- Email: [email protected]
Magnetic moment of a three-dimensional quantum well in a quantizing magnetic field
20 July 1998
PHYSICS LETTERS A
ELSEiX’IER
Physics Letters A 244 ( 1998) 295-302
Magnetic moment of a three-dimensional quantum well in a quantizing magnetic field L.I. Filina, V.A. Geyler, V.A. Margulis, O.B. Tomilin Department
Received
of Physics, Mordovian
State University,
14 April 1997; revised manuscript received Communicated
43oooO Saran&
1 April 1998; accepted by L.J. Sham
Russia
for publication
3 April 1998
Abstract The magnetic moment of an electron gas in a three-dimensional quantum well is studied. It is shown that two kinds of oscillations arise: oscillations caused by a change of the magnetic field strength and those caused by a change in the field direction. @ 1998 Elsevier Science B.V.
1. Introduction
The attract considerable this is true for the magnetic properties of such systems, the size and here [l-12]. This circumstance leads to new physical phenomena and makes it possible to study various parameters of the size quantization potential of the system (i.e. of the confinement potential). Inasmuch as the exact form of the confinement potential in real systems is not determined experimentally, different model potentials are employed in theoretical investigations. In the literature a parabolic potential is widely used for this purpose. The choice of this potential is justified by a number of factors. On the one hand, it has been proved rigorously that any confinement potential for high-energy levels is well approximated by a parabolic potential [ 131. the other even in presence can be obtained with help the phase space. spectrum reduces to algebraic sum the spectra The electron-electron interaction strongly the energy spectrum of quantum dots when Coulomb interaction energy than or comparable the confinement potential [ 141. It leads to new effects in the orbital magnetism of these structures, since the competition between magneto-orbital confinement and Coulomb repulsion leads to a transition in the few-particle ground state and additional level crossing in the excitation spectrum with increasing magnetic field. In this paper we consider the case when the charging energy is much less than the confinement potential (i.e. when the dot size lo and the effective Bohr radius a* satisfy the condition lo/a* < 1 [ 141. 037%9601/98/$19.00 @ 1998 Elsevier Science B.V. All rights resewed. P/f SO375-9601(98)00285-O
296
L.I.
Filina et al. /Physics Letters A 244 (1998) 295-302
-.2
1
Fig. I. Spectrum of
H as a
3
4
~--a
5
omin,
function of the magnetic field strength B; p = r/4, 8 = s/4, 01 = L43= 1012 s-l,
02 = 3 x 1012 s-l.
It follows from considerations based on the density of states (DOS) behaviour that the thermodynamic potential 0 of the system undergoes oscillations of the de Haas-van Alphen type. These oscillations result from the crossing of an energy level with the chemical potential of the system. In this paper we present a theoretical study of the magnetic moment of an electron gas in a three-dimensional quantum well subjected to a uniform magnetic field arbitrarily directed with respect to the potential symmetry axes. The confinement potential of the system is chosen in the form
U(x,y,z)
=
;(.n;1x*+fl;y*+L$z*),
where m is the effective mass of the electron and a,, 02.0~ are the characteristic frequencies of the potential. We note that the particular case of the potential (1) (0, = 02) has been used in Ref. [ 141 for studying the electronic structure of three-dimensional quantum dots in a tilted magnetic field. The spinless one-particle Hamiltonian of an electron in the considered system has the form 2
It is convenient to choose the following gauge for the vector potential A of the magnetic field B, A= ( $B2z - B3y, 0, Bly - ;B2x). By means of a linear canonical transformation of the phase space the Hamiltonian H(p,q) new phase coordinates P, Q in terms of which H has the canonical form H(P,Q,=~(P:+p:+P~)+~(o:e:+o:e:+o:e:,,
can be reduced to
(3)
where the wj are found from the cubic equation [ 151 (~-n>(~~-X)(~-x)-~O:,(~-x>x-w~(~-X)X-~0:,(~-~)x=o.
(4)
Here w,iC= leBj/mcl are the components of the cyclotron frequency. The spectrum of the Hamiltonian (2) by virtue of Eq. (3) has the form snn,/ = fiwl(n + $) +
fio2(m
+
i)
+
rb3(Z+
$>,
n,m,l=O,l,...
