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Quantum transport in semimagnetic HgMnTe inversion layers — Experiment and theory
Surface Science 142 (1984) 588-592
North-Holland. Amsterdam
QUANTUM TRANSPORT IN SEMIMAGNETIC LAYERS - EXPERIMENT AND THEORY G. GRABECKI, Institute
Received
of Physm.
T. DIETL,
J. KOSSUT
Polrsh Accrderql: of Scienws.
10 July 1983: accepted
for publicatton
02
HgMnTe
INVERSION
and W. ZAWADZKI -66X Wur.ww,
2 August
Poland
1983
The first experimental and theoretical study of MIS structures containing localized magnetic moments is reported. Hg, ~, Mn .Te structures with energy gap ranging from 50 to 200 meV have been investigated. In contrast lo MIS structures not containing localized moments. a pronounced temperature dependence of the positions of SdH maxima has been observed. A theory of two-dimensional, semimagnetic, narrow-gap structures has been developed which accounts qualitatively for the observations.
We have investigated experimentally MIS structures which consisted of an aluminum film evaporated on a thin Mylar foil (d = 3.5 pm) attached mechanically to the surface of p-type Hg, ~, Mn ,Te ( IV’~- N, = 2 X 10’” cm ‘). The semiconductor surface was prepared by mechanical polishing and etching in bromine-methanol solution. Capacitance measurements versus gate voltage C( V,) have shown a good stability of the structures. The total electron surface density N, is varied by changing the gate voltage Vg according to the relation: N, = C( V’ - V,,), where V,, is the inversion threshold voltage, corresponding to a crossing of the Fermi level and the conduction band-edge. The differential conductivity du/dVg versus gate voltage for various magnetic fields is shown in fig. 1. A standard sharp increase of the field mobility near V, = 0 corresponds to the inversion threshold. The decrease of the mobility for V, > 150 V is due to an increase of the surface scattering rate, which occurs when the electrons are driven closer to the surface. In magnetic fields B > 2 T magneto-conductance oscillations are observed, indicating a sufficient mobility of the surface electrons p > 5 X 10'cm’/V . s. Two periods of oscillations are observed, indicating that electrons occupy at least two electric subbands. From the data we deduce that for V, > 50 V about 70% of electrons are in the lowest subband. This is in agreement with calculations for Hg, .Cd,Te possessing similar band parameters [l]. We observe an influence of temperature on the positions of oscillation maxima, as shown in fig. 2. This is characteristic of the semimagnetic behavior, in which the paramagnetic magnetization of Mn 2+ ions is strongly temperature 0039-6028/84/$03.00 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
G. Graheckr
Total surface concentrahon
et al. / Qumtum
transport
589
(crnm2)
SamDIe A
I MnxTe P-Hi-: %
15T
f
/-
4.9 T
Gd -100 0 100 200 300 400 500 Gatevoltage (VI Fig. 1. Differential conductivity versus gate voltage for p-Hg,,, Mn, ,Te in various magnetic fields at 4.2 K.
dependent.
It can be seen in fig. 2 that two separate peaks (indicated by arrows) become one at higher temperatures and then again reappear as two. This is in contrast to the usual SdH behavior, for which the oscillations gradually disappear with increasing temperature. We refer to this observation in the theoretical part. A theory of the above system must take into account the following essential features: (1) energy band structure with a small energy gap cg and a strong spin-orbit interaction A; (2) quasi-two-dimensional character of the electron motion near the interface; (3) presence of a magnetic field transverse to the interface; (4) semimagnetic properties related to the exchange interaction between the conduction electrons and the Mn2+ ions. The complete Hamiltonian reads
H = He, + 5, + H,,, +
U(z ) + Hex,, >
where H,, = (1/2m,)( p + eA/c)‘, V, is the is the spin-orbit interaction, U is the U = cc for z < 0 and U = U(z) for z > 0. The where field approximation is Hexch = J(S,)s,, is the thermodynamic mean value of Mn2+ operator [2]. The band structure is described H,,
(1) periodic potential of the lattice, potential of the inversion layer: exchange interaction in the mean J is the exchange operator, (S.) spins and s= is the electron spin using a three-level model of r,,
590
G. Gruhwkr
Total surface
concentration
5.10" ~~ ~~~~
,(0
Sample
10’2 ,
et 01. /
Quonturn ~ransporr
(crr~‘)
~~~15.1tl’2 , ~___
2
0’2
B
p - Hg,_, Mn,Te x=10%
r____-.l
0
100
200
Gate ml tage
300 (VI
400
%I0
Fig. 2. Differential magnetoconductivity versus gate voltage for p-HgC19Mn, ,Te show the peaks which exhibit a strong temperature shift. Note absence of the spin splitting at intermediate temperatures T - 10 K.
