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Landau levels of holes in a two-dimensional channel
Superlattices
and Microstructures,
485
Vol. 70, No, 4, 1991
LANDAULEVELS OF HOLES IN A TWO-DIMENSIONAL CHANNEL Institute
Veronioa E. Bisti of Solid State Physics, USSR Academy of Sciences, 142432 Chernogolovka Moscow District. USSR (Received
12 October
1990)
A method for calculating the Landau levels in a two-dimensional channel is proposed. This method is aimed at construction of an effective two-dimensional Hamiltonian of holes H(k,,k,,B). The Hamiltonian is expanded in powers of the magnetic field B and noncommutlng momenta k, and k,. The results indicate an interconnection between effective It is masses and g-factors of the two-dimensional holes. possible to account for the effects of band warping i;hz non erturbative manner. The shift of phase in Shug nlkov-de Haas oscillations is explained.
1. Introduction Two-dimensional (2D) channels a;; NOS-structures, heterojUbn,C,t,ions duaiea wells have auantum are systems e’xtensively. These oharaoterized by the auantization of the perpendicular motion -and by the free carriers parallel to the motion of interface. In an n-channel layer, the motion parallel to the interface is hardly affected by the quantization of the perpendicular motion. In a normal we obtain the ordinary magnetic field, with the same values of Landau levels, and cyclotron frequency factor, just as in the three-dimensiona 9 case. In a D-channel layer, however. this is not the case. The -degeneracy of the valence bands and the notential V(z) combine to couple strongiy the paraliei The and perpe;g;ular motions. E(k) of the hole ai ersion sub ! ands is then highly nonparabolic.( k = (k, .ky} ) E(k) is calculated from the Shroedinger e uation (I) and consists of subbands doub P y degenerate at the point k=O_ Y = 0 (1) H, (k, , k, , kL ) - the three-dimensional 4x4 Luttinger Hamiltonian for the light and heavy hole bands. The picture of the Landau levels in ohannel layers is also more complex, !-hen in an n-channel. The previous are based on the calculations [l-4 Luttinger work 5 1 . If a normal to the interface and t I!e direction of magnetic field are taken to be along the z axis, {Ho(k,,k,,-f
0749-6036191
k
) - E + V(z)}
I080485 + 04 $02.00/O
the in-plane components k, equation (1) are replaced operator forms
and by
k
68::
where the A,,A, are the oomponents of the vector potential. When the in-plane anisotropy is ne lected, the solut;;; can be express9 f in terms of u,(x,y) harmonic-oscillator functions
‘~N(h+&.2 Y&b-$ qX,YA=
‘~N(Z)UN__~
(3)
‘4N(z)uN+1
So, solving the set of equations for z, the Landau levels were obtained. The anisotropy was taken into aocount by using perturbation theory [2-31. 2.
2D hole field.
Hamiltonian in the magnetio Perturbation theory.
As was written above.the Landau levels of 2D holes obtained were directly from the three-dimensional equation. The value of magnetic field in this method was not limited. A weak magnetio field may be regarded as a perturbation. In this case, a simple method of oaloulating the Landau levels is pr;gzsed.hole degenerate subband structure is described by an effective
0 1991 Academic
Press Limited
Superlattices and Microstructures,
486
whioh is a Hamiltonian z$;$), with elements block-diagonal analytioal in the momentum oomponents k, Blocks of this matrix have and k,. dimension 2x2 and interoonneot doubly degenerate quantum states. The effective 20 hole Hamiltonian in the magnetic field may be constructed by utilizing perturbation theory in the form of expansion in powers of k and B. The 3D Hamiltonian for light and heavy holes is given by 2D
If(K) =h2/2mo{(y1+
zy2)K2’-
(K={k,,k&};
J={J,,Jy,J,).
Ji={J,,J&
matrices for J=3/2) In the magnetic field B, k and k are replaced by the operators (5). an8 the spin-dependent term is added (5) H( B)=u,gn( JB) 3D holes).