(5)
L.I. Filina et al. /Physics
Letters A 244 (1998) 295-302
297
bn,’ 9 8, 7,
I
0
__~-
1
___-a
x/2
Fig. 3. Spectrum of H ils a function of the angle 0; B = 1 T, (p = 7r/4, Rt =
= 0 03
=
lot* s-‘, L!2= 3 x lOI* s-‘.
The E~,~,Iversus B dependence as well as the dependence on the azimuth angle 8 and the polar angle 4p of the direction of B is depicted in Figs. l-3. Clearly, there is strong level crossing in the region w, < L?,, L&, 0s while in the field range where w, > 4, &, 03 (strong magnetic quantization) there is no level crossing at least for low levels.
2. Magnetic
moment
of the quantum
well
We find the classical partition function Z of the system of particles with the spectrum Z = Cnrlr, exp( -e,,,,/T). According to this formula, we obtain
(5) using the formula
Z-t=gsinh(%)sinh(%)sinh(%). We now determine
the thermodynamic
(6) potential
0 of the electron gas using the formula
[ 161
298
L.I. Filina et al. /Physics Letters A 244 (1998) 295-302 afica
s
rr-ice
expti-4
‘(‘)
(7)
&in( xrT() d5’
where ,u is the chemical potential of the gas and 0 < cr < T- ‘. The integrand in Eq. (7) is a meromorphic function which has simple poles at the points tjn = 2rin/huj on the complex &plane and a pole of the multiplicity five at the point 5 = 0. We close the contour of integration in the left half-plane and reduce the considered integral to the sum of the residues in the poles. As a result, we obtain that 0 = Jlmo”+ LPsc, where the monotonic part firnon of the thermodynamic potential is determined by the contribution of the pole at the point ,$ = 0, and the oscillatory part LPsc arises from the contribution of the poles at the points ljn. Using the Vibte relation between the coefficients and the roots of Eq. (4), we obtain
Applying the formula Mj = -80/6’Bj for the components of the magnetic moment vector of the electron gas, we get M,y = 0. The oscillatory part of R has the form
cos( 27rnp/fiiw2)
+ .
slnh(2rr2nT/~iw2)sin(rrnwl/W2)sin(~no3/02)
+
cos( %rnp/ko3)
sinh(27r2nT//&)sin(?mwi/03)sin(
rrnwJog)
>. (9)
It is obvious that formula (9) becomes meaningless if some of the frequencies Wj are comparable. However, even if all the frequencies are not comparable the Fourier series in formula (9) has high-order terms when the quantities nw;/wj (i # j) are close to integers. In this case, when summing the series an obstacle arises similar to the obstacle well known in celestial mechanics as the problem of the small denominators. In the KAM theory this obstacle is overcome by considering only frequencies uj which satisfy a Diophantine incommensurability condition [ 171. We shall use such a kind of incommensurability condition, namely, the Roth condition. A real number Y satisfies the Roth condition if there is a constant C(Y) such that 1~ - n/ml > c/m3 for all integers m and n (m # 0) [ 171. It is known that the Roth condition is valid for all real numbers with probability 1 [ 181. Hence the series (9) converges absolutely with probability 1. Differentiating only the rapidly oscillating factors in Eq. (9)) we find sin(27rnp/tii) w: dtijc sinh(27r*nT/tiwi) sin(rrnoz/oi)
-___
sin(Irnos/oi)
sin (27rnp/hd2) 1 &0* +-w; dwjc sinh( 2r*nT/ZiW2) sin( rnwI /02) sin( rnw3/w2) sin( 27rnp/fiiws) i au3 w: dtijc sinh( 2v2nT/hw3) sin( 7rnwi /w3) sin( rrnoz/ws) > ’
+--
where ps is the Bohr magneton, amI
WjcWI
aojc= (ld-
mo
(6Jf- L$)
o;,
(10)
is the free electron mass,
(11)
and the other derivatives &i/&j, are obtained from Eq. ( 11) by the corresponding permutations of the indices. The convergence of the series ( 10) is proved in the same way as that of the series (9).
