ ɟɟ:

ɄȼȺɇɌɈȼɕȿ ɗɎɎȿɄɌɕ ɏɈɅɅȺ

_________________________________________________________________________________________________________________________________________________________________________________________________

.);(/

;0///)(

M ww

wwuMwwuwwH

Ekvtr

tretkkk

ɉɨɞɫɬɚɜɥɹɹ ɧɢɠɧɢɟ ɨɩɪɟɞɟɥɟɧɢɹ ɜ ɜɟɪɯɧɟɟ ɜɵɪɚɠɟɧɢɟ, ɞɨɤɚɡɵɜɚɟɦ

ɢɫɤɨɦɨɟ ɫɨɨɬɧɨɲɟɧɢɟ.

Ɂɚɞɚɱɚ 1.2. Ⱦɨɤɚɡɚɬɶ, ɱɬɨ ɞɥɹ ɤɜɚɡɢɢɦɩɭɥɶɫɚ ɛɥɨɯɨɜɫɤɨɝɨ ɷɥɟɤɬɪɨɧɚ ɜɨ

ɜɧɟɲɧɟɦ ɦɚɝɧɢɬɧɨɦ ɩɨɥɟ ɫɩɪɚɜɟɞɥɢɜɨ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ:

Hvcdtkd



(ɩɨɜɵɲɟɧɧɨɣ ɬɪɭɞɧɨɫɬɢ).

ɍɤɚɡɚɧɢɟ:

ɢɫɩɨɥɶɡɨɜɚɬɶ ɩɨɫɬɨɹɧɫɬɜɨ ɷɧɟɪɝɢɢ ɜɨ ɜɪɟɦɟɧɢ ɤɚɤ ɮɭɧɤ-

ɰɢɸ ɩɨɥɧɨɝɨ ɢɦɩɭɥɶɫɚ ɜ ɦɚɝɧɢɬɧɨɦ ɩɨɥɟ

PpecArt (/)(,)

Ɂɚɞɚɱɚ 1.3. ɂɦɟɟɬɫɹ ɞɜɭɯɡɨɧɧɚɹ ɫɢɫɬɟɦɚ, ɬ.ɟ. ɷɥɟɤɬɪɨɧɵ ɜ ɤɚɠɞɨɣ ɫɢɫ-

ɬɟɦɟ ɯɚɪɚɤɬɟɪɢɡɭɸɬɫɹ ɫɜɨɢɦ ɬɟɧɡɨɪɨɦ ɦɚɝɧɢɬɨɫɨɩɪɨɬɢɜɥɟɧɢɹ. ɇɚɣɬɢ

ɨɛɳɢɣ ɬɟɧɡɨɪ ɦɚɝɧɢɬɨɫɨɩɪɨɬɢɜɥɟɧɢɹ.

Ɉɬɜɟɬ ɢ ɭɤɚɡɚɧɢɟ

: ɟɫɥɢ ɢɫɯɨɞɧɨ ɢɦɟɥɢ ɩɨ ɤɚɠɞɨɣ ɡɨɧɟ ɦɚɬɪɢɰɭ

ɦɚɝɧɢɬɨɫɨɩɪɨɬɢɜɥɟɧɢɹ ; i=1,2, ɬɨ ɫɤɥɚɞɵɜɚɟɦ

ɫɭɦɦɚɪɧɵɣ ɬɨɤ (ɢ ɦɚɬɪɢɰɵ ɩɪɨɜɨɞɢɦɨɫɬɢ):





jExy

DDEDEE

VV DE V





[],,,;



12 1

;

Ɉɛɪɚɳɚɹ ɦɚɬɪɢɰɵ, ɦɵ ɩɨɥɭɱɚɟɦ ɢɬɨɝɨɜɭɸ ɦɚɬɪɢɰɭ ɦɚɝɧɢɬɨɫɨɩɪɨɬɢɜ-

ɥɟɧɢɹ:



;

 

;



[( )( )]/

[()

()()

VUVVVU

UUUUU U U









 

 

12 1 2 12

12 1 2

RRHA

R R R RRR RH A

ARRH

Ɂɚɞɚɱɚ 1.4. ɂɫɫɥɟɞɨɜɚɬɶ ɩɨɥɭɱɟɧɧɵɟ ɜ ɩɪɟɞɵɞɭɳɟɣ ɡɚɞɚɱɟ ɜɵɪɚɠɟɧɢɹ

ɩɪɢ ɦɚɥɵɯ ɢ ɛɨɥɶɲɢɯ H. ɋɞɟɥɚɬɶ ɜɵɜɨɞ ɨ ɜɨɡɦɨɠɧɨɫɬɢ ɢɡɦɟɪɟɧɢɹ ɤɨɧ-

ɰɟɧɬɪɚɰɢɢ ɧɨɫɢɬɟɥɟɣ ɬɨɤɚ ɩɨ ɩɨɫɬɨɹɧɧɨɣ ɏɨɥɥɚ.

