**3. Theory of structural bands**

Calculations of the spectra of energy E (kz) and E(kp), 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]:

$$\cos k\_x(\mathbf{d}\_1 + \mathbf{d}\_2) = \cos \left(\mathbf{k}\_1 \mathbf{d}\_1\right) \cos \left(\mathbf{k}\_2 \mathbf{d}\_2\right) - \frac{1}{2} \left[ \left(\xi + \frac{1}{\xi}\right) + \frac{\mathbf{k}\_p^\perp}{4\mathbf{k}\_1 \mathbf{k}\_2} \left(\mathbf{r} + \frac{1}{\mathbf{r}} \cdot \mathbf{2}\right) \right] \sin(\mathbf{k}\_1 \mathbf{d}\_1) \sin(\mathbf{k}\_2 \mathbf{d}\_2) \tag{1}$$

where the subscripts 1 and 2 refer to HgTe and CdTe respectively. kz is the superlattice wave vector in the direction parallel to the growth axis z. The two-dimensional wave vector kp(kx, ky) describes the motion of particles perpendicular to kz. Here,

$$\xi = \frac{k\_1}{k\_2} \quad \text{and} \quad \mathbf{r} = \frac{E - \varepsilon\_2}{E + \varepsilon\_1 \parallel - \Lambda} \tag{2}$$

E is the energy of the light particle in the superlattice measured from the top of the Γ<sup>8</sup> va‐ lence band of bulk CdTe, while εi (i =1 or 2) is the interaction band gaps E(Γ6) - E(Γ8) in the bulk HgTe and CdTe respectively. At given energy, the two–band Kane model [10] gives the wave vector (ki ²+kp²) in each host material:

$$\begin{aligned} \frac{2}{3} \text{ P}^2 \hbar^2 \left( \mathbf{k}\_1^2 + \mathbf{k}\_p \mathbf{\hat{z}} \right) &= \left( \mathbf{E} \cdot \Lambda \right) \left( \mathbf{E} \cdot \Lambda + \left\lfloor \mathbf{e}\_1 \right\rfloor \right) & \text{for HgTe} \\ \frac{2}{3} \text{ P}^2 \hbar^2 \left( \mathbf{k}\_2^2 + \mathbf{k}\_p \mathbf{\hat{z}} \right) &= \mathbf{E} \left( \mathbf{E} \cdot \mathbf{e}\_2 \right) & \text{for CdTe} \end{aligned} \tag{3}$$

P is the Kane matrix element given by the relation:

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 behav‐ iour. 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=d1+d2 (124 layers) of d1=8.6 nm (HgTe) /d2=3.2 nm (CdTe). Calculations of the spectres of energy E(d2), E(kz) and E(kp), respectively, in the direction of growth and in plane of the superlattice; were performed in the envelope function formalism. The energy E(d2, Γ, 4.2 K), shown that when d2 increase the gap Eg decrease to zero at the transition semiconduc‐ tor to semimetal conductivity behaviour and become negative accusing a semimetallic con‐ duction. At 4.2 K, the sample exhibits n type conductivity, confirmed by Hall and Seebeck

Haas effect with n = 3.20 1012 cm-2. Using the calculated effective mass (m\*E1(EF) = 0.05 m0) of the degenerated electrons gas, the Fermi energy (2D) was EF=88 meV in agreement with 91 meV of thermoelectric power α. In intrinsic regime, α T-3/2 and RHT3/2 indicates a gap Eg =E1- HH1= 101 meV in agreement with calculated Eg(Γ, 300 K) =105 meV The formalism used here predicts that the system is semiconductor for d1/d2 = 2.69 and d2 < 100 nm. Here, d2=3.2 nm and Eg (Γ,4.2 K) = 48 meV so this sample is a two-dimensional modulated nano-semiconduc‐

In conclusion, we will show that the HgTe/CdTe nano-superlattice is a stable alternative for

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=d1+d2 where d1(HgTe)=8.6 nm and d2(CdTe)= 3.2 nm. It was cut from the epitaxial wafer with a typical sizes of 5x1x1mm3

The ohmic contacts were obtained by chemical deposition of gold from a solution of tetra‐ chloroauric 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 magnetoresist‐ ance 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 magnetore‐

Calculations of the spectra of energy E (kz) and E(kp), respectively, in the direction of growth and in plane of the superlattice; were performed in the envelope function formalism [6-7])

application in infrared optoelectronic devices than the alloys Hg1-xCdxTe.