L.I. Filina et al. /Physics
Letters A 244 (1998) 295-302
Fig. 4. Dependence of M on the angle 8; cp = 7r/6, T = 10 K, Rt = 4 x lo’* s-‘, 02 = 1.1 x lOI The integers I and 2 refer to the values of the magnetic field of 1 T and 2.5 T, respectively.
299
s-‘,
03 = 6
Fig. 5. Magnetic moment as a function of the angle cp; 0 = 7r/6, T = 10 K, L$ = 1012 s-‘, 02 = 3 x lOI p = IO- I3 erg. The integers 1 and 2 refer to the values of the magnetic field of 2 T and 3 T, respectively.
x
1012 s-‘,
s-t,
p = IO-13erg.
03 = 2 x 1012 s-l,
3.Conclusion The classical consideration (when the energy quantization for electrons is absent) corresponds to the conditions Fiw,j<< T,j = 1,2,3. In this case Eq. (6) gives 2 = T3/FiWi~2~s. Taking into account the identity wiw~ws = 0, &Os we get that 2 (and, in virtue of Eq. (7), 0) is independent of the field B. Hence the magnetic moment of the electron gas vanishes. This situation is similar to the absence of the Landau diamagnetism under the classical consideration [ 191. The presence of the magnetic moment in the quantum case bj >> T results from the energy quantization for electrons. At T = 0 there is a simple relation between the electron density of states v(E) in the well and the thermodynamic potential R,
300
L.I. Filina et al. /Physics
Letters
A 244
(1998)
29%302
+
60
-100 :
Fig. 6. Dependence of M on the field strength B; 0 = k/18, (p = 5?r/l’2, f21 = 5 x 10” s-‘, 02 = 8.4 X lOI s-‘, p = IO-‘j erg. The integers 1 und 2 refer to the values of the temperatureof 4.5 K and 6 K. Nqectively.
03 =‘lO’z s-‘,
Fig. 7. Magnetic moment M as a function of temperature;B = 5~/18, cp= 5a/12, 0, = 5 x lo’2 s-‘, 4 = 8.4~ 1OL2 s-‘, a3 = 10” s-*, 1O-l3 erg. The integers 1, 2. 3 refer to fhe foIlowing values of the magnetic field, 1.5 T, 1 T, 2 T (for M, and ML); 1 T. 2 T, 1.5 T (for M,).
,U=
nfim
V(P) =
#R(T=O) au2
1 = -16~i
J
epl dS sinh( Awl l/2)
sinh( ru1)25/2) sinh( fk&+/L?) *
(12)
a-ice
Eqs. (6), (7), and ( 12) show that the oscillations of Y(,u) as well as fl(T = 0) and M(T = 0) have a common origin. Namely, these oscillations are determined by analytical properties of the partition function Z(g); more precisely, by the contribution of its poles on the imaginary axis. On the other hand, V(P) = &,,, S(p - enmr), and hence DOS on the Fermi level varies step-like when an energy level %rnl of the quantum well crosses this Fermi level. Therefore, the physical nature of the oscillations in 0 and M isrelated with the DOS oscillations on the Fermi level under changes of the value or direction of the magnetic field B. It is directly obvious from expression ( 10) that the magnetic moment Mj undergoes two kinds of oscillations, namely, oscillations caused by a change in the strength of a magnetic field and those caused by a change in the direction of the field.
