1 Introduction
Ever since the progress in semiconductor nanotechnology, such as Molecular Beam Epitaxy (MBE), Chemical lithography and etching were developed, it has been made possible to fabricate a wide variety of low-Dimensional Semiconductor Nanostructures like Quantum Well Wires (QWW), Nanowires and Carbon Nanotubes with well controlled shape and composition to achieve the high carrier mobility.[1] Use of Diluted Magnetic Semiconductors (DMS) in such QWW has opened the doors for the researchers to break through the entirely new set of challenges, which had been intimidating in the field of spintronics, since the physical nature of impurity energy levels associated with QWW made of DMS materials can be greatly controlled by the application of external magnetic field, which manifest themselves into fascinating phenomena like Bound Magnetic Polaron,[2] Giant Zeeman Splitting[3] at the band edges, Magneto optical[4]-[5] and Magneto transport.[6]-[7] The shape of the confining potential and the impurity position along these structures mainly determine the spatial confinement of the wavefucntion in these QWW and thereby number of studies concerning QWW with rectangular,[8] Parabolic,[9] V-groove[10] and triangular[11] cross sections have been carried out. Moreover, these 1D systems provide the fantabulous space for the study of Coulomb interaction effects in many body problems because the reduced degree of freedom for the carriers make qualitative changes in the role of interactions which leads to spin-charge separation,[12] charge fractionalization[13] and Wigner crystallization.[14] Many researchers have put their considerable effort to investigate the electron-electron and hole-hole interaction under various confining potentials both in nonmagnetic[15]-[21] and in Semimangetic[22] Semiconducting systems. However, in spite of theoretical activities on the hydrogenic acceptor impurity in DMS Nanostructures,[23]-[26] the studies of impurity states concerning two holes are very few.[27]-[30] Therefore it necessitates to investigate the Coulomb interaction between the two holes confined in a Semimagnetic QWW (SQWW) made of CdTe/Cd$_{1-x}$Mn$_{x}$Te with both square and parabolic band offset under the application of external magnetic field by including the effect of impurity position also for the Mn$^{2+}$ ion composition of $x = 0.3$.
2 Theoretical Formalism
The Hamiltonian of the two holes bound to an acceptor impurity inside the SQWW made of CdTe/Cd$_{1-x}$Mn$_{x}$Te in the effective mass approximation in the presence of applied magnetic field along the direction of growth axis (z-axis) is written as
$$ \textbf{$H_{hh}$} = - (\nabla_{1}^{2} + \nabla_{2}^{2}) - 2z\left(\frac{1}{{r}_{1}} + \frac{1}{{r}_{2}}\right) + V_{B} (x_{1}, y_{1})\nonumber\\ + V_{B}(x_{2}, y_{2})+ \gamma (L_{z_{1}} + L_{z_{2}}) + \frac{\gamma^{2}}{4} (\rho_{1}^{2} + \rho_{2}^{2}) \nonumber\\ + \frac{2}{ |{\vec{r}_{1}} - {\vec{r}_{2}}|}\,, $$
where, $z$ = 2, since it is treated as a helium like impurity and $\gamma = \hbar \omega_{c} / 2R^{*}$ ($\omega_{c}$-cyclotron frequency) is the parameter of the strength of the magnetic field and $\gamma= 1$ corresponds to $\approx$ 1131 Tesla; where, $R^{*}$ is the effective Rydberg of CdTe ($1R^{*} = 78.43$ meV), $r_{1}=\sqrt{\rho_{1}^{2}+z_{1}^{2}}$, $r_{2}=\sqrt{\rho_{2}^{2}+z_{2}^{2}}$ are the mean distance of the parent acceptor atom and the carriers attached to it.