/Vs. This allowed us to observe the Shubnikov-de

.

effects, with a Hall mobility of 2.5 105 cm2

138 Optoelectronics - Advanced Materials and Devices

tor and far-infrared detector (12 μm<λc<28 μm).

**2. Experimental techniques**

**3. Theory of structural bands**

sistance.

$$\frac{2\text{P}^2}{3\parallel\text{e}\_1\parallel} = \frac{1}{2\text{m}^\*}\tag{4}$$

where m\* = 0.03 m0 is the electron cyclotron mass in HgTe [5]. For a given energy E, a super‐ lattice state exists if the right-hand side of Eq. (1) lies in the range [-1,1]. That implies -π/d ≤ kz≤-π/d in the first Brillion zone.

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

$$\xi = \frac{\mathbf{k}\_1}{\mathbf{k}\_2} \mathbf{r}, and \quad \mathbf{r} = \mathbf{1} \qquad \text{with} \begin{cases} \cdot \frac{\hbar^2 (\mathbf{k}\_1 \,^2 \mathbf{k}\_p \mathbf{r})}{2 \text{ m}^\* \text{H} \mathbf{r}} = \mathbf{E} \cdot \boldsymbol{\Lambda} & \text{for } \mathbf{H} \mathbf{g} \, \text{Te} \\\\ \frac{\hbar^2 (\mathbf{k}\_2 \,^2 \mathbf{k}\_p \mathbf{r})}{2 \text{ m}^\* \text{H} \mathbf{r}} = \mathbf{E} & \text{for } \mathbf{C} \, \text{d} \mathbf{T} \mathbf{e} \end{cases} \tag{5}$$

m\*HH =0.3 m0 [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 kz when kp=0 and as function of kp when kz=0 and when kz=π/d. The solving procedure used for studying E as function kz in the case where kp=0 consists of going, with a steep E, through the studied range of energy E and then finding, for each value of E,

for infinite d2 obtained from Eq. (1). Then the superlattice has the tendency to become a lay‐ er group of isolated HgTe wells and thus assumes a semimetallic character. The ratio d1/d2 governs the width of superlattice subbands (i.e. the electron effective mass). A big d1/d2, as in the case of 4.09 in Fig. 2, moves away the material from the two-dimensional behaviour.

Correlation Between Band Structure and Magneto- Transport Properties in n-type HgTe/CdTe Two-Dimensional...

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141

In "Figure 1 (b)" we can see that the band gap Eg(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 va‐ lence band offset Λ between heavy hole band edges of HgTe and CdTe increase. For each Λ, Eg(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 Eg = -8 meV in "Figure 1 (b)" whereas, in intrinsic regime, RH T3/2 indicates a

measured gap Eg ≈ 98 meV in agreement with our calculated Eg(Γ,300 K) = 105 meV.

"Figure 2" shows that, for each d2, Eg (Γ) decreases when d1/d2 increases. For each d1/d2, when d2 increases Eg (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 |λc| diverge at T with d2=54 Å, 100 Å, 150 Å,... respectively for d1/d2=4.09, 2.69, 1.87, … So, the transition goes to high d2 when d1/d2 decreases. In the case of our sample the transition occur at d2

Using the value of ε1 and ε2 at different temperatures between 4.2 K and 300 K [12] and tak‐ ing P temperature independent, this is supported by the fact that from eq.(4) P ≈ εG(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 Eg increases from 48 meV at 4.2 K to 105 meV at 300

*Eg*(*meV* ) (7)

*<sup>λ</sup>c*(*μm*)= <sup>1240</sup>

K. We calculated the detection cut-off wave length by the relation

**Figure 2.** Eg (Γ) and |λc| as function of d2 for various d1/d2 at 4.2 K.

=100 Å.