Ɉɬɜɟɬ:

ɋɈȼɊȿɆȿɇɇɕȿ ɉɊɈȻɅȿɆɕ ɎɂɁɂɄɂ ɌȼȿɊȾɈȽɈ ɌȿɅȺ

_________________________________________________________________________________________________________________________________________________________________________________________________

Ɂɚɦɟɬɢɦ, ɱɬɨ ɭɫɥɨɜɢɟ (ɭɫɥɨɜɢɟ ɫɥɚɛɨɝɨ ɢ ɫɢɥɶ-

ɧɨɝɨ ɩɨɥɹ) ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɜ ɧɚɲɢɯ ɨɛɨɡɧɚɱɟɧɢɹɯ ɭɫɥɨɜɢɸ :

ZW ZW

 !!1( 1)

)

ii ii

RH RH/(/ !!11

ɚ)

2121

)/()(

);/(

:)1(0

UUUU

UUUUU





o

RRR

ɛ)

RRRRRRRRR

of !!

  

():

( )/( ); /( )

UU U

12 12 12

2. ɐɟɥɵɣ ɤɜɚɧɬɨɜɵɣ ɷɮɮɟɤɬ ɏɨɥɥɚ. ɂɫɬɨɪɢɹ ɨɬɤɪɵɬɢɹ.

Ɍɟɨɪɟɬɢɱɟɫɤɨɟ ɨɛɴɹɫɧɟɧɢɟ. Ʉɜɚɧɬɨɜɚɧɢɟ ɭɪɨɜɧɟɣ ɜ

ɦɚɝɧɢɬɧɨɦ ɩɨɥɟ (ɭɪɨɜɧɢ Ʌɚɧɞɚɭ)

ɄȼȺɇɌɈȼɕȿ ɗɎɎȿɄɌɕ ɏɈɅɅȺ

_________________________________________________________________________________________________________________________________________________________________________________________________