r,, r, symmetry levels (8 Luttinger-Kohn functions. solutions in the form: + = Cf,u,,, the set of coupled the envelopes f, is obtained, c
[T,,, . P + H;Fch +(U+~,,--)~&/=o.
cf. ref. [3]). Looking differential equations
for for
(2)
where the matrix elements are calculated using the L-K band-edge functions and E,~ are the band-edge energies (at k = 0). The explicit form of matrix (1) can be found in refs. [2,3]. In the following we put A large and the set (1)
G. Gruhecki
er al. / Quur~tum
transport
591
reduces to 6 equations. The set can be solved to a good approximation by substitution neglecting the commutators [p,, U] (cf. ref. [4]), and small off-diagonal terms of the type b/c,. Here we limit our solutions to a parabolic approximation, which gives
where c,. are the electric subband energies and n = 0, 1, . . are the Landau from the band levels; wC= eB/m* and g * is the spin g-factor resulting structure (g* < 0, cf ref. [3]). The exchange interaction affects the spin splittings in two ways: it diminishes the splittings of all levels (a > 0) and it contributes an increasing term depending on n( b < 0). We denote a = ~x(S].J]S)(S.),
b =ix(
XlJlX)(S,),
(4)
where x is the MnTe mole fraction, (S]J]S) = -0.4 eV, (X]J]X) = 0.6 eV (cf. The latter ref. [5]). The quantity (SZ) < 0 is directly related to magnetization. depends strongly on a magnetic field (increasing function) and the temperature (decreasing function) and it has been measured independently for Hg,,,Mn,,,Te
161. The electric subband energies are calculated for the triangular U(z) = eEz. They are also modified by the semimagnetic terms, $mc(r+~)eE]*
CBT b/3
1 “j
potential
(5)
In the calculations we took the following values: m* = 0.0089 m,, g* = - 112, x lo4 V/cm. Fig. 3 shows calculated electron energies for the first two electric subbands at T = 4.2 K. It can be seen that: (1) the spin splittings exhibit peculiar behavior as functions of a magnetic field; (2) the splittings increase with the increasing Landau number n, which may lead to a rearrangement of the levels. Fig. 4 shows the Landau level energies for the lowest electric subband as functions of the temperature. Due to decreasing magnetization the semimagnetic terms contribute less at higher temperatures. As it can be seen, this leads to level crossings for higher n, as indicated above. We associate the experimental observation indicated in fig. 2 with the level crossing shown in fig. 4, both occurring for higher Landau levels. We have not attempted here a quantitative description of the experiments, since it would require a selfconsistent calculation of electron energies for the nonparabolic band, in which both m* and g* depend on the absolute positions of the electric subbands. E = 1.35
592
I
500
B=5T '
c
200 400
I
I
r=O
I v”:-
2 E 250
s
s t 6 150
w/
m
1
-:r
L
t I
100
0123456 Magnetic
field(T)
Fig. 3. Calculated magnetic field dependence at 4.2 K for the two lowest electric subbands: Fig. 4. Calculated temperature subband (r = 0) in Hg,,,Mn,, exchange mteraction between
z
3+
2-
200
2+ i-
1+
100
oil+
0
5 10 15 Temperature
of the Landau levels in Hg,,,Mn,, ,Te. E, = 100 meV solid lines. r = 0: dashed lines. r = 1.
dependences of the Landau levels associated with the lowest electric ,Te. Es = 100 meV in 5 T. The temperature dependence is due to the conduction electron and localised magnetic moments.
One of us (T.D) is greatly indebted to Professor F. Koch for giving him the opportunity to initiate studies of inversion layers on HgMnTe in the TUM laboratory and for many valuable discussions.
References [I] Y. Takada,
K. Arai and Y. Uemura. in: Lecture Notes in Physics, Vol. 152, Eds. E. Gornik et al. (Springer, Berlin, 1982) p. 101. [2] R.R. Galazka and J. Kossut. in: Lecture Notes in Physics. Vol. 133. Ed. W. Zawadzki (Springer, Berlin. 1980) p. 245. [31 W.. Zawadzki. in: Lecture Notes in Physics. Vol. 133, Ed. W. Zawadrki (Springer. Berlin. 1980) p. 85. [41 W. Zawadzki, J. Phys. C (Solid State Phys.) 16 (1983) 229. W. Dobrowolski, M. Otto, T. Diet1 and R.R. Galazka, in: Proc. 15th Intern. [51 M. Dobrowolska, Conf. on the Physics of Semiconductors [J. Phys. Sot. Japan 49 Suppl. A (1980) 8151. M. van Ortcnberg, A.M. Sandauer, R.R. Galgzka. A. Mycielski and R. 161 W. Dobrowolski. Pauthenet, in: Lecture Notes in Physics, Vol. 152, Eds. E Gornik et al. (Springer. Berlin, 1982) p. 302.