**Figure 1.** a) Energy position and width of the conduction (Ei), heavy-hole (HHi), and the first light-hole (h1) subbands calculated at 4,2 K in the first Brillion zone as a function of layer thickness for HgTe/CdTe superlattice with d1=2.69 d2, where d1 and d2 are the thicknesses of the HgTe and CdTe layers, respectively. T is the point of the transition semicon‐ ductor- semimetal. (b) the band gap Eg (Γ) at Γ, as function of temperature and valence band offset Λ between heavy holes bands edges of HgTe and CdTe for the investigated HgTe/CdTe superlattice.

the value of kz which satisfies the dispersion relations. The same procedure is used for studying E as function kp in the case where kz=0 and kz=π/d. It is noteworthy that, for a giv‐ en value of E, Eq. (1) may have more than one root in kp. It appears, from Eq. (3)-(5), that the carrier wave vectors k1, k2, and kp 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 d2, at 4.2 K, in the first Brillion zone and for d1 = 2.69 d2, is shown in "Figure 1 (a)". The case of our sample (d2 = 32 Å) is indicated by the vertical solid line. Here the cross-over of E1 and HH1 subbands occurs. d2 controls the superlattice band gap Eg = E1-HH1. For weak d2 the sample is semiconductor with a strong coupling between the HgTe wells. At the point T(d2 = 100 Å, E= 38 meV) the gap goes to zero with the transi‐ tion semiconductor- semimetal. When d2 increases, E1 and h1 states drops in the energy gap [0, Λ] and become interface state with energy

$$E\_I = \frac{\Lambda \varepsilon\_2}{\parallel \varepsilon\_1 \parallel + \varepsilon\_2} = \Im 4 \,\mu m \, V \tag{6}$$

for infinite d2 obtained from Eq. (1). Then the superlattice has the tendency to become a lay‐ er group of isolated HgTe wells and thus assumes a semimetallic character. The ratio d1/d2 governs the width of superlattice subbands (i.e. the electron effective mass). A big d1/d2, as in the case of 4.09 in Fig. 2, moves away the material from the two-dimensional behaviour.

**Figure 2.** Eg (Γ) and |λc| as function of d2 for various d1/d2 at 4.2 K.

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 kz when kp=0 and as function of kp when kz=0 and when kz=π/d. The solving procedure used for studying E as function kz in the case where kp=0 consists of going, with a steep E, through the studied

**Figure 1.** a) Energy position and width of the conduction (Ei), heavy-hole (HHi), and the first light-hole (h1) subbands calculated at 4,2 K in the first Brillion zone as a function of layer thickness for HgTe/CdTe superlattice with d1=2.69 d2, where d1 and d2 are the thicknesses of the HgTe and CdTe layers, respectively. T is the point of the transition semicon‐ ductor- semimetal. (b) the band gap Eg (Γ) at Γ, as function of temperature and valence band offset Λ between heavy

the value of kz which satisfies the dispersion relations. The same procedure is used for studying E as function kp in the case where kz=0 and kz=π/d. It is noteworthy that, for a giv‐ en value of E, Eq. (1) may have more than one root in kp. It appears, from Eq. (3)-(5), that the carrier wave vectors k1, k2, and kp are either real or imaginary (i.e. complex) and then using

The energy E as a function of d2, at 4.2 K, in the first Brillion zone and for d1 = 2.69 d2, is shown in "Figure 1 (a)". The case of our sample (d2 = 32 Å) is indicated by the vertical solid line. Here the cross-over of E1 and HH1 subbands occurs. d2 controls the superlattice band gap Eg = E1-HH1. For weak d2 the sample is semiconductor with a strong coupling between the HgTe wells. At the point T(d2 = 100 Å, E= 38 meV) the gap goes to zero with the transi‐ tion semiconductor- semimetal. When d2 increases, E1 and h1 states drops in the energy gap

=34 *meV* (6)

range of energy E and then finding, for each value of E,

140 Optoelectronics - Advanced Materials and Devices

holes bands edges of HgTe and CdTe for the investigated HgTe/CdTe superlattice.

complex numbers in the calculation seems to be more adequate.