ȼ 1980 ɝɨɞɭ ɧɟɦɟɰɤɢɣ ɮɢɡɢɤ Ʉɥɚɭɫ ɮɨɧ Ʉɥɢɬɰɢɧɝ ɫ ɫɨɚɜɬɨ-

ɪɚɦɢ (Klaus von Klitzing, G.Dorda, M.Pepper ) ɜ 1980 ɝɨɞɭ

ɨɛɧɚɪɭɠɢɥɢ, ɱɬɨ ɜ ɫɢɥɶɧɵɯ ɩɨɥɹɯ (~5-10Ɍɥ) ɯɨɥɥɨɜɫɤɚɹ

ɩɪɨɜɨɞɢɦɨɫɬɶ ɤɜɚɧɬɭɟɬɫɹ. Ɉɧɢ ɢɫɫɥɟɞɨɜɚɥɢ ɝɟɬɟɪɨɫɬɪɭɤɬɭ-

ɪɵ ɧɚ ɨɫɧɨɜɟ GaAs ɜ ɫɢɥɶɧɵɯ ɩɨɩɟɪɟɱɧɵɯ ɦɚɝɧɢɬɧɵɯ ɩɨɥɹɯ

ɢ ɩɪɢ ɧɢɡɤɢɯ ɬɟɦɩɟɪɚɬɭɪɚɯ (ɜ ɠɢɞɤɨɦ ɝɟɥɢɢ ɩɪɢ T ~ 5K) ɢ

ɩɨɥɭɱɚɥɢ ɦɚɝɧɢɬɨɫɨɩɪɨɬɢɜɥɟɧɢɟ

ɢ ɯɨɥɥɨɜɫɤɨɟ ɫɨɩɪɨ-

ɬɢɜɥɟɧɢɟ

ɤɚɤ ɮɭɧɤɰɢɢ ɜɧɟɲɧɟɝɨ ɩɪɢɥɨɠɟɧɧɨɝɨ ɦɚɝɧɢɬ-

ɧɨɝɨ ɩɨɥɹ. Ⱥɤɤɭɪɚɬɧɵɟ ɢɡɦɟɪɟɧɢɹ ɩɪɢɜɟɥɢ ɤ ɩɨɪɚɡɢɬɟɥɶɧɨ-

ɦɭ ɪɟɡɭɥɶɬɚɬɭ: ɧɚ ɨɛɵɱɧɨɦ ɞɥɹ ɷɮɮɟɤɬɚ ɏɨɥɥɚ ɥɢɧɟɣɧɨɦ

ɯɨɞɟ ɡɚɜɢɫɢɦɨɫɬɢ

(H) ɛɵɥɢ ɨɛɧɚɪɭɠɟɧɵ ɫɬɭɩɟɧɶɤɢ

(ɪɢɫ.2.1), ɜɵɫɨɬɚ ɤɨɬɨɪɵɯ ɨɤɚɡɚɥɚɫɶ ɪɚɜɧɚ ,

=1,2,3... , ɫ ɨɬɧɨɫɢɬɟɥɶɧɨɣ ɩɨɝɪɟɲɧɨɫɬɶɸ ɧɟ ɯɭɠɟ 10

/2 e!

-7

! ɉɨ-

ɥɨɠɟɧɢɟ ɩɥɚɬɨ ɡɚɜɢɫɟɥɨ ɬɨɥɶɤɨ ɨɬ ɮɭɧɞɚɦɟɧɬɚɥɶɧɵɯ ɮɢɡɢ-

ɱɟɫɤɢɯ ɩɨɫɬɨɹɧɧɵɯ! ɂɫɯɨɞɹ ɢɡ ɷɬɨɝɨ, ɚɜɬɨɪɵ ɩɪɟɞɫɬɚɜɢɥɢ

ɫɜɨɸ ɪɚɛɨɬɭ ɤɚɤ ɧɨɜɵɣ ɩɪɟɰɢɡɢɨɧɧɵɣ ɦɟɬɨɞ ɢɡɦɟɪɟɧɢɹ

ɜɚɠɧɨɣ ɜ ɚɬɨɦɧɨɣ ɮɢɡɢɤɟ ɮɢɡɢɱɟɫɤɨɣ ɜɟɥɢɱɢɧɵ  ɩɨɫɬɨɹɧ-

ɧɨɣ ɬɨɧɤɨɣ ɫɬɪɭɤɬɭɪɵ , ɬ.ɤ. ɫɤɨɪɨɫɬɶ ɫɜɟɬɚ c

ɢɡɜɟɫɬɧɚ ɫ ɛɨɥɶɲɨɣ ɬɨɱɧɨɫɬɶɸ ɢɡ ɧɟɡɚɜɢɫɢɦɵɯ ɷɤɫɩɟɪɢɦɟɧ-

ɬɨɜ (c = 299792458 ɦ/ɫ ).

137/1/

|ce !

Ɋɢɫ. 2.1.

Ɂɚɜɢɫɢɦɨɫɬɶ ɯɨɥɥɨɜɫɤɨɣ

ɩɪɨɜɨɞɢɦɨɫɬɢ

ɢ ɦɚɝ-

ɧɢɬɨɫɨɩɪɨɬɢɜɥɟɧɢɹ

ɨɬ ɧɚɩɪɹɠɟɧɧɨɫɬɢ ɦɚɝ-

ɧɢɬɧɨɝɨ ɩɨɥɹ ɜ ɰɟɥɨɦ

ɤɜɚɧɬɨɜɨɦ ɷɮɮɟɤɬɟ ɏɨɥ-

ɥɚ

ɋɈȼɊȿɆȿɇɇɕȿ ɉɊɈȻɅȿɆɕ ɎɂɁɂɄɂ ɌȼȿɊȾɈȽɈ ɌȿɅȺ

_________________________________________________________________________________________________________________________________________________________________________________________________

ɗɬɨɬ ɷɮɮɟɤɬ ɧɚɡɜɚɥɢ ɰɟɥɵɦ ɤɜɚɧɬɨɜɵɦ ɷɮɮɟɤɬɨɦ ɏɨɥɥɚ

(ɚɛɛɪɟɜɢɚɬɭɪɚ: IQHE  Integer Quantum Hall Effect), ɩɪɢ

ɷɬɨɦ ɭɱɟɧɵɦ-ɩɟɪɜɨɨɬɤɪɵɜɚɬɟɥɹɦ ɜ 1985 ɝɨɞɭ ɛɵɥɚ ɩɪɢɫɭɠ-

ɞɟɧɚ ɇɨɛɟɥɟɜɫɤɚɹ ɩɪɟɦɢɹ. ɂɧɬɟɪɟɫɧɨ, ɱɬɨ ɮɚɤɬɢɱɟɫɤɢ ɨɞɧɨ-

ɜɪɟɦɟɧɧɨ ɥɚɭɪɟɚɬɵ ɇɨɛɟɥɟɜɫɤɨɣ ɩɪɟɦɢɢ 1998 ɝɨɞɚ (Stormer,

Tsui  ɡɚ ɞɪɨɛɧɵɣ ɤɜɚɧɬɨɜɵɣ ɷɮɮɟɤɬ ɏɨɥɥɚ  ɫɦ. ɞɚɥɟɟ)