North-Holland. Amsterdam
QUANTUM TRANSPORT IN SEMIMAGNETIC LAYERS - EXPERIMENT AND THEORY G. GRABECKI, Institute
Received
of Physm.
T. DIETL,
J. KOSSUT
Polrsh Accrderql: of Scienws.
10 July 1983: accepted
for publicatton
02
HgMnTe
INVERSION
and W. ZAWADZKI -66X Wur.ww,
2 August
Poland
1983
The first experimental and theoretical study of MIS structures containing localized magnetic moments is reported. Hg, ~, Mn .Te structures with energy gap ranging from 50 to 200 meV have been investigated. In contrast lo MIS structures not containing localized moments. a pronounced temperature dependence of the positions of SdH maxima has been observed. A theory of two-dimensional, semimagnetic, narrow-gap structures has been developed which accounts qualitatively for the observations.
We have investigated experimentally MIS structures which consisted of an aluminum film evaporated on a thin Mylar foil (d = 3.5 pm) attached mechanically to the surface of p-type Hg, ~, Mn ,Te ( IV’~- N, = 2 X 10’” cm ‘). The semiconductor surface was prepared by mechanical polishing and etching in bromine-methanol solution. Capacitance measurements versus gate voltage C( V,) have shown a good stability of the structures. The total electron surface density N, is varied by changing the gate voltage Vg according to the relation: N, = C( V’ - V,,), where V,, is the inversion threshold voltage, corresponding to a crossing of the Fermi level and the conduction band-edge. The differential conductivity du/dVg versus gate voltage for various magnetic fields is shown in fig. 1. A standard sharp increase of the field mobility near V, = 0 corresponds to the inversion threshold. The decrease of the mobility for V, > 150 V is due to an increase of the surface scattering rate, which occurs when the electrons are driven closer to the surface. In magnetic fields B > 2 T magneto-conductance oscillations are observed, indicating a sufficient mobility of the surface electrons p > 5 X 10'cm’/V . s. Two periods of oscillations are observed, indicating that electrons occupy at least two electric subbands. From the data we deduce that for V, > 50 V about 70% of electrons are in the lowest subband. This is in agreement with calculations for Hg, .Cd,Te possessing similar band parameters [l]. We observe an influence of temperature on the positions of oscillation maxima, as shown in fig. 2. This is characteristic of the semimagnetic behavior, in which the paramagnetic magnetization of Mn 2+ ions is strongly temperature 0039-6028/84/$03.00 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
G. Graheckr
Total surface concentrahon
et al. / Qumtum
transport
589
(crnm2)
SamDIe A
I MnxTe P-Hi-: %
15T
f
/-
4.9 T
Gd -100 0 100 200 300 400 500 Gatevoltage (VI Fig. 1. Differential conductivity versus gate voltage for p-Hg,,, Mn, ,Te in various magnetic fields at 4.2 K.
dependent.
It can be seen in fig. 2 that two separate peaks (indicated by arrows) become one at higher temperatures and then again reappear as two. This is in contrast to the usual SdH behavior, for which the oscillations gradually disappear with increasing temperature. We refer to this observation in the theoretical part. A theory of the above system must take into account the following essential features: (1) energy band structure with a small energy gap cg and a strong spin-orbit interaction A; (2) quasi-two-dimensional character of the electron motion near the interface; (3) presence of a magnetic field transverse to the interface; (4) semimagnetic properties related to the exchange interaction between the conduction electrons and the Mn2+ ions. The complete Hamiltonian reads
H = He, + 5, + H,,, +
U(z ) + Hex,, >
where H,, = (1/2m,)( p + eA/c)‘, V, is the is the spin-orbit interaction, U is the U = cc for z < 0 and U = U(z) for z > 0. The where field approximation is Hexch = J(S,)s,, is the thermodynamic mean value of Mn2+ operator [2]. The band structure is described H,,
(1) periodic potential of the lattice, potential of the inversion layer: exchange interaction in the mean J is the exchange operator, (S.) spins and s= is the electron spin using a three-level model of r,,
590
G. Gruhwkr
Total surface
concentration
5.10" ~~ ~~~~
,(0
Sample
10’2 ,
et 01. /
Quonturn ~ransporr
(crr~‘)
~~~15.1tl’2 , ~___
2
0’2
B
p - Hg,_, Mn,Te x=10%
r____-.l
0
100
200
Gate ml tage
300 (VI
400
%I0
Fig. 2. Differential magnetoconductivity versus gate voltage for p-HgC19Mn, ,Te show the peaks which exhibit a strong temperature shift. Note absence of the spin splitting at intermediate temperatures T - 10 K.