*EI* <sup>=</sup> *Λε*<sup>2</sup> |*ε*<sup>1</sup> | + *ε*<sup>2</sup>

**4. Theoretical results and discussions**

[0, Λ] and become interface state with energy

In "Figure 1 (b)" we can see that the band gap Eg(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 va‐ lence band offset Λ between heavy hole band edges of HgTe and CdTe increase. For each Λ, Eg(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 Eg = -8 meV in "Figure 1 (b)" whereas, in intrinsic regime, RH T3/2 indicates a measured gap Eg ≈ 98 meV in agreement with our calculated Eg(Γ,300 K) = 105 meV.

"Figure 2" shows that, for each d2, Eg (Γ) decreases when d1/d2 increases. For each d1/d2, when d2 increases Eg (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 |λc| diverge at T with d2=54 Å, 100 Å, 150 Å,... respectively for d1/d2=4.09, 2.69, 1.87, … So, the transition goes to high d2 when d1/d2 decreases. In the case of our sample the transition occur at d2 =100 Å.

Using the value of ε1 and ε2 at different temperatures between 4.2 K and 300 K [12] and tak‐ ing P temperature independent, this is supported by the fact that from eq.(4) P ≈ εG(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 Eg 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

$$
\lambda\_c(\mu m) = \frac{1240}{E\_\mathcal{g}(meV)}\tag{7}
$$

In the investigated temperature range 12 μm < λc < 28 μm situates our sample as a far infra‐ red detector.

**Figure 3.** Temperature dependence of the band gap Eg and the cut-off wavelength λc, at the center Γ of the first Bril‐ louin zone.

**Figure 5.** Calculated relative effective mass bands along the wave vector kz and in plane kp of the HgTe/CdTe super‐

Correlation Between Band Structure and Magneto- Transport Properties in n-type HgTe/CdTe Two-Dimensional...

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143

In "Figure 4 (a)" we can see the spectres of energy E(kz) and E(kp), respectively, in the direc‐ tion of growth and in plane of the superlattice at 4.2 K. Along kz the subbands E1 and h1 are large and non-parabolic as shown in "Figure 4. (b)". Along kp, E1 and h1 increase with kp whereas the parabolic HHn decreases in "Figure 4. (b)". This yield to an anti-crossing of HH1

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

By carrying out second derivative of the energy E1, h1 and HH1 along kz and kp in "Figure 4 (a)", we calculated the effective mass bands in "Figure 5". Along kp, the effective mass of heavy holes m\*HH1= - 0.30 m0 and the effective mass of electrons m\*E1 increases from 0.04 m0 to 0.11 m0. In "Figure 4 (a)", the Fermi level across the conduction band E1 at kp=0.014 Å-1

from "Figure 5". Whereas, the effective mass of the light holes h1 deceases from 0.24 m0, to a

<sup>∂</sup><sup>2</sup>*Ekμ<sup>ν</sup>* ∂*kμ*∂*k<sup>ν</sup>*

=0.05 *m*<sup>0</sup> (9)

(8)

= 1 ℏ2

and h1 at kp= 0.0139 Å-1 near the middle of the first Brillouin zone (π/2d).

( 1 *<sup>m</sup>* <sup>∗</sup> )*μ<sup>ν</sup>*

> (*mE*<sup>1</sup> \* )*EF*

lattice at 4.2 K

corresponding to

minimum of

**Figure 4.** a) Calculated bands along the wave vector kz in the right and in plane kp(kx,ky) for kz=0 and π/d in the left,. EF is the energy of Fermi level. (b) Calculated bands along the kz 2 in the right and kp 2 for kz=0 and π/d in the left; of the HgTe/CdTe superlattice at 4.2 K.