ɬɚɤɠɟ ɧɚɛɥɸɞɚɥɢ ɷɬɨ ɤɜɚɧɬɨɜɚɧɢɟ, ɧɨ ɧɟ ɭɫɩɟɥɢ ɨɩɭɛɥɢɤɨ-

ɜɚɬɶ ɪɟɡɭɥɶɬɚɬɵ, ɢ ɜ ɪɚɧɧɢɯ ɫɫɵɥɤɚɯ ɧɚ IQHE ɭɩɨɦɢɧɚɥɢ

ɤɚɤ ɮɨɧ Ʉɥɢɬɰɢɧɝɚ ɫ ɫɨɚɜɬɨɪɚɦɢ, ɬɚɤ ɢ ɧɟɨɩɭɛɥɢɤɨɜɚɧɧɭɸ

ɪɚɛɨɬɭ ɒɬɟɪɦɟɪɚ ɢ Ɍɫɭɢ. Ɍɟɦ ɧɟ ɦɟɧɟɟ, ɜɫɟ ɥɚɜɪɵ ɜ 1985

ɝɨɞɭ ɞɨɫɬɚɥɢɫɶ ɝɪɭɩɩɟ ɮɨɧ Ʉɥɢɬɰɢɧɝɚ.

Ɂɚɦɟɬɢɦ, ɱɬɨ ɨɞɧɨɜɪɟɦɟɧɧɨɟ ɢɡɦɟɪɟɧɢɟ ɦɚɝɧɢɬɨɫɨɩɪɨ-

ɬɢɜɥɟɧɢɹ

ɞɚɟɬ ɪɟɡɤɨɟ ɭɦɟɧɶɲɟɧɢɟ ɷɬɨɣ ɜɟɥɢɱɢɧɵ ɜ ɬɨɱ-

ɤɚɯ ɪɟɚɥɢɡɚɰɢɢ ɫɬɭɩɟɧɟɤ ɤɜɚɧɬɨɜɨɝɨ ɷɮɮɟɤɬɚ ɏɨɥɥɚ. ȼɟɥɢ-

ɱɢɧɚ

ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɢɫɱɟɡɚɸɳɟ ɦɚɥɚ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫɨ

ɫɜɨɢɦ ɡɧɚɱɟɧɢɟɦ ɩɪɢ ɧɭɥɟɜɨɦ ɦɚɝɧɢɬɧɨɦ ɩɨɥɟ (ɪɢɫ.2.1).

Ɍɟɨɪɟɬɢɱɟɫɤɨɟ ɨɛɴɹɫɧɟɧɢɟ ɛɵɥɨ ɞɚɧɨ ɛɭɤɜɚɥɶɧɨ ɜ ɫɥɟ-

ɞɭɸɳɟɦ, 1981 ɝɨɞɭ Ɋ.ȼ. Ʌɚɮɥɢɧɵɦ ɧɚ ɨɫɧɨɜɟ ɪɚɫɫɦɨɬɪɟɧɢɹ

ɞɜɢɠɟɧɢɹ ɧɨɫɢɬɟɥɟɣ ɡɚɪɹɞɚ ɜ ɫɬɪɭɤɬɭɪɟ Al

1-X

As - GaAs

ɤɚɤ ɞɜɭɦɟɪɧɵɯ ɧɟɜɡɚɢɦɨɞɟɣɫɬɜɭɸɳɢɯ ɷɥɟɤɬɪɨɧɨɜ, ɬ.ɤ. ɧɨ-

ɫɢɬɟɥɢ ɦɨɝɭɬ ɞɜɢɝɚɬɶɫɹ ɬɨɥɶɤɨ ɜ ɩɨɥɭɱɚɸɳɢɯɫɹ ɩɥɨɫɤɢɯ

ɤɜɚɧɬɨɜɵɯ ɹɦɚɯ, ɩɨɩɟɪɟɤ ɩɪɢɥɨɠɟɧɧɨɝɨ ɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ.