L.I. Filina et al. /Physics
Letters A 244 (1998) 295-302
301
The behaviour of M as a function of 19, 9 and B is depicted in Figs. 4-6, respectively. Formula (10) shows that the dependence of the components of the magnetic response on the angles 8 and p is defined by two factors: first, by the dependence of tijc on these angles and second, by the dependence of w_i on the angles. In particular, the vanishing of M, and M, at the points 0 and 7r and the vanishing of M, at the point 7r/2 (see Fig. 4) is caused by peculiarities of the value wjc as a function of 6’. The deviation of the curves M, and M, from sine-shaped ones (curve 2 in Fig. 4) and the deviation of M, from cos 6 are caused by peculiarities of the value tij as a function of 8. Fig. 5 shows a similar situation with the Q dependence. Moreover, the pecularities of the plot of M, are determined only by the behaviour of tij, whereas the behaviour of the pecularities of M, and MY is more complicated. In particular, M, vanishes at the points cp = 7r/2,3~/2 and MY vanishes at the point 40 = 0, n-, 27r; this property is related with the behaviour of the Ojc as functions of 40. A deviation of the curves M, and M, from the graphs of the functions cos cp and sin p, respectively, in Fig. 5 is related to the form of the functions wj( 40). Formula ( 10) shows that the peak amplitudes in Mj (0,) must increase as the temperature decreases; Fig. 6 clearly shows this property. The dependence of the magnetic moment on variations of the temperature T is depicted in Fig. 7. It should be noted that at fixed values of the angles 0 and rp, the components Mj(T) of the magnetic moment can be monotonic functions of T (both increasing and decreasing) as well as nonmonotonic ones; the form of the M versus 7’ dependence is conditioned by the values of B (see Fig. 7). In the case of a general parabolic potential, the oscillations determined by the oscillator with the frequency w3 are missing when the direction of the magnetic field coincides with the direction of a symmetry axis of the potential. It should particularly be emphasized that parabolic confining potentials are successfully used for studying the magnetic and thermodynamic properties of low-dimensional systems (quantum wells, quantum dots) [ I ,3, IO- 12,20,2 11. In conclusion, we stress that if a system is in equilibrium with a thermostat, then the chemical potential is useful in describing the magnetic properties of an electron gas in the system, even though the latter has a nanoscale.
Acknowledgement This work was partly supported by the Russian Foundation for Fundamental Researches, the “Universities of Russia” Program, and by a Grant of the Russian Ministry of Education. We wish to thank L. A. Chernozatonskii and I. V. Stankevich for very helpful discussions. We are grateful to the referee for many valuable remarks.
References [ II R. Merlin, Solid State Commun. 64 (1987). 99. 121 A. Feldman, H.A. Kahn, Phys. Rev. B I ( 1970) 4584. [ 3 1 V.A. Geyler, V.A. Margulis, A.G. Nesmelov. 1.1. Chuchaev, Sov. Phys. Solid State 36 (1994) 1090. [ 41 M. Wagner, U. Merkt, A.V. Chaplik, Phys. Rev. B 45 (1992) 1951. [ 5 1 B. Meurer, D. Heitmann. K. Ploog, Phys. Rev. B 48 ( 1993) 11488. 161 U. Merkt. J. Huser, M. Wagner, Phys. Rev. B 43 (1991) 7320. 17 I P.A. Maksym, T. Chakraborty, Phys. Rev. Lett. 65 (1990) 108; Phys. Rev. B 45 (1992) 1947. 181 N.F. Johnson, M.S. Payne, Phys Rev. Len. 67 (1991) 1157. 191 D. Pfannkuche, V. Gudmundsson, P.A. Maksym, Phys. Rev. B 47 (1993) 2244. [ IO/ V.A. Geyler. V.A. Margulis, O.B. Tomilin, JETP Lett. 63 (1996) 578. I I I I Y. Meyr, 0. Entin-Wohlman, Y. Gefen, Phys. Rev. B 42 ( 1990) 8351. 1121 V.A. Geyler, V.A. Margulis, 1.V. Chudaev, JETP 82 (1996) 409. I 13] H.I. Cycon et al., Schrodinger Operators, with Applications to Quantum Mechanics and Global Geometry 1987).
(Springer,
New York,
302 [ 14 I
L.I.
Filina et al. /Physics
Letters A 244 (1998) 295-302
J.H. Oh, K.J. Chang, G. Ihm, S.J. Lee, Phys. Rev. B 50 (1994) 15397.
[ 151Q.P, Li, K. Karai, S.K. Yip, S. Das Sarma, H.D. Drew, Phys. Rev. B 43 (1991) 5151. 1161 A.M. Perelomov, Generalized Coherent states and Their Applications (in Russian) (Nauka, Moscow, 1987) [ 17 / V.l. Arnol’d, Russian Math. Surveys 18 ( 1963) 9. I 18I K.F. Roth, Mathematika 2 ( 1955) 1. [ 19 I D. Shoe&erg, Magnetic Oscillations in Metals (Cambridge University hess, Cambridge, 1984). 120) J.S. De Groote, J.E.M. Hornos, A.V. Chaplik, Phys. Rev. B 46 (1992) 12773. [21 I V.A. Geyler, V.A. Margulis. Phys. Rev. B 55 ( 1997) 2543.
