Double quantum well electronic energy spectrum within a tilted magnetic field

Superlattices and Microstructures, Vol. 26, No. 5,1999 Article No. spmi.1999.0781 Available online at http://www.idealibrary.com on

Double quantum well electronic energy spectrum within a tilted magnetic field S. E LAGOZ , H. E LAGOZ , H. S ARI , Y. E RGÜN , P. K ARASU Cumhuriyet University, Department of Physics, 58140 Sivas, Turkey

I. S ÖKMEN Dokuz Eylül University, Department of Physics, Izmir, Turkey (Received 11 May 1999) The analytical solutions of the Schrödinger equation for a double quantum well structure (DQWS) subjected to an externally applied tilted magnetic field are obtained and the results are discussed. The dependency of the energy spectrum of the system on the applied magnetic field direction is also given. c 1999 Academic Press

Key words: heterostructures, quantum wells, tilted magnetic field.

1. Introduction Over the past decade there has been an increasing interest in the electronic properties of two-dimensional electron gas systems subjected to an externally applied magnetic field due to both technological and academical reasons [1–5]. Most of the theoretical works regarding these systems are done with configurations where the magnetic field is either parallel or perpendicular to interfaces in which electrons or holes are confined. Wang et al. [6] worked on the effects of electronic energy levels in a symmetric quantum well under an in-plane magnetic field and showed that electron energy states will be a mixture of Landau levels (magnetic confinement) and quantum well levels (spatial confinement) depending on the well width and the position of the orbit center. When an applied magnetic field is parallel to the growth direction, Landau levels are formed, but the spatial confinement of the wells does not have a pronounced effect on the electron energy spectrum for this orientation. Furthermore, several authors have done studies on these systems for crossed electric and magnetic fields [7, 8]. Theoretical calculations for a square quantum well becomes difficult when an externally applied magnetic field is tilted with respect to the quantization axis, since in this configuration the variables of the Schrödinger equation become inter-related with each other and except for parabolic [9, 10] and triangular [11] potential profiles, the Schrödinger equation had been thought to be unsolvable analytically. For this reason, several authors have used variational or perturbative approaches to find a solution for this problem. In a brief report of Mitrinovic’ et al. [12] some of the numerical approaches are given and their shortcomings are discussed. In our recently published article [13], without making any approximation or smoothing the potential profile, we have shown that it is possible to obtain the solution analytically by separating the Schrödinger equation of the square well system. 0749–6036/99/110299 + 07 $30.00/0

c 1999 Academic Press

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Here we report the analytical solutions of the Schrödinger equation for a double quantum well structure (DQWS) subjected to an externally applied, uniform, tilted magnetic field.

2. Theory A schematic diagram of the double quantum well structure subjected to an external, tilted magnetic field is shown in Fig. 1A. We have chosen the growth direction along the z-axis and the direction of the magnetic field in the x–z plane B = (cos θ, 0, sin θ ), where θ is the tilt angle from the x-axis. Using the Landau gauge, we define the vector potential as A = (0, x B sin θ − z B cos θ, 0). The Hamiltonian of an electron in such a system can be written as 1 (P + eA)2 + V (z), (1) H= 2µ where µ is the effective mass of the electron and V (z) is the well potential energy of the electron in each region. The formal functional form of V (z) is given as follows, V (z) = V0 {S(z L 1 − z) + S(z − z R1 ) + S(z L 2 − z) + S(z − z R2 ) − 1},

(2)

where S(z) is the step function, and the z L 1 , z L 2 , z R1 , z R2 mark the locations of the well and barrier boundaries as shown in Fig. 1B. Making use of the translational symmetry in the y-direction, the wavefunction of the system can be written as ψ(r) = eik y y φ(x, z), and by using the vector potential we can rewrite the Hamiltonian of the system. 1 (~k y − eB(z cos θ − x sin θ))2 + V (z). (3) H = (Px2 + Pz2 ) + 2µ Applying point canonical transformation,  0    z cos θ − sin θ z = . x0 sin θ cos θ x One can obtain the separated Hamiltonian as follows, where B is now along the x 0 -axis. H = Hx 0 + Hz 0 Px20 + V (x 0 ) 2µ P 20 1 Hz 0 = z + µω2 (z 00 − z 0 )2 + V (z 0 ). 2µ 2