Correlation Between Band Structure and Magneto- Transport Properties in n-type HgTe/CdTe Two-Dimensional... http://dx.doi.org/10.5772/52101 143

In the investigated temperature range 12 μm < λc < 28 μm situates our sample as a far infra‐

**Figure 3.** Temperature dependence of the band gap Eg and the cut-off wavelength λc, at the center Γ of the first Bril‐

**Figure 4.** a) Calculated bands along the wave vector kz in the right and in plane kp(kx,ky) for kz=0 and π/d in the left,. EF

2 in the right and kp

2 for kz=0 and π/d in the left; of the

is the energy of Fermi level. (b) Calculated bands along the kz

HgTe/CdTe superlattice at 4.2 K.

red detector.

142 Optoelectronics - Advanced Materials and Devices

louin zone.

**Figure 5.** Calculated relative effective mass bands along the wave vector kz and in plane kp of the HgTe/CdTe super‐ lattice at 4.2 K

In "Figure 4 (a)" we can see the spectres of energy E(kz) and E(kp), respectively, in the direc‐ tion of growth and in plane of the superlattice at 4.2 K. Along kz the subbands E1 and h1 are large and non-parabolic as shown in "Figure 4. (b)". Along kp, E1 and h1 increase with kp whereas the parabolic HHn decreases in "Figure 4. (b)". This yield to an anti-crossing of HH1 and h1 at kp= 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 ten‐ sor and its elements along μ and ν directions are given by the following expression [12].

$$\left(\frac{1}{m^\*}\right)\_{\mu\nu} = \frac{1}{\hbar^2} \frac{\partial^2 E\_{k\_{\mu\nu}}}{\partial k\_{\mu} \partial k\_{\nu}}\tag{8}$$

By carrying out second derivative of the energy E1, h1 and HH1 along kz and kp in "Figure 4 (a)", we calculated the effective mass bands in "Figure 5". Along kp, the effective mass of heavy holes m\*HH1= - 0.30 m0 and the effective mass of electrons m\*E1 increases from 0.04 m0 to 0.11 m0. In "Figure 4 (a)", the Fermi level across the conduction band E1 at kp=0.014 Å-1 corresponding to

$$\left(m\_{E\_1}^{\*}\right)\_{E\_F} = 0.05 \, m\_0 \tag{9}$$

from "Figure 5". Whereas, the effective mass of the light holes h1 deceases from 0.24 m0, to a minimum of

**Figure 6.** Variation of magnetoresistance of the sample with various angles between the magnetic field and the nor‐ mal to the HgTe/CdTe superlattice surface (a). Temperature dependence of the conductivity (b), weak-field Hall coeffi‐ cient (c) and Hall mobility (d) in the investigated HgTe/CdTe superlattice

$$\left(m\_{\hbar\_1}^{\ast}\right)\_{\text{min}} = 0.16 \, m\_0 \tag{10}$$

1 *Bm*

ured by weak field Hall effect (H=0.5 KOe, I= 5μA) from "Figure 6(c)".

*EF* <sup>=</sup>*nπ<sup>h</sup>* <sup>2</sup>

band is not parabolic with respect to kp

Fermi energy (2D) at 4.2K

gated HgTe/CdTe superlattice

<sup>=</sup>*Δ*( <sup>1</sup>

2

/(*mE*<sup>1</sup> \* )*EF*

deduced from the relaxation time resolution of the Boltzmann equation [15].

*<sup>B</sup>* )(*<sup>n</sup>* ' <sup>+</sup>

The linear line slope gives (1/B) and n = 3.20x1012 cm-2 in good agreement with that meas‐

At low temperature, the superlattice electrons dominate the conduction in plane. The E1

The thermoelectric power (α<0) measurements shown in "Figure 8(b)" indicate n-type con‐ ductivity, confirmed by Hall Effect measurements (RH<0) in "Figure 6(c)". At low tempera‐ ture, α T0.96 (in the top insert of "Figure 8(b)") is in agreement with Seebeck effect theory

**Figure 7.** a) Variation with 1/T of weak-field Hall concentration and (b) Variation with T of Hall mobility in the investi‐

**Figure 8.** a) Variation of transverse magneto-resistance with magnetic field at 4.2 K, (b) Measured thermoelectric

power as a function of temperature in the investigated HgTe/CdTe superlattice.