A”=-B~n,,A~;O,h~~~; H(R.B)=X,+6H
=H(%)+V(z)+H(B)
A$!?$A&l& n
(11)
m ( 12)
R =-3’/2/4((y2+Y3)k2+(y2-73)k~) k,=k +tk ; d+=(dX~fdY)/2; _ xy
matrices. For the21ight 6H1,2
hole
dX,dy,dz-Pauli
levels
,=<1/2,m16HI
we have
1/2,m>
+
<1/2.m~6Hl3/2.n><3/2.nI6H(1/2.m>
=
=h2/2mo(~~-~2+am)~2+,$o(g~-2am)Bd,+ (4)
1)
gh- g factor of T ( Pe =he@mo Landau c gauge ; is chosen:
31/z m
Em - En
BYTE+
t2( y3-y2) ( J;k;+J;k;+J;k;
&=h2/2mot
Vol. 10, No. 4, 1991
(6)
The main part
ives a set of doubly degenerate levels holes) _ 1, and E, ( for heavy and light The perturbation part is
+h2/2mo[ (%++C,fi)d+-(
fi*$_+i;_fi* )d_]p,(
13)
For a, and PI expressions (10) and (11) may be used. if m and n are exchanned. For zero magnetic field, UP(&,k,) is an effective 2D matrix Hamiltonian giving the 20 subband dispersion law: IR(k](14) E1,2=h2/2mo[ (Yl+Y2+a,)k2+1pnI the twofold spin of Tnhe lifting The subband degeneracy takes place. to kS in splitting ilsitl!rop~$.ionYarping (the agreement $plge anisotropy) first appears in terms and therefore can be treated as a perturbation [2.3] In the presence of the external field 8, the effective magnetic Hamiltonian is obtained from 6Il1 (k,,kY) yg)replaclng k, and k by the operators and including the spin-dependent term. The change in the g factor is due and the to the spin-orbit coupling
8H=h2/2mo{(y1+$y2)fi2-2y3(J,i;)2+
of i;, and k, It is noncommutativity determined by the same parame’ter a, which determines the change of effective mass.
+2(y3-Y2)(J~fi~tJ~~~)-2Y3(JZk,J,~+
~,n=i/(Y1+12+an):
+J,fiJ,k,)l
(8)
+closhJzB .
The second-order correction is a 2x2 including terms kl, kS and B. matrix, For the heavy hole levels we have 2
6H3 2 n=<3/2,n)6H13/2,n> +z
=
El7 - Em
=h2/2mo(yl+y2+Ch)j;2t23~0(gh+~‘xn)BdZ+
th2/2m0[(aj;_+~_~)d+-(~*Fi,+~,~*)b_]~, an=h2,/2mo~ 3~2, yEd m n
(9)
( 10) m
(15)
Parameters a and j3 are taken from the dispersion law,expanded in powers of k It makes it possible to avoid summation over the upper-lying subbands , which is needed in calculations of a and p from (11,12). 3. Landau levels.
+
(3 2.r;~6HI1/2.m><1/2.mI6H13/2.n~
g2 n=iah+$an
By using perturbation theory, we have constructed the 2D hole Hamiltonian in a magnetic field (9)) ( 13). The eigenvalues of this Hamiltonian are the Landau levels and may be zlculated by set ordinary solution of for individual differential eq&tions subbands. Let as consider, for example, the lowest heavy hole subband (index n is
Superlattices
and Microstructures,
omitted). &f;iential
The following equations for
{sR2(fix.$,B)-E)o
set y has to
= 0
of be
(16)
( In gauge (5) $,=k, and corresponds to a shift in ;fke,orif;n o;szible he direct to take solution account of the effec P of warping in a nonpertubative manner. band Neglecting (R=-31/2(y2+y3)k!/4),
487
Vol. 70, No. 4, 1991
the
set
kl and ks , B is not the terms limited. If EE falls within the interval where k4 has to be accounted for.butN;:; number of occupied Landau levels weak ) (magnetic field be added% Hamiltonian (17) willishave the terms ykihrreglecting the terms k¶B expression (19) is and a). transformed into
c;?)‘::
a+=R,fi+/2’/2
: ~R,ii_/2’/~
(17)
treat ion annihilation the and operators for the harmonic oscillator with frequency wC= eB/mo c ( ho, =&I B ) and cyclotron radius RC=( hc/eB) ‘j2. The Hamiltonian expressed in term of a and a+
has the form
6H2’a,a+,E)=~oB/mZ(a+a
+aa+)+$o~2BdZ+
+IJ.,E(f B/R,a3d+- t a/R, ( a+ ) 30_ ) (The notat
f
::;;b:;_;
(18)
j = lj31(3/2)‘/2(yt+y,)
