## 1. Introduction

The work of Essaki & Tsu [1] caused a big interest to the study of superlattices made from alternating layers of two semiconductors. The development of molecular beam epitaxy (MBE) was successfully applied to fabricate different quantum wells and superlattices. Among them III-V superlattices [Ga_{1-x}Al_{x}As-GaAs [1-2] - type I], III-V superlattices [InAs/GaSb [3] - type II] and later II-VI superlattice [HgTe/CdTe [4] - type III]. The latter is a stable alternative for application in infrared optoelectronic devices than the Hg_{1-x}Cd_{x}Te alloys. Especially in the region of second atmospheric window (around 10 μm) which is of great interest for communication.

HgTe and CdTe crystallize in zinc –blend lattice respectively. The lattice-matching within 0.3 % yield to a small interdiffusion between HgTe and CdTe layers at low temperature near 200 °C by MBE. HgTe is a zero gap semiconductor (due to the inversion of relative positions of Γ_{6} and Γ_{8} edges [5]) when it is sandwiched between the wide gap semiconductor CdTe (1.6 eV at 4.2 K) layers yield to a small gap HgTe/CdTe superlattice which is the key of an infrared detector.

A number of papers have been published devoted to the band structure of this system [6] as well as its magnetooptical and transport properties [7]. The aim of this work is to show the correlation between calculated bands structures and magneto- transport properties in n type HgTe/CdTe nanostructures superlattices. The interpretation of the experimental data is consistent with the small positive offset Λ=40 meV between the HgTe and CdTe valence bands.

Theoretical calculations of the electronic properties of n-type HgTe/CdTe superlattices (SLs) have provided an agreement with the experimental data on the magneto-transport behaviour. We have measured the conductivity, Hall mobility, Seebeck and Shubnikov-de Haas effects and angular dependence of the magneto-resistance [8]. Our sample, grown by MBE, had a period d=d_{1}+d_{2} (124 layers) of d_{1}=8.6 nm (HgTe) /d_{2}=3.2 nm (CdTe). Calculations of the spectres of energy E(d_{2}), E(k_{z}) and E(k_{p}), respectively, in the direction of growth and in plane of the superlattice; were performed in the envelope function formalism. The energy E(d_{2}, Γ, 4.2 K), shown that when d_{2} increase the gap E_{g} decrease to zero at the transition semiconductor to semimetal conductivity behaviour and become negative accusing a semimetallic conduction. At 4.2 K, the sample exhibits n type conductivity, confirmed by Hall and Seebeck effects, with a Hall mobility of 2.5 10^{5} cm^{2}/Vs. This allowed us to observe the Shubnikov-de Haas effect with n = 3.20 10^{12} cm^{-2}. Using the calculated effective mass (m*_{E1}(E_{F}) = 0.05 m_{0}) of the degenerated electrons gas, the Fermi energy (2D) was E_{F}=88 meV in agreement with 91 meV of thermoelectric power α. In intrinsic regime, α T^{-3/2} and R_{H}T^{3/2} indicates a gap E_{g} =E_{1}-HH_{1}= 101 meV in agreement with calculated E_{g}(Γ, 300 K) =105 meV The formalism used here predicts that the system is semiconductor for d_{1}/d_{2} = 2.69 and d_{2} < 100 nm. Here, d_{2}=3.2 nm and E_{g} (Γ,4.2 K) = 48 meV so this sample is a two-dimensional modulated nano-semiconductor and far-infrared detector (12 μm<λ_{c}<28 μm).

In conclusion, we will show that the HgTe/CdTe nano-superlattice is a stable alternative for application in infrared optoelectronic devices than the alloys Hg_{1-x}Cd_{x}Te.

## 2. Experimental techniques

The HgTe/CdTe superlattice was grown by molecular beam epitaxy (MBE) on a [111] CdTe substrate at 180 °C. The sample (124 layers) had a period d=d_{1}+d_{2} where d_{1}(HgTe)=8.6 nm and d_{2}(CdTe)= 3.2 nm. It was cut from the epitaxial wafer with a typical sizes of 5x1x1mm^{3}. The ohmic contacts were obtained by chemical deposition of gold from a solution of tetrachloroauric acid in methanol after a proper masking to form the Hall crossbar. Carriers transport properties were studied in the temperature range 1.5-300K in magnetic field up to 17 Tesla. Conductivity, Hall Effect, Seebeck effect and angular dependence of the transverse magnetoresistance were measured. The measurements at weak magnetic fields (up to 1.2 T) were performed into standard cryostat equipment. The measurements of the magnetoresistance were done under a higher magnetic field (up to 8 T), the samples were immersed in a liquid helium bath, in the centre of a superconducting coil. Rotating samples with respect to the magnetic field direction allowed one to study the angular dependence of the magnetoresistance.

