Physica B 184 (1993) 318-322 North-Holland PHYSICA[

Electronic structure of the triangular quantum well in a tilted magnetic field

S.J. L e e a, M . J . P a r k a, G . I h m b, M . L . Fa lk b, S .K. N o h b, T.W. K i m c and B . D . C h o e d

aDepartment of Physics, Korea Military Academy, Seoul, South Korea bVacuum Science Laboratory, Korea Research Institute of Standards and Science, Taedok Science Town, South Korea CDepartment of Physics,. Kwangwoon University, Seoul, South Korea ODepartment of Physics, Seoul National University, South Korea

The quantized electron energies in the triangular well of a Si-MOSFET subjected to strong magnetic fields at tilt angles B = (Bx, 0, Bz) are calculated. Approximate but analytic solutions are also provided with numerical solutions. It is shown that the role of the coupling Hamiltonian (~B x • Bz) is crucial in obtaining the correct magnetization, although its contribution to the total energy is minor. This coupling term is responsible for the oscillation of the magnetizations with respect to the chemical potential or the two-dimensional electron density. The latter is also confirmed in the case of a parabolic quantum well.

1. Introduction

Application of a strong magnetic field to a confined electron system has a pronounced effect on spatial quantization. However , most of the theoretical and experimental work regarding these systems is concerned with the configuration in which the magnetic field is perpendicular or parallel to interfaces in which electrons or holes are confined. In this work we deal with a less frequently investigated configuration where an external magnetic field is tilted with respect to the gradient of the confining potential, i.e. to the electric field. It was using this system that Fang and Stiles [1] per formed an ingenious experiment in 1968 to obtain the effective Land6 g factor. To explain this experiment many theoretical works have followed. Among them, Ando and Uemura [2] have explained the experiment as the ex- change effect among electrons in the Landau level. Many experiments have been performed

Correspondence to: S.J. Lee, Department of Physics, Korea Military Academy, P.O. Box 77, Gong-neung-Dong, Nowon- Gu, Seoul 139-799, South Korea.

for different ranges of tilt angles and materials [3] and most of the works have been done by studying quantum oscillations in SdH measure- ments.

However , these quantum oscillations have been tested recently by calculating the magneti- zation of electrons confined in a rectangular quantum well in a G a A I A s - G a A s - G a A I A s sys- tem [4]. They showed that the magnetization parallel to the layer interfaces exhibits oscilla- tions and sharp jumps as a function of the chemi- cal potential. In the G a A I A s - G a A s system, it is not easy to alter the chemical potential with one sample. Motivated by this fact, we have investi- gated this problem with a triangular well which can be approximated as a potential profile in a Si-MOS inversion layer.

2. Asymmetric triangular well in a tilted magnetic field

A rectangular well has symmetry properties and is easier to treat [5] than a triangular well.

0921-4526/93/$06.00 © ~993- Elsevier Science Publishers B.V. All rights reserved

S .J . Lee et al. / On the triangular q u a n t u m well in a tilted magnet ic f i e ld 319

This has been pointed out by Zawadzki [6] even in the parallel-field case.

The initial eigenvalue equation for our system reads

[1 , ] ( p - eA) 2 +- -~ g*tZBB + U ( x ) + U(z ) q,

= E q , ( 1 )

with the triangular potential

U(z) = z < 0 , (2)

and without any additional confining potential in the x-direction; U(x) can be written as U(x) = O. From now on, the spin term in eq. (1) is omitted to simplify the calculation. Since the gauge of the vector potential is not unique, we choose the gauge A = e y ( x B ~ - zBx) ; then the tilted mag- netic field can be expressed as

B = exB ~ + e~B z = exB cos 0 + ezB sin 0

where 0 is the angle between B and the x-axis. If we assume that 0(x, y, z) = eikyY~0(x, z),

then the Schr6dinger equation can be written as

H ( x , z)~O(x, z) = Eql(x, z ) (3)

where the Hamiltonian of the system is given by

H ( x , z ) = Hox + Ho~ + U(z) + Hx~ (4)

with

h 2 0 2 1 2 2 - + mOJzX - ~ < x < ~ ,

H°x 2 m Ox 2 2 ' (5a)

h 2 0 2 1 2 2 - + m~OxZ 0 < z < oo,

H°~ 2 m Oz 2 2 ' (5b)

