ANALELE UNIVERSITĂłII EFTIMIE MURGU REŞIłA ANUL XVI, NR. 1, 2009, ISSN The Ben Daniel-Duke Model Applied to Semiconductor Heterostructure

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1 NLELE UNIVERSITĂłII EFTIMIE MURGU REŞIł NUL XVI, NR., 009, ISSN Cornel HaŃiegan The en Daniel-Due Model pplied to Semiconductor Heterostructure - Part We investigate the semiconductor heterostructure with the en Daniel- Due model applied for the lowest conduction states Ga s-ga () as and for the heavy levels at = 0 in any heterostructures (). In a quantic level we obtained the familiar staircase density of states (). In (3) we calculated the incrgy position of the interface state in a single HgTe-CdTe heterojonction. We also obtained the existence of the interface state relies only on the relative position of the I 8 edges of HgTe and CdTe, their actual energy position, as well as their behavior at 0. Keywords: en Daniel-Due model, heterostructure, quantum well, en Daniel-Due quantum well th Equation(9) defines the in-plane effective mass of the n sub band in the vicinity of =0. It may be remared that if m > m, as is the case in Gas- Ga(l)s or Ga0.47 In0.53 s InP,this in-plane mass m n will increase with increasing sub band index n Using the approximately parabolic in-plane dispersion laws (equation (4)) it is very easy to calculate the density of states p (ε ) associated with the bound states E n. Proceeding exactly as in chapter I we obtain: = n p ( ε ) ( ε ) (0a) p n mns pn ( ε ) = Y ( ε E ) n (0b) πh where Y(x) is the step function. We recover the familiar staircase density of states. The properties of a en Daniel-Due quantum well are summarized in figure 4. 3

2 From the left to the right: conduction band edge profile, energy levels E ande and their associated envelope functions; in-plane dispersions of the E and E sub bands; energy dependence of the heterostructure density of states p (ε ). Figure 4. - recollection of the main properties of the quantum well bound states, solutions of a en Daniel-Due Hamiltonian. 3.Interface states of en Daniel-Due quantum wells ( m m 0 ; = 0 < ) The case m m < 0 is practically realized in HgTe-CdTe heterostructures [0] (see Fig. 5). CdTe is a conventional open gap semiconductor whose level ordering is the same as is found in Gas. HgTe is a symmetry-induced zero gap semiconductor. The Γ 6 band, which is a conduction band in most III-V and II-VI semiconductors, is a light hole band in HgTe. The Γ 6 edge lies ~ 0.3 ev below the Γ 8 edges. s the Γ8 light band Γ6 band are nearly mirror-lie, the Γ 8 light band is a conduction band in HgTe, degenerate at the zone centre with Γ8 heavy hole band(inversion asymmetry splitting having been neglected). Ignoring the absence of centro-symmetry of the zinc-blade lattice, the light particle and heavy hole states decouple at. We can thus treat the problem of the light particle states associated with a I 8 edge as if we were considering a single band. The interesting feature of the HgTe-CdTe heterostructures is that the light particle changes the sign of its effective mass across the interfaces, being electro-lie in the HgTe layer and light hole-lie in the CdTe layers. To be specific, let us consider a CdTe-HgTe-CdTe double heterostructures. 33

3 Figure 5.- and structures of bul HgTe (left panel) and CdTe (right panel) in the vicinity of the I point (schematic). ccording to [7] the bottom of the HgTe I 8 conduction band lies at an Λ ~ 40 mev above the top of the CdTe I 8 valence band. Thus, bound states of the heterostructures only exist if ε Λ (the energy zero being taen at the I 8 edge in HgTe). If Λ ε 0, the states are evanescent in both inds of layers while if ε 0, the carrier wave vector is real (imaginary) in the HgTe (CdTe) layers. Clearly, bound states of positive energies will exist (an infinite number in the oneband description of each host layer). Proceeding as in section. their energies will fulfil m cos ϕ + sinϕ = 0 for even states () m m cos ϕ sinϕ = 0 for odd states () m ϕ = L (3) m m = ε ; = ( ε + Λ) (4) h h 34