Ʌɚɮɥɢɧ ɭɱɟɥ, ɱɬɨ ɷɧɟɪɝɢɹ ɷɥɟɤɬɪɨɧɨɜ ɜ ɦɚɝɧɢɬɧɨɦ ɩɨɥɟ

ɦɨɠɟɬ ɩɪɢɧɢɦɚɬɶ ɬɨɥɶɤɨ ɞɢɫɤɪɟɬɧɵɟ ɡɧɚɱɟɧɢɹ, ɬ.ɟ. ɪɟɚɥɢɡɨ-

ɜɵɜɚɬɶɫɹ ɬɨɥɶɤɨ ɧɚ ɬɚɤ ɧɚɡɵɜɚɟɦɵɯ ɭɪɨɜɧɹɯ Ʌɚɧɞɚɭ. Ɉɧ

ɫɪɚɡɭ ɢɫɩɨɥɶɡɨɜɚɥ ɞɜɭɦɟɪɧɭɸ ɝɟɨɦɟɬɪɢɸ.

ɉɨɥɭɱɢɦ ɹɜɧɵɟ ɜɵɪɚɠɟɧɢɹ ɞɥɹ ɷɧɟɪɝɢɢ ɷɥɟɤɬɪɨɧɚ ɜ

ɦɚɝɧɢɬɧɨɦ ɩɨɥɟ ɜ ɛɨɥɟɟ ɨɛɳɟɦ, ɬɪɟɯɦɟɪɧɨɦ ɫɥɭɱɚɟ, ɪɟɲɢɜ

ɫɬɚɧɞɚɪɬɧɭɸ ɡɚɞɚɱɭ ɒɪɟɞɢɧɝɟɪɚ, ɚ ɡɚɬɟɦ ɭɠɟ ɩɟɪɟɣɞɟɦ ɤ

ɞɜɭɦɟɪɧɨɣ ɫɢɬɭɚɰɢɢ. Ɋɚɫɫɦɨɬɪɢɦ ɷɥɟɤɬɪɨɧ ɜ ɦɚɝɧɢɬɧɨɦ

ɩɨɥɟ ɜ ɝɟɨɦɟɬɪɢɢ ɹɳɢɤɚ L

, L

ɫ ɦɚɝɧɢɬɧɵɦ ɩɨɥɟɦ H ɩɨ

ɨɫɢ Z. Ɋɟɲɢɦ ɭɪɚɜɧɟɧɢɟ ɒɪɟɞɢɧɝɟɪɚ ɞɥɹ ɧɟɝɨ ɫɥɟɞɭɸɳɟɝɨ

ɜɢɞɚ:

ɄȼȺɇɌɈȼɕȿ ɗɎɎȿɄɌɕ ɏɈɅɅȺ

_________________________________________________________________________________________________________________________________________________________________________________________________

])/(/)/][(2/1[

;

AcerimHEH

! 

ww\\

ɉɨɞɪɨɛɧɨɫɬɢ ɫɦ. ɜ ɡɚɞɚɱɚɯ ɜ ɤɨɧɰɟ ɩɚɪɚɝɪɚɮɚ, ɚ ɡɞɟɫɶ ɜɵ-

ɩɢɲɟɦ ɨɬɜɟɬ ɞɥɹ ɷɧɟɪɝɢɢ ɢ ɜɨɥɧɨɜɵɯ ɮɭɧɤɰɢɣ:

eHcl

lyyHlyyy

yzpxpizyx

mpnpE

HnHn

ZCZn

);/]([)2/][exp()(

)()])(/exp[(),,(

;2/)2/1()(







(2.1)

