CC BY
17
Section 12. Physics
Eshpulatov Barat, Doctor science, professor, Samarkand branch of Tashkent University of Information Technologies,
E-mail: barat50@bk.ru Xujanova Dilafruz Shakarbekovna, Assistient, Samarkand branch of Tashkent University of Information Technologies, E-mail: dxujanova@mail.ru Asrarov Shuxrat Abbasovich, Dosent, Samarkand branch of Tashkent University of Information Technologies, E-mail: shuxratasrarov314@gmail.com
QUADRUPLE MAGNETOPOLARONS IN MAGNETOOPTICS OF A QUANTUM-DIMENSIONAL NANOSTRUCTURES
Abstract. A theory of interband magnetooptical light absorption in a nano-dimensional quantum well in a strong magnetic field conditioned by interaction of electrons with optical phonons was developed. We investigated a behavior of magnetooptical peaks corresponded to transitions of electrons from Landau levels with quantum numbers n > 3 in the region of magnetic fields that satisfy the condition of equality of the electron cyclotron frequency in the conduction band to the frequency of longitudinaloptical phonon. For n = 3 in a point of resonance there is an intersection of four terms of the electron-phonon system (electron on Landau level n = 3, electron on the level n = 2 and one optical phonon, electron on the level n = 1 and two phonons, and electron on the level n = 0 and three phonons), which are considered to be a function of the cyclotron frequency of electron. Interaction with phonons removes a degeneracy of the terms, and leads to appearance of four non-intersected branches of electron-phonon spectrum. Accordingly, a magnetooptical peak splits into four peaks. Equations to describe a frequency dependence of a position and intensity of peaks on the magnetic field are presented.
Keywords: interband, magnetooptical, absorption, quantum well, magnetic field, electrons, Landau levels, cyclotron frequency, optical phonons, the electron-phonon system, peak splits, equations, magnetopolaron effect, semiconductor materials.
After pioneer works byJohnson and Larsen [1] a magnetopolaron effect (Johnson-Larsen effect) attracts significant attention of both theoretical [2] and experimental [3] studies.
Polaron states can be formed both in three-dimensional (3D) [1], and quasi-two-dimensional (2D) systems [4-7]. A difference between the systems is in spectra of an electron in the presence of a quantizing magnetic field: in 3D-system these are single-dimensional Landau zones, in 2D-system these are discrete Landau levels. This difference leads to different magnitude of a repulsion of levels of the electron-phonon system.
In [3] Korovin and Pavlov showed that in a case of a bulk
semiconductor (3D) a magneto-polaronic split is propor-
2/
tional to n where a0 - is cutoff frequency of longitudinal optical phonons.
In 2D-systems this effect is stronger, and a distance between the peak split components becomes proportional to rfiha0 [4-7].
In 3D and 2D -systems the magnetopolaron states play important role in formation of a frequency dependence of magnetooptical effects, such as interband light absorption,
[1; 7], cyclotron resonance [3; 4; 8], and Raman scattering [9].
As it was shown in [10], when the following condition is valid
®0 = j®e(h)H , (l)
where
\e\H
(2)
©
(h )H
c •m
(h )
where a>e(h)H - is a cyclotron frequency, c - light speed in a
vacuum, |e| - is charge of the electron, H - isa magnetic field strength, me(h) - is an effective mass of electron (hole), j - is a number. The value j = 1 is corresponding to a double po-laron noted in [10] by A letter. The value j = 2, i.e. coH / co0 = 1/2 is corresponded by two double polarons. The value j = 3, i.e. coH / co0 = 1/3, is corresponded by three double polarons, etc.
Above double polarons there are triple polarons that correspond to intersection of three terms. Quadruple polarons are located above the triple ones, etc. Number of polarons of each type at a given j is equal to j.
Quadruple and higher polarons has not been considered
yet.
This work is devoted to theoretical study of the energy spectrum of the quadruple magnetopolaron, and an influence of the spectrum on formation of the frequency dependence of magnetooptical effects in a rectangular quantum well in a strong magnetic field directed perpendicularly to the well's plane.
As a 2D-system below we consider a single quantum well. In the magnetic field directed normally to its interface the energy levels in the well become discrete ones (with infinite multiple degeneracy), and their classification depends on the ratio of the energy in the well and the cyclotron energy. Below we assume that the energy of quantization in the well is high with respect to cyclotron oneand we take into account only lowest level with adjacent Landau levels.