The profile of both the square and parabolic confining potential $V_{B}$ for the carriers is given as,
$$ \begin{array}{ll} V_B=\left\{\begin{array}{lll} \frac{1}{2} m^*_w \omega^2(\rho_1^2+ \rho_2^2)\,,\quad & |x_1|,|y_1|,|x_2|,|y_2|\leq L/2 & \rightarrow {\rm {Parabolic}}\,,\\ V_0\,, & |x_1|,|y_1|,|x_2|,|y_2|> L/2\,, \end{array}\right.\nonumber\\ V_B=\left\{\begin{array}{ccc} 0\,, \qquad \qquad \quad \quad \quad \quad |x_1|,|y_1|,|x_2|,|y_2|\leq L/2& \rightarrow {\rm {Square}}\,,\\ V_0\,, \qquad \qquad \quad \quad \quad \quad |x_1|,|y_1|,|x_2|,|y_2|> L/2\,. \end{array}\right. \end{array}\\[-8mm] $$
Here, $L$ is the size of the QWW and $V_{0} = 30%\Delta E_{g}^{B}$, where, $\Delta E_{g}^{B}$ is the band gap difference with magnetic field and is given by[28]
$$ \Delta E{g}^{B} = \Delta E_{g}^{0} \Big[ \displaystyle\frac{\eta \, {\exp}^{\zeta \gamma} - 1}{\eta - 1}\Big], $$
and because of which, the strength of the confinement potential is rapidly reduced and results in the modifications of electrical and optical properties. $\eta=\exp^{\zeta \gamma_{0}}$, where, $\alpha$ is a parameter ($\alpha$ = 0.5) and $\gamma _{0}$ as the critical magnetic field. $\Delta E_{g}^{B}$ and$\Delta E_{g}^{0}$ are the band gap difference with and without magnetic field respectively. The band gap of the material is given by Eg (Cd$_{1-x}$Mn$_{x}$Te) = 1606 + 1587$x (\rm{meV})$. The critical magnetic field depends upon the composition of magnetic impurity. This critical field (in Tesla) for various compositions can be obtained using the formula $B_{0} = A_{2} \rm{exp[\it nx]}$, where, $A_{2} = -0.57$ and $n = 16.706$.[28]
The approximate ground state energy for confined acceptor impurity with two holes has been calculated using the variational method. The envelop function $f (z)$ is considered for both square and parabolic confining potentials as,
$$ \nonumber\\[-8mm] \phi_w (x_{1},y_{1},x_{2},y_{2}) =\cos [\alpha_s\;x_{1}] \cos [\alpha_s\;y_{1}] \cos [\alpha_s\;x_{2} ] \cos [\alpha_s\;y_{2}]\rightarrow \rm{Square},\nonumber\\ \phi_b(x_{1},y_{1},x_{2},y_{2}) =B_s\;{\rm e}^{-\beta_s\;(x_{1}+y_{1})}\; {\rm e}^{-\beta_s\;(x_{2}+y_{2})},\nonumber\\ \phi_w(x_{1},y_{1},x_{2},y_{2}) ={\rm e}^{-{1}/{2}\; \alpha_p\;(x_1^2+y_1^2)}\;{\rm e}^{-{1}/{2}\;\alpha_p\; (x_2^2+y_2^2)}\rightarrow \rm{Parabolic},\nonumber\\ \phi_b(x_{1},y_{1},x_{2},y_{2}) =B_p\;{\rm e}^{-\beta_p\;(x_{1}+y_{1})}\; {\rm e}^{-\beta_p\;(x_{2}+y_{2})}, $$
