Basic peculiarities of energy band spectra within generalized kindal's model for semiconductors with one main axis
The analysis of the basic features of the generalized Kildal model had been presented for the semiconductors without the center of symmetry and with one main crystal axis. It had been proved, that the Kramers’ degeneration survives only along and against of the direction of the main crystal axis and...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
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irk-123456789-1206592017-06-13T03:03:21Z Basic peculiarities of energy band spectra within generalized kindal's model for semiconductors with one main axis Chuiko, G.P. Martyniuk, V.V. Bazhenov, V.K. The analysis of the basic features of the generalized Kildal model had been presented for the semiconductors without the center of symmetry and with one main crystal axis. It had been proved, that the Kramers’ degeneration survives only along and against of the direction of the main crystal axis and at point Γ ( 0=k ) within this model under the condition of the absence of the center of symmetry, as rule. The Kramers’ degeneration is possible everywhere, although only for the band of the heavy holes like within Kane’s model, even under the previous condition, but only if casually takes place the special relation between parameters. The typical set of solutions within this model consists of one conductivity band ( 0≥ε ) and of three valence bands ( 0≤ε ). Each of them contains two spin subbands. The additional conductivity band would be possible in principle, but only under special condition ( 0<δ ), which is the obligatory but not enough condition. 2005 Article Basic peculiarities of energy band spectra within generalized kindal's model for semiconductors with one main axis / G.P. Chuiko, V.V. Martyniuk, V.K. Bazhenov // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2005. — Т. 8, № 1. — С. 28-31. — Бібліогр.: 7 назв. — англ. 1560-8034 PACS: 71.20.-b, 71.18.+y http://dspace.nbuv.gov.ua/handle/123456789/120659 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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The analysis of the basic features of the generalized Kildal model had been presented for the semiconductors without the center of symmetry and with one main crystal axis. It had been proved, that the Kramers’ degeneration survives only along and against of the direction of the main crystal axis and at point Γ ( 0=k ) within this model under the condition of the absence of the center of symmetry, as rule. The Kramers’ degeneration is possible everywhere, although only for the band of the heavy holes like within Kane’s model, even under the previous condition, but only if casually takes place the special relation between parameters. The typical set of solutions within this model consists of one conductivity band ( 0≥ε ) and of three valence bands ( 0≤ε ). Each of them contains two spin subbands. The additional conductivity band would be possible in principle, but only under special condition ( 0<δ ), which is the obligatory but not enough condition. |
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Chuiko, G.P. Martyniuk, V.V. Bazhenov, V.K. |
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Chuiko, G.P. Martyniuk, V.V. Bazhenov, V.K. Basic peculiarities of energy band spectra within generalized kindal's model for semiconductors with one main axis Semiconductor Physics Quantum Electronics & Optoelectronics |
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Chuiko, G.P. Martyniuk, V.V. Bazhenov, V.K. |
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Chuiko, G.P. |
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Basic peculiarities of energy band spectra within generalized kindal's model for semiconductors with one main axis |
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Basic peculiarities of energy band spectra within generalized kindal's model for semiconductors with one main axis |
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Basic peculiarities of energy band spectra within generalized kindal's model for semiconductors with one main axis |
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Basic peculiarities of energy band spectra within generalized kindal's model for semiconductors with one main axis |
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Basic peculiarities of energy band spectra within generalized kindal's model for semiconductors with one main axis |
| title_sort |
basic peculiarities of energy band spectra within generalized kindal's model for semiconductors with one main axis |
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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2005 |
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http://dspace.nbuv.gov.ua/handle/123456789/120659 |
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Basic peculiarities of energy band spectra within generalized kindal's model for semiconductors with one main axis / G.P. Chuiko, V.V. Martyniuk, V.K. Bazhenov // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2005. — Т. 8, № 1. — С. 28-31. — Бібліогр.: 7 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
| work_keys_str_mv |
AT chuikogp basicpeculiaritiesofenergybandspectrawithingeneralizedkindalsmodelforsemiconductorswithonemainaxis AT martyniukvv basicpeculiaritiesofenergybandspectrawithingeneralizedkindalsmodelforsemiconductorswithonemainaxis AT bazhenovvk basicpeculiaritiesofenergybandspectrawithingeneralizedkindalsmodelforsemiconductorswithonemainaxis |
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2025-07-08T18:17:46Z |
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Semiconductor Physics, Quantum Electronics & Optoelectronics. 2004. V. 8, N 1. P. 28-31.