PHYSICS LETTERS A
ELSEiX’IER
Physics Letters A 244 ( 1998) 295-302
Magnetic moment of a three-dimensional quantum well in a quantizing magnetic field L.I. Filina, V.A. Geyler, V.A. Margulis, O.B. Tomilin Department
Received
of Physics, Mordovian
State University,
14 April 1997; revised manuscript received Communicated
43oooO Saran&
1 April 1998; accepted by L.J. Sham
Russia
for publication
3 April 1998
Abstract The magnetic moment of an electron gas in a three-dimensional quantum well is studied. It is shown that two kinds of oscillations arise: oscillations caused by a change of the magnetic field strength and those caused by a change in the field direction. @ 1998 Elsevier Science B.V.
1. Introduction
The attract considerable this is true for the magnetic properties of such systems, the size and here [l-12]. This circumstance leads to new physical phenomena and makes it possible to study various parameters of the size quantization potential of the system (i.e. of the confinement potential). Inasmuch as the exact form of the confinement potential in real systems is not determined experimentally, different model potentials are employed in theoretical investigations. In the literature a parabolic potential is widely used for this purpose. The choice of this potential is justified by a number of factors. On the one hand, it has been proved rigorously that any confinement potential for high-energy levels is well approximated by a parabolic potential [ 131. the other even in presence can be obtained with help the phase space. spectrum reduces to algebraic sum the spectra The electron-electron interaction strongly the energy spectrum of quantum dots when Coulomb interaction energy than or comparable the confinement potential [ 141. It leads to new effects in the orbital magnetism of these structures, since the competition between magneto-orbital confinement and Coulomb repulsion leads to a transition in the few-particle ground state and additional level crossing in the excitation spectrum with increasing magnetic field. In this paper we consider the case when the charging energy is much less than the confinement potential (i.e. when the dot size lo and the effective Bohr radius a* satisfy the condition lo/a* < 1 [ 141. 037%9601/98/$19.00 @ 1998 Elsevier Science B.V. All rights resewed. P/f SO375-9601(98)00285-O
296
L.I.
Filina et al. /Physics Letters A 244 (1998) 295-302
-.2
1
Fig. I. Spectrum of
H as a
3
4
~--a
5
omin,
function of the magnetic field strength B; p = r/4, 8 = s/4, 01 = L43= 1012 s-l,
02 = 3 x 1012 s-l.
It follows from considerations based on the density of states (DOS) behaviour that the thermodynamic potential 0 of the system undergoes oscillations of the de Haas-van Alphen type. These oscillations result from the crossing of an energy level with the chemical potential of the system. In this paper we present a theoretical study of the magnetic moment of an electron gas in a three-dimensional quantum well subjected to a uniform magnetic field arbitrarily directed with respect to the potential symmetry axes. The confinement potential of the system is chosen in the form
U(x,y,z)
=
;(.n;1x*+fl;y*+L$z*),
where m is the effective mass of the electron and a,, 02.0~ are the characteristic frequencies of the potential. We note that the particular case of the potential (1) (0, = 02) has been used in Ref. [ 141 for studying the electronic structure of three-dimensional quantum dots in a tilted magnetic field. The spinless one-particle Hamiltonian of an electron in the considered system has the form 2
It is convenient to choose the following gauge for the vector potential A of the magnetic field B, A= ( $B2z - B3y, 0, Bly - ;B2x). By means of a linear canonical transformation of the phase space the Hamiltonian H(p,q) new phase coordinates P, Q in terms of which H has the canonical form H(P,Q,=~(P:+p:+P~)+~(o:e:+o:e:+o:e:,,
can be reduced to
(3)
where the wj are found from the cubic equation [ 151 (~-n>(~~-X)(~-x)-~O:,(~-x>x-w~(~-X)X-~0:,(~-~)x=o.
(4)
Here w,iC= leBj/mcl are the components of the cyclotron frequency. The spectrum of the Hamiltonian (2) by virtue of Eq. (3) has the form snn,/ = fiwl(n + $) +
fio2(m
+
i)
+
rb3(Z+
$>,
n,m,l=O,l,...