We assume that the valence band (v) and the conduction band (c) are situated in the center of Brillouin zone, and a direct dipole transition is between them. Interaction with LO-phonons, which determines in our case the split of peaks, we assume to be weak. In many semiconductors the condition mjmh << 1 is valid. We consider an interband optical transition, as a result of which there appears an electron in the quantum well of the conduction band and a hole in the valence band on Landau level (we neglect a possibility of formation of
exciton states). If temperature is low, and the magnetic field is close to the resonant one (see Eq.1), then the hole states will be standard in frames of the chosen mechanism of interaction, since the hole cannot really emit LO-phonon due to energy insufficiency. The hole cannot absorb phonons due to their absence. The electron in such conditions can really emit LO-phonon and to go from the level with quantum number «to the level with quantum number n-1.
The absorption will be characterized by a rate of absorbed energy [11]
W = Wo®eH ^ReiG, (a,v-®ha) (3)
a
Where Gr (a,e) = _e-am -Z(a,fi) + i5]-1;5 ^+0 (4) is a one-particle retarded Green function of electron,
®ea =®eH ^ K + 1) + 0/ fl),
®ha = (Eg / + ®hH ^ na+ 1 j + (£h0 IK) , (5)
/(( + ^)2](2 /ch)(|py\2 /m0Eg)( /m„)(6)
Here Py - is an interband matrix momentum element calculated from Bloch modulating factors, Eg - is a value of band gap, e^ - is statistical dielectric permittivity of semiconductor (dielectric), £e(h)0 = n2h2 / 2me^h)d2 - is the energy of l = 1 dimensionally-quantized level in the quantum well.
Functions ^(a,s) in (4) can be easily calculated by using the Feynman diagram technique. The rules are standard ones.
We'll start from the energy part of ^(a,s) equal to
Z(a,e) = £ X r2 C |2|/aai@|V®0 +
? ai
In case of electron transition to Landau level with n = 3, in the sum by na a resonant will a summand with na = 2 since it corresponds to a real resonant transition between neighboring Landau levels. Assuming na = 3, na = 2 we will have for the resonant summand
W =
Z(3,£)=©0c (y + X + iS
Y =
f É ^ ÉeH
É
Ve o /
®„t
®n
C =-3
1 ™
-1du 4ue ~
3 - 3u + - u2 T ¿J"*
®n
(8) (9)
To find out a picture qualitatively we neglect firstly the graphs containing apical parts. In this case it's enough to take into account a series of graphs presented in (Fig. 1).
Figure 1. A series of graphs that leads to equation (10)
Figure 2. The spectrum of electron with taking into account its interaction with optical phonons versus external magnetic field
Non-resonant members in all graphs are neglected since they are small in comparison with resonant ones. In this case a summation the series in (Fig. 3) leads to equation
Cn
Y + X--
Bn
- = o,
Y + 2X-
An
Y + 3X
where A =4n /2, B = (n /16), C = (Wn /128). The equation can be written in the form
x4 + 6yx3 + [11y2-(A + B + C ) x2 +
+ [6y2 - (A + 3B + 5C) yx - 6Cy2 + A • C = 0
(10)
(11)
where x = (y /-J^), Y = (x/yfc) (12a)
When X = 0 (resonant case, a>eH =©0) roots of (12) will be:
Y = 1.44^/n, Y2 = 0,55^,73 = -0,55^, /4 = -1.44^(13)
The spectrum of electron-phonon system versus magnetic field is presented in (Fig. 2).
Four terms of electron-phonon system, which in the absence of interaction are intersected in the point (oeH = (O, after account of electron-phonon interaction is split, according to results of Section 2, into four branches of electron-phonon spectrum. A peak of magnetooptical absorption, corresponding to a throw of electron by light to Landau level with quantum number n = 3, is also split at a>eH = a>O into four S -like peaks. The absorption at coeH = coO will be determined by four branches of electron-phonon spectrum in conduction band, since holes do not contribute to nonstationarity of levels when condition (me / mh)«: 1 is valid. We calculate the absorption as a function of light frequency in the region of the studied peak by neglecting small contribution from apical parts. Be taking into account (3),(4) and (10), as well as inequality of effective masses, we have
Cn
W = W„ n
r--
r + X--
Bn r + 2X-
An
r+3X
r = I co -co,,- (0„„—c I/o
(14)
W n
w„,
wr
(12)
( ) By using properties of S -function from (14) we have
^4=i{r,3 + 6Xr2 +[|U2 -(A + B)]rt +X[_6X2 -(A + 3B)n]}
^ (15)
4r3 + 18Xr2 + 2[||X2 - (A + B + C)n]rt + X[6X2 - (A + 3B + 5C
xS {r4 + 6ir3 + [IU2 - (A + B + C)ri~\ r2 +2 [6X2 - (A + 3B + 5C) r - 6Cl^ + ACrf
By substitution ofA, B and C from (11) into (15) we have that peaks are situated at ri = -1.44^, !2 = -0.55^, r3 = 0.55^, r4 = 1.44^/n, the magnitudes of peak intensity are If = If = 0.068 and I2C = Ic3 = 1.18 and middle peaks are 17 times more intensive than lateral peaks. A distance between lateral peaks is 2.88^/n, between middle peaks it is 1.1^/n, and the distance between lateral and middle peaks is 0.89-Tn
At A = -Jn, intensities of peaks are, consequently, I1 = 0.002 I2 = 0.0079, I3 = 0.2799 and I4 = 0.6534 . Their positions are: r1 = -3.66^, I2 =-2.083,/)!. r3 = -0.83^ and r4 = 0.57^/n .