© 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
28
PACS: 71.20.-b, 71.18.+y
Basic peculiarities of energy band spectra within generalized
Kildal’s model for semiconductors with one main axis
G.P. Chuiko , V.V. Martyniuk and V.K. Bazhenov
Kherson National Technical University, Department of General and Applied Physics
24, Beryslavske shosse, 73008 Kherson, Ukraine, e-mail: chuiko@public.kherson.ua
Abstract. The analysis of the basic features of the generalized Kildal model had been
presented for the semiconductors without the center of symmetry and with one main
crystal axis. It had been proved, that the Kramers’ degeneration survives only along and
against of the direction of the main crystal axis and at point Γ ( 0=k ) within this model
under the condition of the absence of the center of symmetry, as rule. The Kramers’
degeneration is possible everywhere, although only for the band of the heavy holes like
within Kane’s model, even under the previous condition, but only if casually takes place
the special relation between parameters. The typical set of solutions within this model
consists of one conductivity band ( 0≥ε ) and of three valence bands ( 0≤ε ). Each of
them contains two spin subbands. The additional conductivity band would be possible in
principle, but only under special condition ( 0<δ ), which is the obligatory but not
enough condition.
Keywords: energy bands, degeneration, Kramers, splitting, spin sub-bands.
Manuscript received 31.03.05, accepted for publications 18.05.05.
1. Introduction
A generalization of the known Kildal model had been
reported [1] recently. This version describes the energy
band spectra of semiconductors with one main axis as
well as prior models [2, 3]. On the other hand, the new
version is applicable not only to the crystals with the
center of symmetry as [2, 3], but also to those, where
such centers are missing. Moreover, there takes into
consideration oneself and the lattice deformation along
the main axis.
The Hamiltonian with its exact dispersion law had
been presented in [1]. Authors reported that it had been
possible to present the characteristic polynomial of the
Hamiltonian as a product of two different polynomial
factors. Both had the same degree. Indeed the dispersion
law [1] looks like following with spherical coordinates
( ϕθ ,,k ):
( ) ( ) ( )( )( )
( )
22 2 2
1 2
2 2 2
3
( ) ( )sin ( )cos
( ) ( ) sin 0
Pk f f
Pk f
ε ε θ ε θ
ε θ
Γ − + −
− =
(1)
This characteristic polynomial has the quite evident
decomposition into the product of two factors. Let us to
explain the sense of symbols. There:
( ) ( )( )
( )
2 2
2
2 /3 /3 2 /9
( )
2 /3
g
d
ε ε ε ε δ η
ε ε
ε
⎧ ⎫⎡ ⎤− + Δ + +Δ − Δ −⎪ ⎪⎣ ⎦Γ = ⎨ ⎬
− + Δ⎪ ⎪⎩ ⎭
, (2)
( )( )( )222
1 9/3/3/)( ηδεεε Δ−Δ++Δ+= Pf , (3)
( ) ( )3/224
2 Δ+= − εεηε Pf , (4)
d
P
f Δ= ε
η
ε )
3
2
()(3 . (5)
All these are polynomials as to the energy of car-
riers – ε . Thereto: Pg ,, Δε are three well-known
Kane’s parameters [3] (the energy gap, the spin-splitting
parameter and the matrix element of the impulse).
Further δ is the known parameter of the crystal field
[2] and d is another parameter of the crystal field,
which describes the absence of the symmetry center [1].
Lastly η is the scalar factor taking into account the
deformation of the lattice [4]. The expressions (2 to 5)
are equally correct both for semiconductors with
0>gε and for others with 0<gε . Zero of energy
( 0=ε ) is located at the top of the band of heavy holes
at any of both situations.
However, the paper [1] did not contain the deep
analysis of the equation (1) as well as the investigation
of the influence of its features as for the energy bands
spectra. Only the few details could be as exclusion from
this statement. They were visible in a figure showing
the typical dependences of )(kε for these two
directions. On the other hand, there had been presented
only one concrete material (by the way with 0<gε ),
and therefore it remained not lucid even as far as these
details are general like other similar crystals.
The main aim of this paper is the search of
mentioned basic peculiarities of the energy band spectra
.