r,, r, symmetry levels (8 Luttinger-Kohn functions. solutions in the form: + = Cf,u,,, the set of coupled the envelopes f, is obtained, c
[T,,, . P + H;Fch +(U+~,,--)~&/=o.
cf. ref. [3]). Looking differential equations
for for
(2)
where the matrix elements are calculated using the L-K band-edge functions and E,~ are the band-edge energies (at k = 0). The explicit form of matrix (1) can be found in refs. [2,3]. In the following we put A large and the set (1)
G. Gruhecki
er al. / Quur~tum
transport
591
reduces to 6 equations. The set can be solved to a good approximation by substitution neglecting the commutators [p,, U] (cf. ref. [4]), and small off-diagonal terms of the type b/c,. Here we limit our solutions to a parabolic approximation, which gives
where c,. are the electric subband energies and n = 0, 1, . . are the Landau from the band levels; wC= eB/m* and g * is the spin g-factor resulting structure (g* < 0, cf ref. [3]). The exchange interaction affects the spin splittings in two ways: it diminishes the splittings of all levels (a > 0) and it contributes an increasing term depending on n( b < 0). We denote a = ~x(S].J]S)(S.),
b =ix(
XlJlX)(S,),
(4)
where x is the MnTe mole fraction, (S]J]S) = -0.4 eV, (X]J]X) = 0.6 eV (cf. The latter ref. [5]). The quantity (SZ) < 0 is directly related to magnetization. depends strongly on a magnetic field (increasing function) and the temperature (decreasing function) and it has been measured independently for Hg,,,Mn,,,Te
161. The electric subband energies are calculated for the triangular U(z) = eEz. They are also modified by the semimagnetic terms, $mc(r+~)eE]*
CBT b/3
1 “j
potential
(5)
In the calculations we took the following values: m* = 0.0089 m,, g* = - 112, x lo4 V/cm. Fig. 3 shows calculated electron energies for the first two electric subbands at T = 4.2 K. It can be seen that: (1) the spin splittings exhibit peculiar behavior as functions of a magnetic field; (2) the splittings increase with the increasing Landau number n, which may lead to a rearrangement of the levels. Fig. 4 shows the Landau level energies for the lowest electric subband as functions of the temperature. Due to decreasing magnetization the semimagnetic terms contribute less at higher temperatures. As it can be seen, this leads to level crossings for higher n, as indicated above. We associate the experimental observation indicated in fig. 2 with the level crossing shown in fig. 4, both occurring for higher Landau levels. We have not attempted here a quantitative description of the experiments, since it would require a selfconsistent calculation of electron energies for the nonparabolic band, in which both m* and g* depend on the absolute positions of the electric subbands. E = 1.35
592
I
500
B=5T '
c
200 400
I
I
r=O
I v”:-
2 E 250
s
s t 6 150
w/
m
1
-:r
L
t I
100
0123456 Magnetic
field(T)
Fig. 3. Calculated magnetic field dependence at 4.2 K for the two lowest electric subbands: Fig. 4. Calculated temperature subband (r = 0) in Hg,,,Mn,, exchange mteraction between
z
3+
2-
200
2+ i-
1+
100
oil+
0
5 10 15 Temperature
of the Landau levels in Hg,,,Mn,, ,Te. E, = 100 meV solid lines. r = 0: dashed lines. r = 1.
dependences of the Landau levels associated with the lowest electric ,Te. Es = 100 meV in 5 T. The temperature dependence is due to the conduction electron and localised magnetic moments.
One of us (T.D) is greatly indebted to Professor F. Koch for giving him the opportunity to initiate studies of inversion layers on HgMnTe in the TUM laboratory and for many valuable discussions.
References [I] Y. Takada,
K. Arai and Y. Uemura. in: Lecture Notes in Physics, Vol. 152, Eds. E. Gornik et al. (Springer, Berlin, 1982) p. 101. [2] R.R. Galazka and J. Kossut. in: Lecture Notes in Physics. Vol. 133. Ed. W. Zawadzki (Springer, Berlin. 1980) p. 245. [31 W.. Zawadzki. in: Lecture Notes in Physics. Vol. 133, Ed. W. Zawadrki (Springer. Berlin. 1980) p. 85. [41 W. Zawadzki, J. Phys. C (Solid State Phys.) 16 (1983) 229. W. Dobrowolski, M. Otto, T. Diet1 and R.R. Galazka, in: Proc. 15th Intern. [51 M. Dobrowolska, Conf. on the Physics of Semiconductors [J. Phys. Sot. Japan 49 Suppl. A (1980) 8151. M. van Ortcnberg, A.M. Sandauer, R.R. Galgzka. A. Mycielski and R. 161 W. Dobrowolski. Pauthenet, in: Lecture Notes in Physics, Vol. 152, Eds. E Gornik et al. (Springer. Berlin, 1982) p. 302.