## 3. Theory of structural bands

Calculations of the spectra of energy E (k_{z}) and E(k_{p}), respectively, in the direction of growth and in plane of the superlattice; were performed in the envelope function formalism [6-7]) with a valence band offset Λ between heavy holes bands edges of HgTe and CdTe of 40 meV determined by the magneto-optical absorption experiments [9].

The general dispersion relation of the light particle (electron and light hole) subbands of the superlattice is given by the expression [6]:

where the subscripts 1 and 2 refer to HgTe and CdTe respectively. k_{z} is the superlattice wave vector in the direction parallel to the growth axis z. The two-dimensional wave vector k_{p}(k_{x}, k_{y}) describes the motion of particles perpendicular to k_{z}. Here,

E is the energy of the light particle in the superlattice measured from the top of the Γ_{8} valence band of bulk CdTe, while εi (i =1 or 2) is the interaction band gaps E(Γ_{6}) - E(Γ_{8}) in the bulk HgTe and CdTe respectively. At given energy, the two–band Kane model [10] gives the wave vector (k_{i}²+k_{p}²) in each host material:

P is the Kane matrix element given by the relation:

where m* = 0.03 m_{0} is the electron cyclotron mass in HgTe [5]. For a given energy E, a superlattice state exists if the right-hand side of Eq. (1) lies in the range [-1,1]. That implies -π/d ≤ k_{z}≤-π/d in the first Brillion zone.

The heavy hole subbands of the superlattice are given by the same Eq. (1) with :

m*_{HH} =0.3 m_{0} [5] is the effective heavy hole mass in the host materials.

The band structure computation consists of solving Eq. (1) which represents the dispersion relations (i.e. finding the values of energy E which are roots of the Eq. (1) for a given value of the carrier wave vector). Here, we are interested in studying the states of energy of light particles and heavy holes in HgTe/CdTe superlattice as function of k_{z} when k_{p}=0 and as function of k_{p} when k_{z}=0 and when k_{z}=π/d. The solving procedure used for studying E as function k_{z} in the case where k_{p}=0 consists of going, with a steep E, through the studied range of energy E and then finding, for each value of E,

the value of k_{z} which satisfies the dispersion relations. The same procedure is used for studying E as function k_{p} in the case where k_{z}=0 and k_{z}=π/d. It is noteworthy that, for a given value of E, Eq. (1) may have more than one root in k_{p}. It appears, from Eq. (3)-(5), that the carrier wave vectors k_{1}, k_{2}, and k_{p} are either real or imaginary (i.e. complex) and then using complex numbers in the calculation seems to be more adequate.

## 4. Theoretical results and discussions

The energy E as a function of d_{2}, at 4.2 K, in the first Brillion zone and for d_{1} = 2.69 d_{2}, is shown in “Figure 1 (a)”. The case of our sample (d_{2} = 32 Å) is indicated by the vertical solid line. Here the cross-over of E_{1} and HH_{1} subbands occurs. d_{2} controls the superlattice band gap E_{g} = E_{1}-HH_{1}. For weak d_{2} the sample is semiconductor with a strong coupling between the HgTe wells. At the point T(d_{2} = 100 Å, E= 38 meV) the gap goes to zero with the transition semiconductor- semimetal. When d_{2} increases, E_{1} and h_{1} states drops in the energy gap [0, Λ] and become interface state with energy

for infinite d_{2} obtained from Eq. (1). Then the superlattice has the tendency to become a layer group of isolated HgTe wells and thus assumes a semimetallic character. The ratio d_{1}/d_{2} governs the width of superlattice subbands (i.e. the electron effective mass). A big d_{1}/d_{2}, as in the case of 4.09 in Fig. 2, moves away the material from the two-dimensional behaviour.