ȼɟɥɢɱɢɧɚ l

, ɤɚɤ ɨɬɦɟɱɚɥɨɫɶ ɜ ɩɪɟɞɵɞɭɳɟɦ ɩɚɪɚɝɪɚɮɟ, ɧɚ-

ɡɵɜɚɟɬɫɹ ɦɚɝɧɢɬɧɨɣ ɞɥɢɧɨɣ ɢ ɹɜɥɹɟɬɫɹ ɦɚɫɲɬɚɛɨɦ ɩɪɨ-

ɫɬɪɚɧɫɬɜɟɧɧɨɣ ɥɨɤɚɥɢɡɚɰɢɢ ɷɥɟɤɬɪɨɧɚ ɜ ɦɚɝɧɢɬɧɨɦ ɩɨɥɟ

ɜɦɟɫɬɨ ɤɥɚɫɫɢɱɟɫɤɨɝɨ ɪɚɞɢɭɫɚ Ʌɚɪɦɨɪɚ. ɍɪɨɜɧɢ E

ɧɚɡɵɜɚ-

ɸɬɫɹ ɭɪɨɜɧɹɦɢ (ɢɥɢ ɩɨɞɭɪɨɜɧɹɦɢ, ɩɨɞɡɨɧɚɦɢ) Ʌɚɧɞɚɭ. Ɉɧɢ

ɷɤɜɢɞɢɫɬɚɧɬɧɵ, ɢ ɨɬɫɬɨɹɬ ɞɪɭɝ ɨɬ ɞɪɭɝɚ ɧɚ

ɂɡ ɪɟɲɟɧɢɹ (2.1) ɜɢɞɢɦ, ɱɬɨ ɷɧɟɪɝɢɹ j-ɝɨ ɭɪɨɜɧɹ Ʌɚɧɞɚɭ

ɜ ɩɥɨɫɤɨɣ ɝɟɨɦɟɬɪɢɢ (ɤɨɝɞɚ p

=0) ɢɦɟɟɬ ɜɢɞ, ɩɨɯɨɠɢɣ ɧɚ

ɷɧɟɪɝɢɸ ɤɜɚɧɬɨɜɨɝɨ ɨɫɰɢɥɥɹɬɨɪɚ:

!)2/1( 

, j = 0, 1,

2, ... Ɋɚɞɢɭɫ ɨɪɛɢɬɵ (ɚɧɚɥɨɝɚ ɪɚɞɢɭɫɚ Ʌɚɪɦɨɪɚ ɜ ɤɜɚɧɬɨɜɨɦ

ɫɥɭɱɚɟ) ɷɥɟɤɬɪɨɧɚ ɧɚ j-ɦ ɭɪɨɜɧɟ Ʌɚɧɞɚɭ ɪɚɫɬɟɬ ɫ ɧɨɦɟɪɨɦ

ɭɪɨɜɧɹ: r

=2(j+1/2)l

(ɫɦ. ɡɚɞɚɱɭ). ȼɢɞɧɨ, ɱɬɨ ɪɚɞɢɭɫ ɨɪɛɢ-

ɬɵ ɞɥɹ ɧɚɢɧɢɡɲɟɝɨ ɭɪɨɜɧɹ Ʌɚɧɞɚɭ (j=0) ɬɨɱɧɨ ɪɚɜɟɧ ɦɚɝ-

ɧɢɬɧɨɣ ɞɥɢɧɟ.

ȼɚɠɧɚɹ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚ ɷɥɟɤɬɪɨɧɧɨɝɨ ɝɚɡɚ  ɩɥɨɬɧɨɫɬɶ

ɫɨɫɬɨɹɧɢɣ  ɜɟɥɢɱɢɧɚ, ɯɚɪɚɤɬɟɪɢɡɭɸɳɚɹ ɡɚɫɟɥɟɧɧɨɫɬɶ ɪɚɡ-

ɪɟɲɟɧɧɵɯ ɭɪɨɜɧɟɣ ɷɧɟɪɝɢɢ. ȼɫɟ ɮɢɡɢɱɟɫɤɢɟ ɜɟɥɢɱɢɧɵ (ɜ

ɬɨɦ ɱɢɫɥɟ ɢ ɩɪɨɜɨɞɢɦɨɫɬɶ) ɥɢɛɨ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɵ, ɥɢɛɨ

ɹɜɥɹɸɬɫɹ ɦɨɧɨɬɨɧɧɵɦɢ ɮɭɧɤɰɢɹɦɢ ɩɥɨɬɧɨɫɬɢ ɫɨɫɬɨɹɧɢɣ

ɷɥɟɤɬɪɨɧɨɜ ɜɛɥɢɡɢ ɭɪɨɜɧɹ Ɏɟɪɦɢ E

 ɝɪɚɧɢɱɧɨɣ ɷɧɟɪɝɢɢ,

ɨɬɞɟɥɹɸɳɟɣ ɫɜɨɛɨɞɧɵɟ ɫɨɫɬɨɹɧɢɹ ɨɬ ɡɚɧɹɬɵɯ (ɩɨ ɤɪɚɣɧɟɣ

ɦɟɪɟ, ɩɪɢ ɧɢɡɤɢɯ ɬɟɦɩɟɪɚɬɭɪɚɯ).

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

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