(5)
L.I. Filina et al. /Physics
Letters A 244 (1998) 295-302
297
bn,’ 9 8, 7,
I
0
__~-
1
___-a
x/2
Fig. 3. Spectrum of H ils a function of the angle 0; B = 1 T, (p = 7r/4, Rt =
= 0 03
=
lot* s-‘, L!2= 3 x lOI* s-‘.
The E~,~,Iversus B dependence as well as the dependence on the azimuth angle 8 and the polar angle 4p of the direction of B is depicted in Figs. l-3. Clearly, there is strong level crossing in the region w, < L?,, L&, 0s while in the field range where w, > 4, &, 03 (strong magnetic quantization) there is no level crossing at least for low levels.
2. Magnetic
moment
of the quantum
well
We find the classical partition function Z of the system of particles with the spectrum Z = Cnrlr, exp( -e,,,,/T). According to this formula, we obtain
(5) using the formula
Z-t=gsinh(%)sinh(%)sinh(%). We now determine
the thermodynamic
(6) potential
0 of the electron gas using the formula
[ 161
298
L.I. Filina et al. /Physics Letters A 244 (1998) 295-302 afica
s
rr-ice
expti-4
‘(‘)
(7)
&in( xrT() d5’
where ,u is the chemical potential of the gas and 0 < cr < T- ‘. The integrand in Eq. (7) is a meromorphic function which has simple poles at the points tjn = 2rin/huj on the complex &plane and a pole of the multiplicity five at the point 5 = 0. We close the contour of integration in the left half-plane and reduce the considered integral to the sum of the residues in the poles. As a result, we obtain that 0 = Jlmo”+ LPsc, where the monotonic part firnon of the thermodynamic potential is determined by the contribution of the pole at the point ,$ = 0, and the oscillatory part LPsc arises from the contribution of the poles at the points ljn. Using the Vibte relation between the coefficients and the roots of Eq. (4), we obtain
Applying the formula Mj = -80/6’Bj for the components of the magnetic moment vector of the electron gas, we get M,y = 0. The oscillatory part of R has the form
cos( 27rnp/fiiw2)
+ .
slnh(2rr2nT/~iw2)sin(rrnwl/W2)sin(~no3/02)
+
cos( %rnp/ko3)
sinh(27r2nT//&)sin(?mwi/03)sin(
rrnwJog)
>. (9)
It is obvious that formula (9) becomes meaningless if some of the frequencies Wj are comparable. However, even if all the frequencies are not comparable the Fourier series in formula (9) has high-order terms when the quantities nw;/wj (i # j) are close to integers. In this case, when summing the series an obstacle arises similar to the obstacle well known in celestial mechanics as the problem of the small denominators. In the KAM theory this obstacle is overcome by considering only frequencies uj which satisfy a Diophantine incommensurability condition [ 171. We shall use such a kind of incommensurability condition, namely, the Roth condition. A real number Y satisfies the Roth condition if there is a constant C(Y) such that 1~ - n/ml > c/m3 for all integers m and n (m # 0) [ 171. It is known that the Roth condition is valid for all real numbers with probability 1 [ 181. Hence the series (9) converges absolutely with probability 1. Differentiating only the rapidly oscillating factors in Eq. (9)) we find sin(27rnp/tii) w: dtijc sinh(27r*nT/tiwi) sin(rrnoz/oi)
-___
sin(Irnos/oi)
sin (27rnp/hd2) 1 &0* +-w; dwjc sinh( 2r*nT/ZiW2) sin( rnwI /02) sin( rnw3/w2) sin( 27rnp/fiiws) i au3 w: dtijc sinh( 2v2nT/hw3) sin( 7rnwi /w3) sin( rrnoz/ws) > ’
+--
where ps is the Bohr magneton, amI
WjcWI
aojc= (ld-
mo
(6Jf- L$)
o;,
(10)
is the free electron mass,
(11)
and the other derivatives &i/&j, are obtained from Eq. ( 11) by the corresponding permutations of the indices. The convergence of the series ( 10) is proved in the same way as that of the series (9).
L.I. Filina et al. /Physics
Letters A 244 (1998) 295-302
Fig. 4. Dependence of M on the angle 8; cp = 7r/6, T = 10 K, Rt = 4 x lo’* s-‘, 02 = 1.1 x lOI The integers I and 2 refer to the values of the magnetic field of 1 T and 2.5 T, respectively.