At X = integral intensities of peaks are, consequently, I1 = 0.002, I2 = 0.0058, I3 = 0.0769 and I4 = 0.9292 . Positions of peaks: ri = —9.3^, I ^ —r3 = —3y/q and r4 = 0.23^/n .
When X = the peaks are shifted towards smaller frequencies of the exciting light. Intensity of the very right peak is I1 = 0.98, and, further to the left, the magnitudes of peak intensity are I2 = 0.02, I3 = 0.0016, I4 = 0.424-10-3. Positions of peaks are r1 = -181/77, r2 = -1-2, I =-6^,
rA - 0,01^.
At X = -6yjn the picture becomes inverse.
Thus, at X > 0 with rising X the peaks shift towards low-frequency region and the intensities of left peaks is decreased, while the intensity of very right peak increases. At X » 1 there is only one last peak. At X < 0 the picture is inverse, i.e. peaks are shifted towards high-frequency region, the intensity of the first peak increases and the intensity of other peaks decreases. Only one peak remains at |X| » 1.
5
References:
1. Johnson E. I. I., Larsen D. M. Polaron Induced Anomalies in the Interband Magnetoabsorption of In Sb. // Phys. Rev.Let-ters,1966. 16.- P. 655-659.
2. Korovin L. I., Pavlov S. T. On the role of optical phonons in interband magnetooptical absorption of a semi-conductor // JETP, 1967. 53. - No. 5.- P. 1708-1716. [in Russian].
3. Ivanov-Omskii V. I., Korovin L. I., Shereghi E. M. Phonon - Assisted Cyclotron Resonance in Semiconductors // Phys. Stat. Sol. (b), 1978. 90.- No. 1.- P. 11-32.
4. Horst M., Merkt U., Kotthaus J. P. Magneto-Polarons in a Two-Dimensional Electron Inversion Layer on In Sb. // Phys. Rev. Letters, 1983. 50.- No. 10.- P. 754-757.
5. Korovin L. I., Pavlov S. T., Eshpulatov B. E. Magnetooptical absorption in a semiconductor at a surface layer with participation of optical phonons // FTT, 1978. 20.- No. 12.- P. 3594-3598. [inRussian].
6. Das Sarma S., Madhukar A. Study of electron-phonon interaction and magneto-optical anomalies in two-dimensionally confined systems // Phys.Rev., 1980.- B22.- No. 6.- P. 2823-2836.
7. Rogers D. C., Nicholas R. J., Ben Amor S., Portal J. C., Cho A. Y., Sivso D. Inter-band magneto-absorption in a quantum well // Solid State Comm. 1986.- 60.- No. 2.- P. 83-86.
8. Korovin L. I., Eshpulatov B. E. Theory of polaron effect in cyclotron resonance in two-dimensional electron gas. // FTT, 1987.-29.- No. 7.- P. 1963-1969. [in Russian].
9. Korovin L. I., Pavlov S. T., Eshpulatov B. E. Two- and three-phonon resonant Raman scattering in quasi-two-dimensional electronic systems in a strong magnetic field // FTT, 1993. 35.- No. 6.- P. 1562-1576. [in Russian].
10. Lang I. G., Korovin L. I., Pavlov S. T. Wave-functions and energies of magnetopolarons in semiconductor quantum wells. // FTT,2005.- 47.- No. 9.- P. 1704-1710. [inRussian].
11. Korovin L. I., Eshpulatov B. E. Interband magnetooptical absorption in surface layer of a semiconductor // FTT, 1979.21.- No. 12.- P. 3703-3712. [inRussian].