Semiconductor Physics, Quantum Electronics & Optoelectronics. 2004. V. 8, N 1. P. 28-31.
© 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
29
within the model [1], and this investigation should be
leaning mostly against the detailed algebraic analysis of
the expressions (1 to 5). It should provide the selection
of most common features of the energy band structures
for all the semiconductors described by this model.
2. The analysis of the dispersion law
Let us first present two factors of (1) as:
( ) ( ) ( )( )
( )
2 2 2
1 2
3
( , ) ( ) ( )sin ( )cos
( )sin
P Pk f f
Pkf
α ε θ ε ε θ ε θ
ε θ
= Γ − + +
+
(6)
( ) ( ) ( )( )
( )
2 2 2
1 2
3
( , ) ( ) ( )sin ( )cos
( )sin
P Pk f f
Pkf
β ε θ ε ε θ ε θ
ε θ
= Γ − + −
−
(7)
Let us rewrite further these polynomials to their
canonical (alias monic) forms as regard regard to the
main variable (ε ) by using the expressions (2 to 5):
( )
( )
4 3 2 4 2
2 1 2
2
2 2 2
( , ) ((sin cos )
2 2( ) sin( )
3 3
2 ( (1 )))
3 3
( (1 ))sin ( )
3 3
g
g
P
Pk d Pk d
Pk
α ε θ ε ε δ ε θ η θ
η θ
ε δ η ε
δ η θ
−
−
−
−
= − − ∆ − − + ×
⎛ ⎞ ⎛ ⎞× − ∆ + ∆ +⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
∆⎛ ⎞+ ∆ + − −⎜ ⎟
⎝ ⎠
∆ ∆
− + −
( )
( )
4 3 2
4 2 2 1
2 2
2 2 2
( , ) ((sin
2cos )( ) sin( )
3
2 2 ( (1 )))
3 3 3
( (1 ))sin ( )
3 3
g
g
P
Pk d Pk
d
Pk
β ε θ ε ε δ ε θ
η θ η θ
ε δ η ε
δ η θ
− −
−
−
= − − ∆ − − +
⎛ ⎞+ + ∆ +⎜ ⎟
⎝ ⎠
∆⎛ ⎞ ⎛ ⎞+ ∆ + ∆ + − −⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
∆ ∆
− + −
(9)
The companion matrixes corresponding to both
polynomials are almost identical:
⎟⎟
⎟
⎟
⎟
⎠
⎞
⎜⎜
⎜
⎜
⎜
⎝
⎛
−
−
−
−
=
⎟⎟
⎟
⎟
⎟
⎠
⎞
⎜⎜
⎜
⎜
⎜
⎝
⎛
−
−
−
−
=
3
2
1
0
3
2
1
0
100
010
001
000
;
100
010
001
000
a
a
a
a
H
a
a
a
a
H β
β
α
α ; (10)
here ja are the coefficients at jε in the polynomials
(8, 9) ( 3,2,1,0=j ) and the additional index (at βα 11 ,aa )
shows the only two different coefficients in (8), (9) and
only two different elements in matrixes (10). Therefore,
the Hamiltonian that corresponds to the characteristic
polynomial (1), acquires the block-diagonal structure
with two sub-matrixes (10):
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
β
α
H
H
H
0
0
. (11)
Such a shape of Hamiltonian means that its space of
the eigenvectors is divided onto the two invariant sub-
spaces that are non-equivalent generally speaking [5].
Let us find the resultant of polynomials, ),( βα PPres
by using of the standard algebraic method [6]. The
result may be written in the form:
( )( )2 2
5 4 6 6 6
256( , ) 1 3
729
( ) sin
res P P
d Pk
α β η η δ
η θ−
⎛ ⎞= − − ∆ + ×⎜ ⎟
⎝ ⎠
×∆
(12)
Any two polynomials have a mutual root if and only
if their resultant is equal to zero [6, 7]. Here are below
few physically reasonable conditions providing a zero
of above-mentioned expression:
k = 0, (13)
0sin =θ , (14)
0=d . (15)
Every one of them provides also the automatic
execution of the following condition:
βα 11 aa = . (16)
In addition, as it follows from (16):
βαβα HHPP == , .