299
s-‘,
03 = 6
Fig. 5. Magnetic moment as a function of the angle cp; 0 = 7r/6, T = 10 K, L$ = 1012 s-‘, 02 = 3 x lOI p = IO- I3 erg. The integers 1 and 2 refer to the values of the magnetic field of 2 T and 3 T, respectively.
x
1012 s-‘,
s-t,
p = IO-13erg.
03 = 2 x 1012 s-l,
3.Conclusion The classical consideration (when the energy quantization for electrons is absent) corresponds to the conditions Fiw,j<< T,j = 1,2,3. In this case Eq. (6) gives 2 = T3/FiWi~2~s. Taking into account the identity wiw~ws = 0, &Os we get that 2 (and, in virtue of Eq. (7), 0) is independent of the field B. Hence the magnetic moment of the electron gas vanishes. This situation is similar to the absence of the Landau diamagnetism under the classical consideration [ 191. The presence of the magnetic moment in the quantum case bj >> T results from the energy quantization for electrons. At T = 0 there is a simple relation between the electron density of states v(E) in the well and the thermodynamic potential R,
300
L.I. Filina et al. /Physics
Letters
A 244
(1998)
29%302
+
60
-100 :
Fig. 6. Dependence of M on the field strength B; 0 = k/18, (p = 5?r/l’2, f21 = 5 x 10” s-‘, 02 = 8.4 X lOI s-‘, p = IO-‘j erg. The integers 1 und 2 refer to the values of the temperatureof 4.5 K and 6 K. Nqectively.
03 =‘lO’z s-‘,
Fig. 7. Magnetic moment M as a function of temperature;B = 5~/18, cp= 5a/12, 0, = 5 x lo’2 s-‘, 4 = 8.4~ 1OL2 s-‘, a3 = 10” s-*, 1O-l3 erg. The integers 1, 2. 3 refer to fhe foIlowing values of the magnetic field, 1.5 T, 1 T, 2 T (for M, and ML); 1 T. 2 T, 1.5 T (for M,).
,U=
nfim
V(P) =
#R(T=O) au2
1 = -16~i
J
epl dS sinh( Awl l/2)
sinh( ru1)25/2) sinh( fk&+/L?) *
(12)
a-ice
Eqs. (6), (7), and ( 12) show that the oscillations of Y(,u) as well as fl(T = 0) and M(T = 0) have a common origin. Namely, these oscillations are determined by analytical properties of the partition function Z(g); more precisely, by the contribution of its poles on the imaginary axis. On the other hand, V(P) = &,,, S(p - enmr), and hence DOS on the Fermi level varies step-like when an energy level %rnl of the quantum well crosses this Fermi level. Therefore, the physical nature of the oscillations in 0 and M isrelated with the DOS oscillations on the Fermi level under changes of the value or direction of the magnetic field B. It is directly obvious from expression ( 10) that the magnetic moment Mj undergoes two kinds of oscillations, namely, oscillations caused by a change in the strength of a magnetic field and those caused by a change in the direction of the field.