Hx 0 =

Here we have used z 00 =

~k y eB

(4)

= a 2H k y for the position of the orbit center, a H =

q

~

µω

for the magnetic

length, ω = for the cyclotron frequency, and the transformed coordinates = x cos θ + z sin θ and 0 z = −x sin θ + z cos θ . Corresponding one-dimensional Schrödinger equations for eqn (4) are given as eB µ

x0

Hx 0 X (x 0 ) = E x 0 X (x 0 )

(5)

H φ(z ) = E φ(z ). z0

V (x 0 , z 0 )

Where the potential term V (z) = boundaries along the x 0 -axis are given as, V (x 0 ) = V0 sin2 θ

z0

0

= V (x 0 ) + V (z 0 ) becomes separable, the potential and the

{S(x L0 1 − x 0 ) + S(x 0 − x 0R1 ) + S(x L0 2 − x 0 ) + S(x 0 − x 0R2 )} + V0

x L0 1,2 ,R1,2 = z L 1,2 R1,2 sin θ + x cos θ z 0L 1,2 ,R1,2

0

= z L 1,2 ,R1,2 cos θ − x sin θ.

(6)

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z 80

A B

B

60 ∼

θ V

∼

Z 0 (k y ) = 5

40 ∼

Z 0 (k y ) = 3

AlxGa1 – xAs GaAs AlxGa1 – xAs

x

GaAs AlxGa1 – xAs

∼

20

Z 0 (k y ) = 0

0 –8

–4

4

0

∼

8

Z

Fig. 1. A, Schematic representation of the DQWS under an externally applied tilted magnetic field. B, The z˜ 00 dependence of the potential after the coordinate transformation.

This form of the potential reveals that, aside from a constant energy shift term (V0 ), Hx 0 describes a onedimensional symmetric double quantum well system (without any magnetic field effect) with an effective height Veff = V0 sin2 θ and an effective width L eff = L sin θ , resulting in an energy spectrum independent of k y . For a fixed θ value, the E x 0 contribution to the total energy is constant and its solution is well known. We will focus on the z 0 term since this term contains the magnetic field term. Similarly, we can write V (z 0 ) by using transferred boundaries, V (z 0 ) = V0 cos2 θ {S(z 0L 1 − z 0 ) + S(z 0 − z 0R1 ) + S(z 0L 2 − z 0 ) + S(z 0 − z 0R2 )}.

(7)

Equation (5), when considered with eqn (7), describes a symmetric double quantum well system with an effective height Veff = V0 cos2 θ and an effective width L eff = L cos θ that is subjected to an external magnetic field which is applied along the z 0 -axis so that the problem is now reduced into a well-known problem, namely, a double quantum well structure under a magnetic field applied parallel to the growth direction; finally, the solution of the system is given by writing the separated wavefunction as ψ(F) = eik y y χ (x 0 )φ(z 0 ) with corresponding energy spectrum E = E x 0 +E y 0 +E z 0 . As mentioned above, contributions from E x 0 + E y 0 are known; to find out the contribution from E z 0 we follow the treatment of Lee et al. [7] on √ 0 E0 2 0 0 0 the z equation. Changing variables from z to u˜ = (z − z 0 ), E˜ z 0 = z , and V˜ (˜z 0 ) = V (z ) , we obtain the aH

0

~ω

~ω

Schrödinger equation corresponding to the z 0 motion in terms of dimensionless variables as   d 2 φ(u) ˜ ˜ z 0 − u˜ − 1 u˜ 2 φ(u) + E ˜ = 0. 4 d u˜ 2

The solution of this equation is given by the well-known Weber functions √ √      π m 1 1 2 2π u˜ 1 m 3 1 2 m/2 −u˜ 2 /4 Dm (u) ˜ =2 e F − u˜ − F − − u˜ , 5(1/2 − m/2) 2 2 2 0(−m/2) 2 2 2 2

(8)

(9)

where 0(x) is the Gamma function and F(a|b|2) is the confluent hypergeometric function and the parameter m is related to the eigenvalues as E˜ z 0 = m + 21 .