The first two conditions (13) and (14) simply signify
that each of the energy levels is always twice
degenerated at the point Γ ( 0=k ) firstly, and along
and against the direction of the main crystal axis
secondly. These two statements are independent of the
presence or absence of the symmetry center in a
semiconductor. All energy levels are twice degenerated
for any set of ( θ,k ) at presence of the symmetry center
thirdly. It follows from the condition (15). Validity of
these assertions is independent of the magnitudes or the
signs of other model parameters. Thereby they are
faithful for all semiconductors described by this model.
Although the formal possibility to think about the
other trivial conditions (for instance, like 0=P and so
on) as it is following from (12), but it seems senseless
physically. Nevertheless, there is one non-trivial
condition making the expression (12) equal to zero,
specially:
( )
2
2
3
1
η
ηδ −∆
= . (17)
This condition looks like a relation between
parameters of the model and may be satisfied, or be
almost satisfied, randomly or artificially. That is why
the condition (17) has in principle another quality as
compared to the above conditions. The condition (17)
leads to another but an equivalent condition:
00 =a . (18)
.
,
,
,
(8)
.
Semiconductor Physics, Quantum Electronics & Optoelectronics. 2004. V. 8, N 1. P. 28-31.
© 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
30
The polynomials (8), (9) and matrixes (10) are still
different, but get now one mutual root: 0=ε . Thus, this
mutual root is twice degenerated, and thereto is
independent of the module ( k ) as well as of the
direction (θ ) of the wavevector. It signifies physically,
that we deal with the infinitely narrow band of the
heavy holes like the same within Kane’s model [3]. This
analogy does not surprise, if to note that the condition of
applicability of the Kane’s model (i.e., 1=η and
0=δ ) converts immediately the expression (17) into
the identity. Such a kind of degeneration should be
acknowledged as parametric dependent and con-
sequently casual by the nature. That is why it has
another quality and origin in contrast with the above-
considered kinds of the genuine degeneration and even
with same within Kane’s model.
At the same time, the other roots of characteristic
polynomials all are different despite (17) and thus are
single. It follows from the resultant between shortened
polynomials, obtained from αP and βP after the divi-
ding of both on the energyε . This resultant is equal to:
3
sin
27
64),( ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ ∆
⎟
⎠
⎞
⎜
⎝
⎛−=
η
θ
εε
βα dkPPP
res . (19)
The expression is not equal to zero obviously under
the condition (17) and returns us to conditions (13 to
15) if we keep in mind their degeneration.
One more theorem of the algebra of polynomials
gives warranty that all roots of a polynomial are
different each to other (and thus are single). It would be
true, if only a polynomial is mutually simple with own
first derivative as for main variable [5]. The direct
calculation within well-known Euclid’s algorithm
shows that both greatest common divisors of the
polynomials (8), (9) as for their derivatives are trivial,
and thereby are equal to 1. Thus, all the roots of poly-
nomials are single under the condition, of course, that
neither of the above-mentioned conditions (13 to 17) is
satisfied. Moreover, the roots are different not only
“inside polynomials”, but “between them”, too.
What type of the degeneration we are keeping in
mind? The time inversion operator K̂ , for instance, can
convert polynomials one into another as it turns of a
motion and a spin state into own opposites. So from
πθθ +=K̂ and θθ sin)(sinˆ −=K follows, that βα PPK =ˆ
and vice versa αβ PPK =ˆ . The subscripts βα , should be
recognized consequently as the indicators of the
different spin states that are degenerated or not
degenerated in dependence on the satisfaction of the
terms (13 to 17).
It can be proved doubtless by the transformation of
the Hamiltonian matrix (11) with operator K̂ . Indeed:
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⋅−=⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ −
α
β
β
α
H
H
K
H
H
K
0
0
)1(ˆ
0
0ˆ 1 (20)
Here C
i
i
CK ˆ
0
0ˆ,ˆ
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
== σ , and Ĉ is the operator
of the complex conjugation, and also takes into account
that both blocks (10) have only the real matrix elements.
The time inversion operator just permutes the blocks of
Hamiltonian and there any phase multiplier does not
matter physically, firstly. The time inversion and
Hamiltonian are not commutative operators, so the well-
known Kramers theorem about the double degeneration
does not act under condition βα HH ≠ , secondly.
Well, it means we deal presently with the Kramers
degeneration, if opposite βα HH = [4].