In “Figure 1 (b)” we can see that the band gap E_{g}(Γ) increases, presents a maximum at 10, 15 and 25 meV respectively for 4.2, 77 and 300K, near Λ=40 meV and decreases when the valence band offset Λ between heavy hole band edges of HgTe and CdTe increase. For each Λ, E_{g}(Γ) increases with T. Our chosen value of 40 meV is indicated by a vertical solid line. This offset agrees well with our experimental results and 0 meV used by [4] for all temperatures, indicated by a horizontal solid line in “Figure 1 (b)”, contrary to 360 meV given by [11]. The later offset give E_{g} = -8 meV in “Figure 1 (b)” whereas, in intrinsic regime, R_{H} T^{3/2} indicates a measured gap E_{g} ≈ 98 meV in agreement with our calculated E_{g}(Γ,300 K) = 105 meV.

“Figure 2” shows that, for each d_{2}, E_{g} (Γ) decreases when d_{1}/d_{2} increases. For each d_{1}/d_{2}, when d_{2} increases E_{g} (Γ) decreases, go to zero at the transition point T and became negative for a semimetal conductivity. In the right of “Figure 2”, the cut-off wavelength |λ_{c}| diverge at T with d_{2}=54 Å, 100 Å, 150 Å,... respectively for d_{1}/d_{2}=4.09, 2.69, 1.87, … So, the transition goes to high d_{2} when d_{1}/d_{2} decreases. In the case of our sample the transition occur at d_{2} =100 Å.

Using the value of ε_{1} and ε_{2} at different temperatures between 4.2 K and 300 K [12] and taking P temperature independent, this is supported by the fact that from eq.(4) P ≈ ε_{G}(T)/m*(T) ≈ Cte, we get the temperature dependence of the band gap Eg, in the centre Γ of the first Brillouin zone in “Figure 3”. Note that E_{g} increases from 48 meV at 4.2 K to 105 meV at 300 K. We calculated the detection cut-off wave length by the relation

In the investigated temperature range 12 μm < λ_{c} < 28 μm situates our sample as a far infrared detector.

In “Figure 4 (a)” we can see the spectres of energy E(k_{z}) and E(k_{p}), respectively, in the direction of growth and in plane of the superlattice at 4.2 K. Along k_{z} the subbands E_{1} and h_{1} are large and non-parabolic as shown in “Figure 4. (b)”. Along k_{p}, E_{1} and h_{1} increase with k_{p} whereas the parabolic HH_{n} decreases in “Figure 4. (b)”. This yield to an anti-crossing of HH_{1} and h_{1} at k_{p}= 0.0139 Å^{-1} near the middle of the first Brillouin zone (π/2d).

For an anisotropic medium, such as the HgTe/CdTe superlattices, the effective mass is a tensor and its elements along μ and ν directions are given by the following expression [12].

By carrying out second derivative of the energy E_{1}, h_{1} and HH_{1} along k_{z} and k_{p} in “Figure 4 (a)”, we calculated the effective mass bands in “Figure 5”. Along k_{p}, the effective mass of heavy holes m*_{HH1}= - 0.30 m_{0} and the effective mass of electrons m*_{E1} increases from 0.04 m_{0} to 0.11 m_{0}. In “Figure 4 (a)”, the Fermi level across the conduction band E_{1} at k_{p}=0.014 Å^{-1} corresponding to

from “Figure 5”. Whereas, the effective mass of the light holes h_{1} deceases from 0.24 m_{0}, to a minimum of

at k_{p}=0.14 Å^{-1}, and increase to 0.30 m_{0} assuming an electronic conduction.

## 5. Experimental results and discussions

In “Figure 6.(a)” we can see that the angular dependence of the magnetoresistance vanishes, when the field is parallel to the plane of the SL (at 90°), indicating a two dimensional (2D) behaviour supported by the observation of SDH oscillations in “Figure 8 (a)”.

We have also measured the conductivity, Hall mobility and Seebeck effect. At low temperatures, the sample exhibits n type conductivity (R_{H} < 0) with a concentration n=1/e R_{H} =3.24x10^{12} cm^{-2} from “Figure 6 (c)” and in “Figure 7. (a)”; and a Hall mobility μ_{n} = 2.5x10^{5} cm^{2}/Vs in “Figure 6 (d)”. The plot log(μ_{n})-log(T) in the “Figure 7.(b)” of the Hall mobility shows a scattering of electrons by phonons in the intrinsic regime with μ_{H}T^{1.58} and week activation energy at low temperature with μ_{H} T^{0.05}.