U ( z ) = e F z , 0 < z < o o , (5c)

In a paper of Zawadzki et al. [7], quantization of electron energies was developed for the case of a triangular well in a parallel magnetic field. Taking into account this method, we derive a solution up to the second-order perturbation energy of eq. (5d). In this case the energy eigenvalue of the total Hamiltonian H is ex- pressed as

e . , = e °, + e ; , = E ° + e ° + e ; , (6)

where E ° = h % ( l + ½), 1 = 0 , 1, 2, 3 , . . . , and E ° is known numerically. The perturbed term is given by

EL, = E I(nllnx~ln'l')l~-=6---~--- r ,n ' Ent - - E n T •

It is noted that

(7)

( n l l H , z ln ' l ' ) :

m % % (6,,,,+a~/(l + 1)/2 + 6 r , , _ l W T / 2 ) ( n l z l n ' ) Ol z

(8)

where % = (mOOz/h) 1/2. After summation over l' in eq. (7), we get

E ' t = -½mOo2xl(nlzln)l 2 + E S... n ' # n

where

ann , 2 = m h ~ o x % (l + 1 /2 ) (E ° - E° , ) + ( 1 / 2 ) h w z

(9)

× I < n l z l n ' ) ? .

(E o o 2 - E . , ) - ( h , o ~ ) 2

(10)

Substituting eq. (9) into eq. (6), we get the final expression of Enl ,

E . t = E ° + E ° - ½mo~Z~l(nlzln)l 2 + E ann'' n '~n

(111

Hxz = - m % % x z . (5d)

Here we put % = e B x / m , % = e B z / m and have substituted x for x - (hky/mOgz).

Typically the third term is one order of mag- netitude larger than the last term. However, the contribution to the magnetization is not trivial.

Figures l(a) and (b) show the tilt angle depen-

320 S.J. Lee et al. / On the triangular quantum well in a tilted magnetic field

1° F 75

o J- 0 30 60 90 0 30 60 90

0 0 Fig. 1. Ene rgy spec t rum of a tr iangular quan tum well in a tilted magnet ic field (B = 15 T) as a function of the tilt angle 0 taking into account Hxz (b) and neglecting Hx~ (a). The q u a n t u m well cor responds to the electric field E = 5 x 104 V/ cm and we put m = 0.195 m~.

dence of the energy spectrum of a triangular quantum well taking into account Hxz (b) and neglecting Hxz (a). Here we used typical data, a magnetic field B = 15T, an electric field E = 5 x 10 4 V / c m , and an effective electron mass m = 0.195me, where m e is bare electron mass. In fig. 2 energy eigenvalues and eigenfunctions in the z-direction are shown and the potential in the 15 ° case is closer to the parabolic shape with larger to x than that in the 60 ° case.

Now we consider the parabolic well in a tilted magnetic field. In this case, the potential in eq. (1) is expressed as U ( z ) = 0 , U ( x ) = ½mto~x 2

126.7 meV

106.1 meV

84.3 meV

60.6 meV

33.4 meV

b)

_ _ ~ : ~ 113.0 meV

/ 95.7 meV

~ 2 : : : ~ 77.2 meV

56.7 rneV

_ _ 32.1 rneV

Fig. 2. Effective potential energy (H0z + U(z)), energy eigenvalues E ° and cor responding wave funct ions 4~,(z) of the z-direct ion for two different tilt angles 0 = 15 ° (a), 0 = 60 ° (b).

with a parabolic confinement in the x-direction. The y -componen t of the momen tum Py is a constant of motion. After a proper shift of the origin of coordinates, the Hamil tonian can be written as

P~ 1 2 2 P~ 1 2 2 H = ~ + ~ mS2 1 x + ~ m + -2 m t °x z

- m w x o~zxz (12)

where 0 2 = 0021 + tO2z . The above Hamiltonian represents two coupled harmonic oscillators. This can be easily diagonalized [8] by an appro- priate rotation of the coordinates x and z and the energy eigenvalues are easily formed as