L.I. Filina et al. /Physics
Letters A 244 (1998) 295-302
301
The behaviour of M as a function of 19, 9 and B is depicted in Figs. 4-6, respectively. Formula (10) shows that the dependence of the components of the magnetic response on the angles 8 and p is defined by two factors: first, by the dependence of tijc on these angles and second, by the dependence of w_i on the angles. In particular, the vanishing of M, and M, at the points 0 and 7r and the vanishing of M, at the point 7r/2 (see Fig. 4) is caused by peculiarities of the value wjc as a function of 6’. The deviation of the curves M, and M, from sine-shaped ones (curve 2 in Fig. 4) and the deviation of M, from cos 6 are caused by peculiarities of the value tij as a function of 8. Fig. 5 shows a similar situation with the Q dependence. Moreover, the pecularities of the plot of M, are determined only by the behaviour of tij, whereas the behaviour of the pecularities of M, and MY is more complicated. In particular, M, vanishes at the points cp = 7r/2,3~/2 and MY vanishes at the point 40 = 0, n-, 27r; this property is related with the behaviour of the Ojc as functions of 40. A deviation of the curves M, and M, from the graphs of the functions cos cp and sin p, respectively, in Fig. 5 is related to the form of the functions wj( 40). Formula ( 10) shows that the peak amplitudes in Mj (0,) must increase as the temperature decreases; Fig. 6 clearly shows this property. The dependence of the magnetic moment on variations of the temperature T is depicted in Fig. 7. It should be noted that at fixed values of the angles 0 and rp, the components Mj(T) of the magnetic moment can be monotonic functions of T (both increasing and decreasing) as well as nonmonotonic ones; the form of the M versus 7’ dependence is conditioned by the values of B (see Fig. 7). In the case of a general parabolic potential, the oscillations determined by the oscillator with the frequency w3 are missing when the direction of the magnetic field coincides with the direction of a symmetry axis of the potential. It should particularly be emphasized that parabolic confining potentials are successfully used for studying the magnetic and thermodynamic properties of low-dimensional systems (quantum wells, quantum dots) [ I ,3, IO- 12,20,2 11. In conclusion, we stress that if a system is in equilibrium with a thermostat, then the chemical potential is useful in describing the magnetic properties of an electron gas in the system, even though the latter has a nanoscale.
Acknowledgement This work was partly supported by the Russian Foundation for Fundamental Researches, the “Universities of Russia” Program, and by a Grant of the Russian Ministry of Education. We wish to thank L. A. Chernozatonskii and I. V. Stankevich for very helpful discussions. We are grateful to the referee for many valuable remarks.
References [ II R. Merlin, Solid State Commun. 64 (1987). 99. 121 A. Feldman, H.A. Kahn, Phys. Rev. B I ( 1970) 4584. [ 3 1 V.A. Geyler, V.A. Margulis, A.G. Nesmelov. 1.1. Chuchaev, Sov. Phys. Solid State 36 (1994) 1090. [ 41 M. Wagner, U. Merkt, A.V. Chaplik, Phys. Rev. B 45 (1992) 1951. [ 5 1 B. Meurer, D. Heitmann. K. Ploog, Phys. Rev. B 48 ( 1993) 11488. 161 U. Merkt. J. Huser, M. Wagner, Phys. Rev. B 43 (1991) 7320. 17 I P.A. Maksym, T. Chakraborty, Phys. Rev. Lett. 65 (1990) 108; Phys. Rev. B 45 (1992) 1947. 181 N.F. Johnson, M.S. Payne, Phys Rev. Len. 67 (1991) 1157. 191 D. Pfannkuche, V. Gudmundsson, P.A. Maksym, Phys. Rev. B 47 (1993) 2244. [ IO/ V.A. Geyler. V.A. Margulis, O.B. Tomilin, JETP Lett. 63 (1996) 578. I I I I Y. Meyr, 0. Entin-Wohlman, Y. Gefen, Phys. Rev. B 42 ( 1990) 8351. 1121 V.A. Geyler, V.A. Margulis, 1.V. Chudaev, JETP 82 (1996) 409. I 13] H.I. Cycon et al., Schrodinger Operators, with Applications to Quantum Mechanics and Global Geometry 1987).
(Springer,
New York,
302 [ 14 I
L.I.
Filina et al. /Physics
Letters A 244 (1998) 295-302
J.H. Oh, K.J. Chang, G. Ihm, S.J. Lee, Phys. Rev. B 50 (1994) 15397.
[ 151Q.P, Li, K. Karai, S.K. Yip, S. Das Sarma, H.D. Drew, Phys. Rev. B 43 (1991) 5151. 1161 A.M. Perelomov, Generalized Coherent states and Their Applications (in Russian) (Nauka, Moscow, 1987) [ 17 / V.l. Arnol’d, Russian Math. Surveys 18 ( 1963) 9. I 18I K.F. Roth, Mathematika 2 ( 1955) 1. [ 19 I D. Shoe&erg, Magnetic Oscillations in Metals (Cambridge University hess, Cambridge, 1984). 120) J.S. De Groote, J.E.M. Hornos, A.V. Chaplik, Phys. Rev. B 46 (1992) 12773. [21 I V.A. Geyler, V.A. Margulis. Phys. Rev. B 55 ( 1997) 2543.