This relatively electrons high mobility allowed us to observe the Shubnikov-de Haas effect until 8 Tesla in “Figure 8 (a)”. Its well knows that the oscillations of the magnetoresistance are periodic with respect to 1/B [14]. The period (1/B) is related to the concentration n of the electrons by the relation:

In the insert of “Figure 8(a)” we have plotted the inverse of the minima’s 1/B_{m} as a function of the entire n’ following the formula:

The linear line slope gives (1/B) and n = 3.20x10^{12} cm^{-2} in good agreement with that measured by weak field Hall effect (H=0.5 KOe, I= 5μA) from “Figure 6(c)”.

At low temperature, the superlattice electrons dominate the conduction in plane. The E_{1} band is not parabolic with respect to k_{p}^{2} in “Figure 4 (b)”. That permits us to estimate the Fermi energy (2D) at 4.2K

The thermoelectric power (α<0) measurements shown in “Figure 8(b)” indicate n-type conductivity, confirmed by Hall Effect measurements (R_{H}<0) in “Figure 6(c)”. At low temperature, α T^{0.96} (in the top insert of “Figure 8(b)”) is in agreement with Seebeck effect theory deduced from the relaxation time resolution of the Boltzmann equation [15].

For our degenerate electrons gas the Seebeck constant is described by the formula:

where the collision time τ E^{s-(1/2)}. This permits us to estimate the Fermi energy at E_{F} = 91 meV (in “Figure 4(a)”), with s= 2.03 corresponding to electrons diffusion by ionized impurities, in agreement with the calculated E_{F}=88 meV in formula (13). In intrinsic regime for T>150 K, the measure of the slope of the curve R_{H} T^{3/2} indicates a gap E_{g} = 98 meV which agree well with calculated E_{g} (Γ, 300 K) = E_{1}-HH_{1}= 105 meV. Here αT^{-3/2} indicates electrons scattering by phonons.

This HgTe/CdTe superlattice is a stable alternative for application in far infrared optoelectronic devices than the random alloys Hg_{0.99}Cd_{0.01}Te because the small composition x=0.01, with E_{g} (Γ, 300 K) =100 meV given by the empiric formula for Hg_{1-x}Cd_{x}Te [16]

is difficult to obtain with precision while growing the ternary alloys and the transverse effective masse in superlattice is two orders higher than in the alloy. Thus the tunnel length is small in the superlattice [17].

## 6. Conclusions

The fundamental main ideas of this work are:

- HgTe is a zero gap semiconductor (or semimetal) when it is sandwiched between the wide gap semiconductor CdTe (1.6 eV at 4.2 K) layers yield to a narrow gap HgTe/CdTe superlattice which is the key of an infrared detector.

- Before growing our superlattice, we calculated the bands structures E(d_{2}) and the gap for each ratio thickness d_{1}/d_{2}. After we choosed the SL in the semiconductor conductivity zone.

We reported here remarkable correlations between calculated bands structures and magneto- transport properties in HgTe/CdTe nanostructures superlattices. Our calculations of the specters of energy E(d_{2}), E(k_{z}) and E(k_{p}), respectively, in the direction of growth and in plane of the superlattice; were performed in the envelope function formalism.

The formalism used here predicts that the system is semiconductor, for our HgTe to CdTe thickness ratio d_{1}/d_{2} = 2.69, when d_{2} < 100 nm. In our case, d_{2}=3.2 nm and E_{g} (Γ, 4.2 K) = 48 meV. In spite of it, the sample exhibits the features typical for the semiconductor n-type conduction mechanism. In the used temperature range, this simple is a far-infrared detector, narrow gap and two-dimensional n-type semiconductor. Note that we had observed a semimetallic conduction mechanism in the quasi 2D p type HgTe/CdTe superlattice [18].

In conclusion, the HgTe/CdTe superlattice is a stable alternative for application in infrared optoelectronic devices than the alloys Hg_{1-x}Cd_{x}Te.

Measurements performed by us on others’ samples indicate an improvement of quality of the material manifested by higher mobility.