on
is
(23)
+ +oBY s/Rc2
terms of the solved explicitly in harmonic-oscillator functions. Following Luttinger, we define
4 Discussion. In the e eriments of Dorozhkin and Olshanetzkii channel Sl(ll0) 2D layers, a p~~eon&ft by n in the Shubnikov-de Haas oscillations over a narrow range of magnetic fields has been found. The authors explained it by the linear k term ;Fththe dispersion law of 2D systems spin,;;;;; ~;~~!ing (the Bych%%shba However the effect may be explained within the scope of the effective-mass approximation. Our calculation of the Landau levels, when applied to the experiment 7 , enable6 one to relate the phase 8ill i t to ka terms in the 2D hole Hamiltonian.
;us:y~_(N+l)v”~+l
and making the substitution A"N-3 @=
[
cuN
1
leads to the algebraic equation Landau levels. For N=3,4.., we have Ei .2= 2p,B/m2( N-l ) !:
(19)
for
the
3 )2+i2/~N(~-i)(N-2))1/2. +P,B(( 3/m2-2s2 For N=O,1,2,
we have
El= 2~&3/m,(N+l/;?) Within the linear becomes
- &,g2B B-approximation
(20) (21) this
El ,2=2uoB/m2( N+1/2) 7 23~~9~8 . (22) The pro osed method may be used under the fo Plowing limitations. If the density of 2D holes is such that the Fermi level EF falls within the interval in which El ,f (k) are approximated by
Fig.1.
Landau levels
in St p-channel.
mr=0.23 ; -=0.26 n,=0.9 lo1 d cm-2;
; y=-0.04 EF=7.2meV;
Superlattices
::bbt%
,“” V(z)=
and Microstructures,
sufficient
Vol. IO, No. 4, 1997
for
the
lowest
2nn,ez/c
11 = 4.0
: 12 = 1.2
(24) : g,, = 1.2
The
values of m2, fi. 1 and EF are presented in Figs 1 and 2. As is seen, the phase shift in the Shubnikov-de Haas oscillations takes plaCe for f&h about 4 T. B‘h being slightly dependent on n,.The experiments are explained qualitatively by the Laloulat ions but the Byschkov-Rashba effect cannot be ruled cut.
REFERENCES
Fig.2.
Landau levels in St p-channel. ma=0.23 ; -=0.18 ; I=-0.02 n,=2.7 1OI! cm-‘; EF=19.5 mf3V;
The Landau levels layers are pi;tteto;; different
for p-channel St Figs. 1 and 2 for densities n,
Parameters - and 1 are calculated from equationml ’ ? 1) in the spherically symmetric approximation by assuming the well potential V(z) to be triangular (it
1. D.A.Broido, L.J.Sham. Phys.Rev.831, 888 (19851. D.A.Broido. L.J.Sham. 2. SIR.Eric-Yang, Ph s.Rev.832. 6630 (1985). G.Landvehr. Surf.Science. 3. E. # an ert 170. !%3 (1986) Y.Altarelli. Phys.Rev. 4. U.Ekenberg, 832. 3712 (1985). 5. J.Y.Luttinger, W.Kohn. Phys.Rev.97, 869 (1955). 6. S.V. Yeshkov S.N. Molotkov. Poverkhnost kl . 5 (1989) E.B.Olshanetzkii. 7. S.I.Dorozhkin, J.Ex .Theor.Phys.,Pfs’ma v redakcfyu 46, $99 (1987). E.I.Rashba. J.Ex .Theor 8. Y.A.Byohkov. Phys.,Pfs’mo v redakclyu 39.6 %(1984)
and Microstructures,
485
Vol. 70, No, 4, 1991
LANDAULEVELS OF HOLES IN A TWO-DIMENSIONAL CHANNEL Institute
Veronioa E. Bisti of Solid State Physics, USSR Academy of Sciences, 142432 Chernogolovka Moscow District. USSR (Received
12 October
1990)
A method for calculating the Landau levels in a two-dimensional channel is proposed. This method is aimed at construction of an effective two-dimensional Hamiltonian of holes H(k,,k,,B). The Hamiltonian is expanded in powers of the magnetic field B and noncommutlng momenta k, and k,. The results indicate an interconnection between effective It is masses and g-factors of the two-dimensional holes. possible to account for the effects of band warping i;hz non erturbative manner. The shift of phase in Shug nlkov-de Haas oscillations is explained.