E , a = h o ) + ( n + ½) + ho)_(l + ½) (13)

where

2 1 2 2 2 to_+= ~(to x + t o z + t o l+__t),

2 2 l -~ (((.O 2 -- (.O~ -- O) 1 ) .+ .I 2 2 \1 /2 t'l'fiO ¢D !____x__z., 4

3. Jumping magnetization

From eqs. (11) and (13) we can obtain the x and z components of the magnetization M x and M z per unit area at tempera ture T = 0 as a function of the chemical potential /~. We first discuss the parabolic-well case. The magnetiza- tion Mx, z can be expressed as

eB~ ~, 0 ( ~ - E . , t ) OE"'l - M x ' z - ~ h ,,i OBx,z"

(14)

To calculate the magnetization, we find the fol- lowing relation:

Oto+ etOxtO + 2 2 2 - - + - - - o , 1 )

OB x m

+ 4to2o0 ~ ) - ]/z . (15)

The oscillations in M . appear as if discontinuous (steplike) oscillations of a small period hto_ aris-

S.J. Lee et al. / On the triangular quantum well in a tilted magnetic f ield 321

ing from the nonzero o~_ term were superim- posed upon smooth oscillations of a period hto+. These discontinuous oscillations can be under- stood by examining the quantity OEn,t/OB x given in eq. (14). Since Ow+/aB x > 0 and Oto_/OB x < 0 , increasing the index l with fixed index n makes the quantity OEn,t/OB x smaller (it can be nega- tive), showing the oscillations. In contrast, we note that the quantity Oto+_/OB z is always positive and thus M z shows a type of oscillations different from Mx, i.e. steplike jumps accumulate in one direction and M z has no extrema.

We now discuss the magnetization of the tri- angular-well problem. The overall picture of ex- planation is very similar to that of the parabolic well but in this case we can not directly see the oscillation analytically. So we made a numerical calculation of eq. (14) with En. t given by eq. (11), which is already demonstrated in figs. 1 and 2. The magnetization is related to the quantity dEn, t /dBx , which can be written as

dEn,t _ etOx ( n lz2ln ) - etoxl ( n l z l n ) [ 2 dBx

2 '[- n x nvan Snn, + mho~2t%

× ~ ( l + 1 /2)e tOx((n[z2ln) - ( n ' [ z2 [n ' ) ) (E ° - EO) - (h z)

x I < n l z l n ' > l 2 . (16)

Since 0 dE~ /dB~ = e o ~ ( n l z 2 l n ) , d E ° / d B x = 0, the last two terms in eq. (16) come from S~, terms in eq. (11). The contribution to M~ from the last two terms in eq. (16) (which is negative) become comparable to those from the first two terms (which is positive) as the index l increases. There is no l dependence in dE~, t /dB ~ except the last term.

In fig. 3 the negative magnetization per unit area along the x-direction is given as a function of the chemical potential at the temperature T = 0 K. At 0 = 15 ° jumpings follow the old path (0 = 0 , parallel-field case) and have stepwise modulations. Upon increasing the degree of tilt angles, the oscillating pattern in the magnetiza- tion shows features more departed from the

# .2

.9

0 25 50 75 100 1.1, / meV

Fig. 3. Osc i l l a t ing nega t ive m a g n e t i z a t i o n pe r uni t a rea for

t h r e e d i f fe ren t t i l t angles 0 as a func t ion of the chemica l p o t e n t i a l / ~ at B = 15 T. For the p a r a m e t e r s see fig. 1 . 0 = 15 ° ( t o p ) , 0 = 30 ° (middle) and 0 = 60 ° (bo t tom) .

0 = 0 case and the width of the steps increases. This originates from the wider energy gaps be- tween the branches as 0 increases. These up and down jumps continue as the chemical potential passes through the consecutive energy levels shown in fig. 1.

Although our calculations have been carried out for a triangular potential well with a constant slope, we are quite positive that oscillations of magnetization will also occur in real MOSFETs when changing the gate voltage. Inclusions of many-body effects, finite-temperature effects and self-consistency between the gate voltage and the triangular potential through electron density are ongoing research subjects. However, no ex- perimental data is available so far on this subject.

Acknowledgements

This work has been supported by the Center for Theoretical Physics at Seoul National Uni- versity, Korea Science and Engineering Founda- tion, and the Wharagdae Research Institute. One of the authors (B.D.C.) is very grateful for the support of the ETRI, Korea Telecommunica- tion Authority.

322 S.J. Lee et al. / On the triangular quantum well in a tilted magnetic field

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