HAL Id: jpa-00217975
https://hal.archives-ouvertes.fr/jpa-00217975
Submitted on 1 Jan 1978
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of
sci-entific research documents, whether they are
pub-lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
OSCILLATIONS OF THE RELAXATION TIME IN
STRONG MAGNETIC FIELDS
P. Středa, L. Smrćka
To cite this version:
JOURNAL DE PHYSIQUE Colloque C6, suppliment au no 8, Tome 39, aolit 1978, page C6-1110
O S C I L L A T ! O N S O F THE R E L A X A T I O N T I M E I N STRONG MAGNETIC F I E L D S
P. StFeda and L. SmrEka
I n s t i t u t e of Solid State Physics, CzechosZovak Acad. Sci., 162 53 Prague, CzechosZoualcia
R6sumb.- Les oscillations quantiques de la magn6toconductivit6 et de la force magn6to6lectrique sont calculbes dans un modSle de bande de conduction parabolique. L'on montre que la contribution principale 1 ces effets est due aux ascillations du temps de relaxation.
Abstract.- Quantum oscillations of the longitudinal magnetoconductivity and magnetothermopower were calculated for a simple model of solid with parabolic electron band. It is shown that the main con- tribution originates in oscillations of the relaxation time.
INTRODUCTION.- A strong magnetic field applied to the system of electrons causes the quantization of the electron energy spectrum. The quantum effects be- come particularly important at low temperatures and manifest themselves e.g. in the oscillatory behaviour of the magnetic susceptibility and transport coeffi- cients as functions of the magnetic field H. The theory predicts that both the density of electron states p(EF) and the electron mean lifetime =(EF) at the Fermi energy EF will oscillate while changing magnetic field /1,2/. The density of states give the main contribution to the oscillations of susceptibi- lity. In this communication we shall demonstrate
that, at least for a simple model, the oscillations of the longitudinal magnetoconductivity are mainly due to the oscillations in the relaxation time. This effect will even be more pronounced in the longitu-
dinal magnetothennopower S which is related to the longitudinal conductivity all by the Mott formula
written in the well known form
where the effective number of carriers NI, is given /2,4/ by relation
Here G denotes the resolvent
a I ~fi~k:
G~ = [E~
-
E~-
Z(E~)]-'
; E~ = HU(~ + +-
2m(4) where E is the eigenenergy of electron state a, a
a
substitutes for (n,kx,kz). The effective number N
I I
becomes close to the number of electrons only inthelow field limit w~ << 1 and if the Landau-Peierls criterion EF >> l? is well satisfied 141. Let us af- so note that for the short-range scattering poten- tial of impurities the relaxation time appearing in (2) is just equal to the mean lifetime.
(l) RESULTS AND DISCUSSION.- In order to analyse semi- quantitatively the behaviour of conductivity, it is If the scattering of electrons is elastic, and Hw useful to separate all quantities of interest into >> kT holds 131 (L is the Lorenz number, w = eH/mc the smooth and oscillatory parts using the Poisson is the cyclotron frequency). summation formula
+oJ +m +a
THE MODEL AND BASIC FORMULAE
.-
Let the electron stru-1
f (n) =+l
£(X)+
2I
[
f(x)cos(2~x)dx. (5) cture is given by a parabolic dispersion law, and n=o.b
n=16the randomly distributed impurities are described by Then we can write
YI
= No + NosCS T = 'lo+ 'losc
6-fynction-like potentials. The scattering of elec- and the longitudinii conductivity reads trons calculated self-consistently leads to the dam-
Nose
=osc )
,
ping of-singularities in the density of states and l/' = 0' +
-
No ) ( l +-
=o
yields the oscillatory self-energy of electrons C, where smooth n o = e 2 ~ o ~ o / m is multiplied by two oscil- which is related to the mean lifetime r.by expres- lating factors proportional to the relative magni- sion T = %/2r,
r
= -Id. The longitudinal conducti- t u d e ~ of oscillations in N and inx.vity was calculated via the Kubo formula and can be Since ~ ~ ~is approximately equal to the ~ /
11
' l ~p6sc/~o (posc/po is the relative magnitude of osci- llations in the density of states p(E F ) =
-
;
ImC G ) equations (33 and (5) yield the expression
CL a
Condition EF >> implies T ~ ~>>Nosc/~O ~ / T ~and so the oscillations in the relaxation time dominate in the oscillations of conductivity.
This conclusion is confirmed by the numeri- cal results presented in figure 1 which were obtai- ned for the model with parameters corresponding to the case of a strongly doped semiconductor with electron density 10"
Fig.1 : The relative oscillation amplitudes of T ~ ~ (full line) and Nosc/No (broken line) as ~ / T ~ functions of inverse magnetic field.
Though the quantity EF/lfiw ranges only from 2 to 7
and the condition EF >>
h
is only approximately valid, the% those of
relative changes of N I , do not exceed 10 'Cosc/~o '
The relative magnitude of oscillations of the longitudinal magnetoconductivity itself is shown on figure 2, together a' the derivative with
/l'
respect to the EF, which forms the main contribu- tion to the thermopower (1). The very large oscil- lations, which even change the sign of a;,, can be understood on the basis of intimate relation bet- ween the derivatives of a with respect to EF and
l/
with respect to 1 /H.
Fig. 2 : The relative oscillation amplitudes of uosc/~o (broken line) and U
'
/U: (full line) asOSC
functions of inverse magnetlc field.
References
/l/ Tanaka, S. and Morita, T., Progr. Theor. Phys. 47 (1972) 378.
121 Gerhardts, R. and Hajdu, J.Z., Physik 245 (1971) 126.
131 SmrEka, L. and ~t;eda, P., J. Phys. C 10 (1977) 2153.