where, $\alpha_{s}=({m_{w}^{*} E_{s}}/{\hbar^{2}})^{({1}/{2})}$, $\beta_{s}=({m_{b}^{*} (V_{0}-E_{s})}/{\hbar^{2}})^{({1}/{2})}$;$\alpha_{p} = \hbar\omega$, $\beta_{p}=({m_{b}^{*} (V_{0}-E_{p})}/{\hbar^{2}})^{({1}/{2})}$. $E_{s}$ and $E_{p}$ are the lowest subband energy for square and parabolic potentials respectively. $E_{s}$, $E_{p}$ and the constants $B_{s}$, $B_{p}$ are obtained by choosing the proper boundary conditions. The trial wavefunction of the ground state of the two holes bound to an acceptor impurity is chosen as
$$ \nonumber\\[-5mm]\Psi(x_{1},y_{1},x_{2},y_{2}) = N_{hh} \left\{\begin{array}{ll} \phi_w (x_1,y_1,x_2,y_2) {\rm e}^{-\lambda |{\vec{r}}_1-{\vec{r}}_2|}, \quad & |x_1|,|y_1|,|x_2|,|y_2|\leq L/2\rightarrow \rm{With Correlation},\\ \phi_b (x_1,y_1,x_2,y_2) {\rm e}^{-\lambda |{\vec{r}}_1-{\vec{r}}_2|}, \quad & |x_1|,|y_1|,|x_2|,|y_2|>L/2,\\ \phi_w (x_1,y_1,x_2,y_2) {\rm e}^{-\lambda (r_1+r_2)}, \quad & |x_1|,|y_1|,|x_2|,|y_2|\leq L/2 \rightarrow \rm{Without Correlation},\\ \phi_b (x_1,y_1,x_2,y_2) {\rm e}^{-\lambda (r_1+r_2)}, \quad & |x_1|,|y_1|,|x_2|,|y_2|> L/2,\end{array}\right. $$
where, $N_{hh}$ is the normalization constant and $\lambda$ is the variational parameter.
The expectation value of ${H}_{hh}$ is minimized with respect to $\lambda$ and the hole-hole interaction energy is obtained as follows:
$$ \nonumber\\[-8mm]H_{\rm{min}}=\left< \Psi |{H}_{hh}| \Psi \right>\,,\quad E_{hh}=\langle \Psi \Big|\frac{2}{|{\vec{r}}_1-{\vec{r}}_2}\Big|\Psi\rangle\,. $$
The binding energy of the acceptor impurity with two holes in the presence of magnetic field is found by solving the Schr$\ddot{\rm o}$dinger equation variationally and is given by
$$ E_{B}=2E_{\rm{sub}}+\gamma-{\langle H\rangle}_{\rm min}. $$
3 Results and Discussions
3.1 Hole-Hole Interaction in a Square Well Confinement
The results for the variation of two-hole binding energy of the acceptor impurity and the Coulomb interaction between them, which are confined in a semimagnetic QWW with square band offset in various applied magnetic fields ($\gamma = 0$, $\gamma = 0.04$, $\gamma = 0.06$) by neglecting the correlation between the two holes (Case I) in the chosen wavefuntion as given in Eq. (5) are presented in Figs. 1(a) and 1(b). The solid lines represent the results for the On Centre (OC; $Z_i=0$) acceptor impurity and the dotted lines show for On Edge (OE; $Z_i=L/2$) acceptor impurity. One observes a fall in binding energy and interaction energy when a magnetic field is applied. The possible explanation for this fall of binding energy and the interaction with the applied magnetic field may be due to the reduction of the potential barrier (142 meV, 66 meV, and 29 meV for $\gamma = 0$, $\gamma = 0.04$, $\gamma = 0.06$ respectively), which confines the interacting carriers (holes) and thereby the impurity energy levels become shallower which causes the tunnelling of the carriers through the barrier Cd$_{1-x}$Mn$_{x}$Te.