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From the asymptotic properties of the Weber functions, the solutions of eqn (8) can be expressed for each region as follows.  C1 Dm 0 (−u), ˜ u˜ < u L1    ˜ + C3 Dm (u), ˜ u L1 < u˜ < u R1  C2 Dm (−u) φ(u) ˜ = C4 Dm 0 (−u) (10) ˜ + C5 Dm 0 (u), ˜ u R1 < u˜ < u L2   C D (− u) ˜ + C D ( u), ˜ u < u ˜ < u  6 m 7 m L2 R2  C8 Dm 0 (u), ˜ u R2 < u, ˜ 0 where the parameters m and m are related to each other by the dimensionless potential height V˜ , namely m − m 0 = V˜ cos2 θ . Applying the continuity conditions of the wavefunctions and its first derivative at the boundaries we obtain a set of linearly independent equations. Furthermore, to have a unique solution, we impose the condition that the determinant of the coefficients C1 through C8 must be zero which gives us the master equation (not given here due to space limitation).

3. Results and discussion From the master equation, one can obtain the quantum number m in terms of k y and θ . Therefore, treating θ as a parameter we can calculate m values as a function of k y numerically, if the parameters of the quantum structure and the magnitude of B are given. Without loss of generality, and  for  simplicity, we have chosen the L δ i,l boundaries of the double well structure as z i j = (−1) (2δ j,2 + cos θ ) 2 , where i = r, l (for right, left), j = 1, 2 (first well, second well) and δi j represents Kronecker’s delta so that in dimensionless variables the  

boundaries can be written as u˜ i j = (−1)δ δ j2 + 12 cos θ . In numerical calculations we have used the parameters that belonged to the Alx Ga1−x As/ GaAs/ Alx Ga1−x As/ GaAs/Alx Ga1−x As system, in which for x = 0.3 we obtain V0 = 225 meV and we use L = 36 Å for the width of the wells and the barrier, and the magnetic field value B is taken to be 26 T (this choice of the field value might seem awkward, however, calculations are done with dimensionless and scaled parameters so we would find the same m behavior for, say, B = 6 T and L = 105 Å but then we would have spaghetti-like energy states too close to each other to allow any clear picture of physics), a compatible length scale to that of well width. Thus, the energy levels are expected to be quantized by the combination of spatial and magnetic confinement (Landau levels). From Figs 2 to 5, the numerical solutions of the m’s are plotted versus the orbit center position z˜ 00 for ◦ 0 , 30◦ , 45◦ and 60◦ , which denote the values of the angle between the magnetic field direction and the x-axis. From these figures, we note that beyond z˜ 00 ≥ 6 the energy levels are flat and the same on both sides, these are Landau levels, elevated exactly by an amount V˜0 cos2 θ to that of bulk Landau levels. So we conclude that, in this region, the orbit center and the orbit radius are far away from the well region, therefore no effect of spatial confinement can be seen here. In region B, the story is different, we see very strong spatial localization effects, since z˜ 00 begins right in the middle of the barrier no matter what the energy (so the radius) is, the wells and the barrier are felt by the particle resulting in energy levels reflecting strong spatial confinement effects; as the energy increases, we see that energy levels start showing a mixed character between spatial and magnetic confinement. Another important behavior of the energy levels is in the region 3 ≤ z˜ 00 ≤ 6 where transition from localized to extended states occurs. Here we see that as the energy or z˜ 00 values increase, levels start oscillating to find the most suitable Landau levels and finally become an extended state as described by Lee et al. [5]. For θ = 0◦ , the first energy state’s flat bottom looks somewhat puzzling and it is worth a closer examination. In DQWSs, for z˜ 00 = 0 the orbit center is in the center of the barrier, and no matter what the radius of the orbit is, the electron motion must be strongly affected by the barrier structure, therefore the spatial confinement effects should be much stronger than the magnetic confinement effects, therefore we should see levels reflecting double well characters not flat Landau-like levels, for this reason we have done a more precise calculation for the first state with more data points covering the region il