The characteristic polynomials (8 and 9) have both
the fourth degree. This means that their roots could be
obtained in radicals. However, it would be very
tiresome and scarcely usefully to review these bulky
expressions. It seems more important to find the
distribution of roots of these polynomials on signs,
especially in an asymptotic limit: ∞→k . The
coefficients of both polynomials should be identical, if
it is possible to neglect almost all elements, except for
those which contain senior (leading) degrees of ∞→k .
Let us write their asymptotic expressions:
4 3
2 2 2 2 2
2 1
2 2 2
0
1; ;
2; ( sin( ) ) ;
3
sin( )
3
ga a
a P k a P k
a P k
δ ε
δ θ
δ θ
= = ∆ + −
∆
= − = − +
∆⎛ ⎞= −⎜ ⎟
⎝ ⎠
It is accepted in these expressions an insignificant
simplifying 1≈η for the sake of greater simplicity and
transparency. The amount of changes of signs in the
sequence (21) determines the amount of positive roots
of the proper polynomial ( N ). This assertion is known
as Cartesian rule of signs [7].
Would we want to make no assumption about
parameters of (21), so the distribution of signs should
adopt the kind as following a pseudo-vector: (+,?,–,?,?).
There are only six variants of sign distributions turning
out a substitution in place of every indefinite sign (?) of
one of certain signs (±). It is easily to check that they
correspond to three possible values of 3,2,1=N , in
pairs. Well, such super-generalized approach is not too
fruitful. All, what is allowed to say is that every
polynomial has at least one negative root (i.e. a valence
band).
Let us accept only one, but physically quite obvious
assumption that 0>∆ . As a result, we can immediately
write two different pseudo-vectors: (+,?,–,–,–) if 0>δ ,
and (+,?,–,?,+), if 0<δ . Moreover, it is 1=N (if 0>δ )
or 2=N (if 0<δ ), independently of the variants of the
any substitutions instead of indefinite signs. Thus each
polynomial has at least one (if 0>δ ) or even two (if
0<δ ) positive roots, but no more. One of them is the
conductivity band, whereas three negative roots of
polynomial correspond to three valence bands, of
course. Note that these assertions base just on one,
.
Semiconductor Physics, Quantum Electronics & Optoelectronics. 2004. V. 8, N 1. P. 28-31.
© 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
31
physically evident assumption about sign of one
parameter.
However, the possibility of two positive roots as the
alternative to the typical conditions a bit washes out this
analysis. The nature of the second positive root needs
the additional study in this case, in dependence off other
parameters of model. Thus, it as yet is few outside of
the area of the presented investigation.
3. Conclusions
The Kramers degeneration survives only along and
against of the direction of the main crystal axis and at
point Γ ( 0=k ) within generalized Kildal’s model
under the condition of the absence of the center of
symmetry, as a rule.
The Kramers degeneration is possible everywhere,
although only for the band of the heavy holes like those
within Kane’s model, even under the condition of pt.1,
but only if casually takes place the special relation
between parameters: ( )
2
2
3
1
η
ηδ −∆
= .
The typical set of solutions within this model consist
of one conductivity band ( 0≥ε ) and of three valence
bands ( 0≤ε ). Each of them contains two spin sub-
bands. The additional conductivity band would be
possible in principle, but only under special condition
( 0<δ ), which is the obligatory but not enough
condition.
References
1. G. Chuiko, N. Don, O. Dvornik, V. Ivchenko,
A. Sergeev, Simple inverted band structure model
for cadmium arsenide (Cd3As2) // Moldavian Jour-
nal of the Physical Sciences, 2(1), p. 88-94 (2003).
2. H. Kildal, Band structure of 2CdGeAs near k = 0 //
Phys. Rev., 10(12), p. 5082-5087 (1974).
3. E.O. Kane, Band structure of indium antimonide // J.
Phys.Chem. Solids 1, p. 249-261 (1958).
4. G.L. Bier, G.E. Pikus, Symmetrya i deformatsionnye
effekty w poluprovodnikach, Nauka, Moscow (1972)
(in Russian).
5. V.V. Woyevodin, Y.A. Kuznetsov, Matricy i vychis-
lenya, Nauka, Moscow (1984) (in Russian).
6. A.G. Kurosh, Kurs vysshey algebry, Nauka,
Moscow (1968) (in Russian).
7. G.A. Korn, T.M. Korn. Mathematical Handbook for
scientists and engineers, McGraw Hill Book
Company, New York–San Francisco–Toronto-
London–Sydney (1968).
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