1. Introduction Two-dimensional (2D) channels a;; NOS-structures, heterojUbn,C,t,ions duaiea wells have auantum are systems e’xtensively. These oharaoterized by the auantization of the perpendicular motion -and by the free carriers parallel to the motion of interface. In an n-channel layer, the motion parallel to the interface is hardly affected by the quantization of the perpendicular motion. In a normal we obtain the ordinary magnetic field, with the same values of Landau levels, and cyclotron frequency factor, just as in the three-dimensiona 9 case. In a D-channel layer, however. this is not the case. The -degeneracy of the valence bands and the notential V(z) combine to couple strongiy the paraliei The and perpe;g;ular motions. E(k) of the hole ai ersion sub ! ands is then highly nonparabolic.( k = (k, .ky} ) E(k) is calculated from the Shroedinger e uation (I) and consists of subbands doub P y degenerate at the point k=O_ Y = 0 (1) H, (k, , k, , kL ) - the three-dimensional 4x4 Luttinger Hamiltonian for the light and heavy hole bands. The picture of the Landau levels in ohannel layers is also more complex, !-hen in an n-channel. The previous are based on the calculations [l-4 Luttinger work 5 1 . If a normal to the interface and t I!e direction of magnetic field are taken to be along the z axis, {Ho(k,,k,,-f
0749-6036191
k
) - E + V(z)}
I080485 + 04 $02.00/O
the in-plane components k, equation (1) are replaced operator forms
and by
k
68::
where the A,,A, are the oomponents of the vector potential. When the in-plane anisotropy is ne lected, the solut;;; can be express9 f in terms of u,(x,y) harmonic-oscillator functions
‘~N(h+&.2 Y&b-$ qX,YA=
‘~N(Z)UN__~
(3)
‘4N(z)uN+1
So, solving the set of equations for z, the Landau levels were obtained. The anisotropy was taken into aocount by using perturbation theory [2-31. 2.
2D hole field.
Hamiltonian in the magnetio Perturbation theory.
As was written above.the Landau levels of 2D holes obtained were directly from the three-dimensional equation. The value of magnetic field in this method was not limited. A weak magnetio field may be regarded as a perturbation. In this case, a simple method of oaloulating the Landau levels is pr;gzsed.hole degenerate subband structure is described by an effective
0 1991 Academic
Press Limited
Superlattices and Microstructures,
486
whioh is a Hamiltonian z$;$), with elements block-diagonal analytioal in the momentum oomponents k, Blocks of this matrix have and k,. dimension 2x2 and interoonneot doubly degenerate quantum states. The effective 20 hole Hamiltonian in the magnetic field may be constructed by utilizing perturbation theory in the form of expansion in powers of k and B. The 3D Hamiltonian for light and heavy holes is given by 2D
If(K) =h2/2mo{(y1+
zy2)K2’-
(K={k,,k&};
J={J,,Jy,J,).
Ji={J,,J&
matrices for J=3/2) In the magnetic field B, k and k are replaced by the operators (5). an8 the spin-dependent term is added (5) H( B)=u,gn( JB) 3D holes).