4 Figure 6.- Evolution of ground and first excited bound states (labeled and respectively) versus the HgTe slab thicness in a CdTe-HgTe-CdTe double heterostructure. The bound state wave functions are all characterized by cusps at the interfaces due to the change in the carrier effective mass at the hetero-interfaces. This sign reversal also implies that equation () can be fulfilled at ε = 0 for a certain L while equation () can not. This means that at least one state (even in z) should lie below the bottom of the HgTe conduction band edge. This state is an interface level, built from evanescent states in each of the host layers, whose wave function peas at the interface. More precisely, we can write: ( z) = cos( z) z L (5) ( z) = exp z L z L (6) z (7) with: ( ) ( ) = z m m = ( ε ); = ( ε + Λ) (8) h h 35

5 d y matching ( z) and ( z) µ at z = L, we find that ε should be dz the root of the implicit equation m tanh L = (9) m It is very easy to chec that equation (9) always admits one solution E (and only one) which extrapolates to Λ when L 0. second state may actually exist in the energy segment [-Λ,0] if the HgTe layer is thic enough. It corresponds to an odd envelope function: ( z) = sinh( z) ; z L (30) ( z) = exp z L ; z L (3) = z (3) ( ) ( ) z The E energy is the solution of the implicit equation: m cot anh L = (33) m which admits a solution if m h L > m m Λ (34) gain, the solution of equation (33), if it exists, is unique. When L becomes very large the energies E and E converge to the value: Λ E = (35) m + m which is the energy position of the interface state in a single HgTe-CdTe heterojunction [3, 4]. Clearly, at large L (i.e. L >) the two states E and E are very well approximated by the symmetric and antisymmetric combinations of the two interface states centred at ± L respectively. The behavior of E and E versus 36

6 L presented in figure 6 to illustrate the previous discussion. In figure 7 we show z envelope functions in Hg -x Cd x Te-HgTe-Hg -x Cd x Te quantum the calculated ( ) wells to illustrate the interface nature of the E state. lthough the existence of the interface state relies only on the relative position of the I 8 edges of HgTe and CdTe, their actual energy position, as well as their behavior at 0 (where they strongly couple to the heavy hole states), remains a subject of active research. Fig. 7. Dimensionless envelope functions of the ground states in Hg -x Cd x Te- HgTe-Hg -x Cd x Te double heterostructures (x= and x=0.) for two different HgTe slab thicnesses. References. Duggan G., The Journal of Vacuum Science and Technology. 3 (985) 4.. Guldner Y., astard G., Vieren J.P., Voos M., Faurie J.P., Million., Physics Review, 5 (983) Yia-Chung Chang, Schulman J.N., astard G., Guldner Y., Voods M., Physics Review, 3 (985) Lin Liu Y.R., Sham L.J., Physics Review, 3 (985)

7 5. Voisin P. Two dimensional Systems, Heterostructures and Superlattices edited by G. auer, F. Kuchar and H. Heinrich, Springer Series in Solid State Sci. 53,Springer Verlag, erlin, 984, p ir G.L., Pius G.E., Symmetry and Strain-induced Effects in Semiconductors, Wiley, New Yor, Marzin J.Y., Heterojunctions and Semiconductor Superlattices edited by G. llan, G. astard, N. occara, M. Lannoo and M. Voods, Springer Verlag, erlin, 986, p. 6. ddress: Dr. Cornel HaŃiegan Resita, l reazova, nr. /, ap. 9, 30067, Resita, 38

wave mechanics applied to semiconductor heterostructures

wave mechanics applied to semiconductor heterostructures GERALD BASTARD les editions de physique Avenue du Hoggar, Zone Industrielle de Courtaboeuf, B.P. 112, 91944 Les Ulis Cedex, France Contents PREFACE