Fig. 1 Variation of (a) binding energy and (b) interaction energy of the two holes bound to an acceptor impurity in an SQWW with square well type confining potential for different impurity locations and magnetic field by neglecting the correlation between the carriers. |
This reduction in the binding and the interaction energy as a function of applied magnetic field is clearly seen only in the narrower wire size as one can see from the figure that both the energies converges irrespective of the applied magnetic field, when the wire size is increased from narrow to bulk limit. Moreover, the effect of magnetic field on the binding energy and the interaction energy is less pronounced when the impurity is placed at the interface between the non-magnetic and semimagnetic layers i.e. for OE impurity, which has less binding inside the wire as compared to OC impurity. Though the impact of magnetic field on the strength of the interaction between the two holes is small for OE impurity case, it is not entirely negligible as one can see from the numerical value of interaction energy which scales down as the magnetic field increases. It is surprising to note that the strength of the Coulomb interaction between the two holes is comparatively negligible for OE impurity. This prodigious fall of interaction energy for OE impurity may be well understood from the fact that there is a fluctuation in the correlation of Mn$^{2+}$ ions along the interface which produces the local changes in the total height of the potential barrier formed between the nonmagnetic and magnetic layers of SQWW. Owing to this fluctuation, the carriers bound to the wire are driven closer to the interface region. In addition to this, the non-accounting of the correlation between the two holes, which is chosen in wavefunction may also contribute to the substantial reduction in the strength of Coulomb interaction between the two holes.
Figures 2(a) and 2(b) depict the variation of binding energy of the acceptor impurity with two holes and the interaction energy against the wire size in various magnetic field, when the correlation between the two holes is accounted (Case II) in the wavefunction. The continuous lines represent the variation for OC impurity and the dotted lines for OE impurity.
Fig. 2 Variation of (a) binding energy and (b) interaction energy of the two holes bound to an acceptor impurity in an SQWW with square well type confining potential for different impurity locations and magnetic field by accounting the correlation between the carriers. |
It is noted from Fig. 2(a) that the trend of the binding energy against wire size for various magnetic field is as same as the trend of the binding energy in Case I, but with lower in magnitude. The contribution of the correlation between the two holes to the binding energy is about 20%-30% and to the Coulomb interaction is about 70%-80%.The reason for this reduced binding and the Coulomb interaction can be explained through the kinetic energy and the potential energy variation of the carriers for both with and without including the correlation which has been compared in Fig. 3. Despite an increase in the kinetic energy of the carriers is observed for the Case I, a large difference in the potential energy variation between the two cases is also noticed. The calculated potential energy is more negative for the Case I compared to the Case II, which means that the carriers are strongly attached to the parent acceptor atom only when the correlation effect is ignored. Therefore the binding energy of the carriers will become very less when the correlation effect is taken into account between the two holes. It is also noted from Fig. 2(a) that the fall of binding energy with respect to magnetic field is not as rapid as the fall of binding energy observed in the Case I as one can compare the numerical values of the binding energy in both the cases from Figs. 1(a) and 2(a). It is interesting to note the increase of Coulomb interaction between the two holes as the wire size is shrunk towards the narrower region and it attains the maximum around the wire size of 80 Å and beyond which it starts to fall again which is contrary to the Case I where no such turnover is noticed against the wire size. It clearly indicates that the wire of size 50 Å makes the carriers to interact more with each other when the correlation is not considered between them. But, for Case II, a wire of size 80 Å is needed for carriers to repel each other to a greater extent. Indeed, the applied magnetic field shows its prominent effect by suppressing the amplitude of the peak observed for the interaction energy at around 80 Å. The effect of magnetic field on the binding of the carriers inside the QWW for OE impurity is not much appealing as expected. But, by comparing the results of Coulomb interaction for both the impurities from Fig. 2(b), it is noted that the effect of impurity location is not very significant on the Coulomb interaction between the two holes as the strength of the interaction is consistently maintained for smaller wire size, when the impurity moves from the center to the edge of the wire, which reflects the importance of correlation dependence of the Coulomb interaction between the two holes.