A”=-B~n,,A~;O,h~~~; H(R.B)=X,+6H
=H(%)+V(z)+H(B)
A$!?$A&l& n
(11)
m ( 12)
R =-3’/2/4((y2+Y3)k2+(y2-73)k~) k,=k +tk ; d+=(dX~fdY)/2; _ xy
matrices. For the21ight 6H1,2
hole
dX,dy,dz-Pauli
levels
,=<1/2,m16HI
we have
1/2,m>
+
<1/2.m~6Hl3/2.n><3/2.nI6H(1/2.m>
=
=h2/2mo(~~-~2+am)~2+,$o(g~-2am)Bd,+ (4)
1)
gh- g factor of T ( Pe =he@mo Landau c gauge ; is chosen:
31/z m
Em - En
BYTE+
t2( y3-y2) ( J;k;+J;k;+J;k;
&=h2/2mot
Vol. 10, No. 4, 1991
(6)
The main part
ives a set of doubly degenerate levels holes) _ 1, and E, ( for heavy and light The perturbation part is
+h2/2mo[ (%++C,fi)d+-(
fi*$_+i;_fi* )d_]p,(
13)
For a, and PI expressions (10) and (11) may be used. if m and n are exchanned. For zero magnetic field, UP(&,k,) is an effective 2D matrix Hamiltonian giving the 20 subband dispersion law: IR(k](14) E1,2=h2/2mo[ (Yl+Y2+a,)k2+1pnI the twofold spin of Tnhe lifting The subband degeneracy takes place. to kS in splitting ilsitl!rop~$.ionYarping (the agreement $plge anisotropy) first appears in terms and therefore can be treated as a perturbation [2.3] In the presence of the external field 8, the effective magnetic Hamiltonian is obtained from 6Il1 (k,,kY) yg)replaclng k, and k by the operators and including the spin-dependent term. The change in the g factor is due and the to the spin-orbit coupling
8H=h2/2mo{(y1+$y2)fi2-2y3(J,i;)2+
of i;, and k, It is noncommutativity determined by the same parame’ter a, which determines the change of effective mass.
+2(y3-Y2)(J~fi~tJ~~~)-2Y3(JZk,J,~+
~,n=i/(Y1+12+an):
+J,fiJ,k,)l
(8)
+closhJzB .
The second-order correction is a 2x2 including terms kl, kS and B. matrix, For the heavy hole levels we have 2
6H3 2 n=<3/2,n)6H13/2,n> +z
=
El7 - Em
=h2/2mo(yl+y2+Ch)j;2t23~0(gh+~‘xn)BdZ+
th2/2m0[(aj;_+~_~)d+-(~*Fi,+~,~*)b_]~, an=h2,/2mo~ 3~2, yEd m n
(9)
( 10) m
(15)
Parameters a and j3 are taken from the dispersion law,expanded in powers of k It makes it possible to avoid summation over the upper-lying subbands , which is needed in calculations of a and p from (11,12). 3. Landau levels.
+
(3 2.r;~6HI1/2.m><1/2.mI6H13/2.n~
g2 n=iah+$an
By using perturbation theory, we have constructed the 2D hole Hamiltonian in a magnetic field (9)) ( 13). The eigenvalues of this Hamiltonian are the Landau levels and may be zlculated by set ordinary solution of for individual differential eq&tions subbands. Let as consider, for example, the lowest heavy hole subband (index n is
Superlattices
and Microstructures,
omitted). &f;iential
The following equations for
{sR2(fix.$,B)-E)o
set y has to
= 0
of be
(16)
( In gauge (5) $,=k, and corresponds to a shift in ;fke,orif;n o;szible he direct to take solution account of the effec P of warping in a nonpertubative manner. band Neglecting (R=-31/2(y2+y3)k!/4),
487
Vol. 70, No. 4, 1991
the
set
kl and ks , B is not the terms limited. If EE falls within the interval where k4 has to be accounted for.butN;:; number of occupied Landau levels weak ) (magnetic field be added% Hamiltonian (17) willishave the terms ykihrreglecting the terms k¶B expression (19) is and a). transformed into
c;?)‘::
a+=R,fi+/2’/2
: ~R,ii_/2’/~
(17)
treat ion annihilation the and operators for the harmonic oscillator with frequency wC= eB/mo c ( ho, =&I B ) and cyclotron radius RC=( hc/eB) ‘j2. The Hamiltonian expressed in term of a and a+
has the form
6H2’a,a+,E)=~oB/mZ(a+a
+aa+)+$o~2BdZ+
+IJ.,E(f B/R,a3d+- t a/R, ( a+ ) 30_ ) (The notat
f
::;;b:;_;
(18)
j = lj31(3/2)‘/2(yt+y,)
on
is
(23)
+ +oBY s/Rc2
terms of the solved explicitly in harmonic-oscillator functions. Following Luttinger, we define
4 Discussion. In the e eriments of Dorozhkin and Olshanetzkii channel Sl(ll0) 2D layers, a p~~eon&ft by n in the Shubnikov-de Haas oscillations over a narrow range of magnetic fields has been found. The authors explained it by the linear k term ;Fththe dispersion law of 2D systems spin,;;;;; ~;~~!ing (the Bych%%shba However the effect may be explained within the scope of the effective-mass approximation. Our calculation of the Landau levels, when applied to the experiment 7 , enable6 one to relate the phase 8ill i t to ka terms in the 2D hole Hamiltonian.