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10

A

303

C

B

8

∼ m

6 θ = 0º

4 2.2

2 2.0 –2 –1 0

0 –10 –8 –6 –4 –2 0

∼

2

4

6

1

2

8

10

Z0 (ky) Fig. 2. A plot of the electronic energy spectrum m˜ versus dimensionless orbit center z˜ 00 for a symmetric DQWS subjected to an externally applied tilted magnetic field for tilt angle θ = 0◦ . The dashed rectangle drawn over the energy minimum is enlarged and shown in the inset.

10 A

C

B

8

∼ m

6

4 θ = 30º 2

0 –10 –8 –6 –4 –2

0

2

4

6

8

10

∼ Z0 (ky)

Fig. 3. A plot of the electronic energy spectrum m˜ versus dimensionless orbit center z˜ 00 for a symmetric DQWS subjected to an externally applied tilted magnetic field for tilt angle θ = 30◦ .

|˜z 00 | ≤ 2. The inset in Fig. 2 shows the result of this calculation and, as expected, we see that the first energy level is not flat, rather, it has the behavior of a particle trapped in a double well structure. Finally, an important point worth mentioning is in Figs 2 and 3 we have two bound states, whereas in Fig. 4 this drops to one bound state and remains as such till the well structure totally disappears. This is an interesting result since here the only parameter that changes is the tilt angle and it is an external, easily

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10

8 A

B

C

∼ m

6

4

2

θ = 45 º

0 –10 –8 –6 –4 –2

0

2

4

6

8

10

∼ Z0 (ky)

Fig. 4. A plot of the electronic energy spectrum m˜ versus dimensionless orbit center z˜ 00 for a symmetric DQWS subjected to an externally applied tilted magnetic field for tilt angle θ = 45◦ .

10

8

A

C

B

∼ m

6

4

2 θ = 60º 0 –10 –8 –6 –4 –2

0

2

4

6

8

10

∼ Z0 (ky)

Fig. 5. A plot of the electronic energy spectrum m˜ versus dimensionless orbit center z˜ 00 for a symmetric DQWS subjected to an externally applied tilted magnetic field for tilt angle θ = 60◦ .

changeable parameter. So it suggests that, by just changing the tilt angle of the magnetic field, we can change the character of electronic states from confined to extended and vice versa. As the tilt angle theta increases we see that the spatial confinement effects are less and less pronounced, for this reason the Landau levels start taking over for smaller z˜ 00 and m values. This is exactly what we

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expect since with increasing θ , the well (barrier) width (L cos θ) and the well (barrier) height (V˜ cos2 θ) drop reducing the effects of spatial confinement; as θ goes to 90◦ , the quantum structure totally disappears leaving its place to bulk structure, therefore energy levels smoothly go to bulk Landau levels.

4. Summary The complete and exact solution of a symmetric DQWS under an externally applied, homogeneous, tilted magnetic field is given. Eigenvalues and eigenstates of the system are obtained. This is followed by a discussion of the orbit center position dependency of the eigenvalue behavior of the system as a function of θ . Furthermore, it is pointed out that, by just changing the tilt angle θ it is possible to change the character of a state from confined to extended or vice versa, an important result promising future field direction sensitive device designs. Acknowledgements—This work has been partly supported by TUBITAK (TBAG-1646) and Cumhuriyet University AFS.

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