Fig. 3 Comparison between the Case I and Case II by plotting kinetic energy and the potential energy variation against wire size for $\gamma = 0$. |
3.2 Hole-Hole Interaction in a Parabolic Well Confinement
The obtained results for both the binding energy and the interaction energy of the two holes as a function of wire size are plotted in Figs. 4(a) and 4(b) for the QWW with parabolic confinement. It is observed from the results that both the binding energy and the interaction energy are much larger in the parabolic confinement than in the square well confinement and it is again clear from the Figs. 4(a) and 4(b) that both the energies decrease as the wire size increases as expected. When the applied magnetic field is increased ($\gamma = 0.06$) near to the critical magnetic field ($\gamma = 0.075$), a turnover is seen in the binding energy against the wire size. The reason for this turnover may be attributed to the fact that the confining potential realized by the carriers is substantially reduced because of the applied magnetic field as discussed in Subsec. 3.1. Therefore, the wire of size 50 Å does not favor for the larger confinement of the carriers inside the QWW for $\gamma = 0.06$ as in the case of zero ($\gamma = 0$) and lower magnetic field ($\gamma = 0.04$), since there is a finite probability for the carriers to tunnel through the barrier when they are bound to the wire with reported size. The mutual repulsion between the two holes is raised at the wire of size 60 Å and it is noticed that this is the optimized wire size for the carriers to repel each other to greater extent irrespective of the applied magnetic field.
Fig. 4 Carriers in Parabolic type confining potential. Variation of (a) binding energy and (b) interaction energy of the two holes bound to an acceptor impurity in an SQWW for various magnetic field by neglecting the correlation between the carriers. |
This may be due to the omission of correlation in the wave function. But the inclusion of correlation in the two hole wavefucntion shows a peculiar trend of the binding as well as the interaction energy as a function of both wire size and applied magnetic field as shown in Figs. 5(a) and 5(b).
Fig. 5 Carriers in Parabolic type confining potential. Variation of (a) binding energy and (b) interaction energy of the two holes bound to an acceptor impurity in an SQWW for various magnetic field by including the correlation between the carriers. |
With increasing magnetic field the Coulomb repulsion increases, since the two holes are closer together. This behavior once again elucidates the importance of accounting the correlation in the wavefunction. But, by neglecting this correlation effect one would erroneously assume that with the applied magnetic field the Coulomb interaction between the holes is decreased instead of concluding that it is an increasing function of applied magnetic field. Moreover, one would clearly notice from Fig. 5(b) that the interaction energy is a decreasing function of wire size and when the wire size is increased beyond 80 Å, $E_{hh}$ is nearly a constant. However, interestingly the wire size at which the onset of saturation of $E_{hh}$ occurs is shifted to higher values of wire size when the applied magnetic field is increased from $\gamma = 0$ to $\gamma = 0.06$. The results presented in Fig. 5(a) astonish as the confinement of the carriers in low-dimensional region is very much less under such parabolic confinement when the correlation effect is included as compared to the bulk limit. The plots for the variation of kinetic energy, potential energy and the barrier potential against the wire size for both the cases are presented in Figs. 6(a) and 6(b).
Fig. 6 Comparision between the Case I and Case II by plotting kinetic energy and the potential energy variaton against wire size for $\gamma = 0$ when carriers are confined in a parabolic potential. (a) Without including the Correlation and (b) With including the Correlation. |
Even though the interaction energy between the two holes reaches a constant value as seen from Fig. 5(b) and the bulk limit may provide more space for the two holes to depart from each other, the Coulomb interaction of both the carriers with their parent acceptor atom is very strong as compared to the low-dimensional region as shown in Fig. 6(b). This may be due to the Gaussian nature of the wavefunction in parabolic confinement. The reliability of these results could not be verified as there are no experimental results in such SQWW emphasized with hole-hole interaction. But, the results have been verified to some extent by reducing the two particle Hamiltonian to a single particle Hamiltonian by treating the carriers as non-interacting.
4 Concluding Remarks
An investigation of two holes confined in an SQWW has been made under the effect of applied magnetic field and also the impurity location. Irrespective of the nature of the confining potential, the correlation effect contributes to a greater extent on the Coulomb interaction between the two holes confined in an SQWW. While, in the case of parabolic confinement, the percentage of contribution by correlation is about $\approx40%$ to the binding energy and is about $\approx90%$ to the Coulomb interaction energy for smaller wire size, in the case of square confinement, it is about 20%—30% and 70%—80% respectively. It is not possible to check the reliability of our results as the experimental results are not available.