;us:y~_(N+l)v”~+l
and making the substitution A"N-3 @=
[
cuN
1
leads to the algebraic equation Landau levels. For N=3,4.., we have Ei .2= 2p,B/m2( N-l ) !:
(19)
for
the
3 )2+i2/~N(~-i)(N-2))1/2. +P,B(( 3/m2-2s2 For N=O,1,2,
we have
El= 2~&3/m,(N+l/;?) Within the linear becomes
- &,g2B B-approximation
(20) (21) this
El ,2=2uoB/m2( N+1/2) 7 23~~9~8 . (22) The pro osed method may be used under the fo Plowing limitations. If the density of 2D holes is such that the Fermi level EF falls within the interval in which El ,f (k) are approximated by
Fig.1.
Landau levels
in St p-channel.
mr=0.23 ; -=0.26 n,=0.9 lo1 d cm-2;
; y=-0.04 EF=7.2meV;
Superlattices
::bbt%
,“” V(z)=
and Microstructures,
sufficient
Vol. IO, No. 4, 1997
for
the
lowest
2nn,ez/c
11 = 4.0
: 12 = 1.2
(24) : g,, = 1.2
The
values of m2, fi. 1 and EF are presented in Figs 1 and 2. As is seen, the phase shift in the Shubnikov-de Haas oscillations takes plaCe for f&h about 4 T. B‘h being slightly dependent on n,.The experiments are explained qualitatively by the Laloulat ions but the Byschkov-Rashba effect cannot be ruled cut.
REFERENCES
Fig.2.
Landau levels in St p-channel. ma=0.23 ; -=0.18 ; I=-0.02 n,=2.7 1OI! cm-‘; EF=19.5 mf3V;
The Landau levels layers are pi;tteto;; different
for p-channel St Figs. 1 and 2 for densities n,
Parameters - and 1 are calculated from equationml ’ ? 1) in the spherically symmetric approximation by assuming the well potential V(z) to be triangular (it
1. D.A.Broido, L.J.Sham. Phys.Rev.831, 888 (19851. D.A.Broido. L.J.Sham. 2. SIR.Eric-Yang, Ph s.Rev.832. 6630 (1985). G.Landvehr. Surf.Science. 3. E. # an ert 170. !%3 (1986) Y.Altarelli. Phys.Rev. 4. U.Ekenberg, 832. 3712 (1985). 5. J.Y.Luttinger, W.Kohn. Phys.Rev.97, 869 (1955). 6. S.V. Yeshkov S.N. Molotkov. Poverkhnost kl . 5 (1989) E.B.Olshanetzkii. 7. S.I.Dorozhkin, J.Ex .Theor.Phys.,Pfs’ma v redakcfyu 46, $99 (1987). E.I.Rashba. J.Ex .Theor 8. Y.A.Byohkov. Phys.,Pfs’mo v redakclyu 39.6 %(1984)