1

Correlation Between Band Structure and Magneto- Transport Properties in n-type HgTe/CdTe Two-Dimensional...

<sup>2</sup> ) (12)

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145

in "Figure 4 (b)". That permits us to estimate the

=88 meV (13)

at kp=0.14 Å-1, and increase to 0.30 m0 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 tempera‐ tures, the sample exhibits n type conductivity (RH < 0) with a concentration n=1/e RH =3.24x1012 cm-2 from "Figure 6 (c)" and in "Figure 7. (a)"; and a Hall mobility μn = 2.5x105 cm2 /Vs in "Figure 6 (d)". The plot log(μn)-log(T) in the "Figure 7.(b)" of the Hall mobility shows a scattering of electrons by phonons in the intrinsic regime with μHT1.58 and week acti‐ vation energy at low temperature with μH T0.05.

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:

$$\tau\_H = \frac{e}{\pi^\* \hbar \Delta \left(\frac{1}{B}\right)}\tag{11}$$

In the insert of "Figure 8(a)" we have plotted the inverse of the minima's 1/Bm as a function of the entire n' following the formula:

Correlation Between Band Structure and Magneto- Transport Properties in n-type HgTe/CdTe Two-Dimensional... http://dx.doi.org/10.5772/52101 145

$$\frac{1}{\frac{1}{B\_m}} = \Delta \left( \frac{1}{B} \right) \left( n^\* + \frac{1}{2} \right) \tag{12}$$

The linear line slope gives (1/B) and n = 3.20x1012 cm-2 in good agreement with that meas‐ ured 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 E1 band is not parabolic with respect to kp 2 in "Figure 4 (b)". That permits us to estimate the Fermi energy (2D) at 4.2K

$$E\_F = n\pi\hbar^2 \left/ \left(m\_{E\_1}^\*\right)\_{E\_F} = 88\,\text{meV} \tag{13}$$

The thermoelectric power (α<0) measurements shown in "Figure 8(b)" indicate n-type con‐ ductivity, confirmed by Hall Effect measurements (RH<0) in "Figure 6(c)". At low tempera‐ ture, α T0.96 (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].

**Figure 6.** Variation of magnetoresistance of the sample with various angles between the magnetic field and the nor‐ mal to the HgTe/CdTe superlattice surface (a). Temperature dependence of the conductivity (b), weak-field Hall coeffi‐

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)

We have also measured the conductivity, Hall mobility and Seebeck effect. At low tempera‐ tures, the sample exhibits n type conductivity (RH < 0) with a concentration n=1/e RH =3.24x1012 cm-2 from "Figure 6 (c)" and in "Figure 7. (a)"; and a Hall mobility μn = 2.5x105

/Vs in "Figure 6 (d)". The plot log(μn)-log(T) in the "Figure 7.(b)" of the Hall mobility shows a scattering of electrons by phonons in the intrinsic regime with μHT1.58 and week acti‐

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

In the insert of "Figure 8(a)" we have plotted the inverse of the minima's 1/Bm as a function

*<sup>n</sup>* <sup>=</sup> *<sup>e</sup> <sup>π</sup>* <sup>ℏ</sup> *<sup>Δ</sup>*( <sup>1</sup>

\* )min =0.16 *<sup>m</sup>*<sup>0</sup> (10)

*<sup>B</sup>* ) (11)

cient (c) and Hall mobility (d) in the investigated HgTe/CdTe superlattice

144 Optoelectronics - Advanced Materials and Devices

**5. Experimental results and discussions**

vation energy at low temperature with μH T0.05.

electrons by the relation:

of the entire n' following the formula:

cm2

(*mh* <sup>1</sup>

at kp=0.14 Å-1, and increase to 0.30 m0 assuming an electronic conduction.

behaviour supported by the observation of SDH oscillations in "Figure 8 (a)".

**Figure 7.** a) Variation with 1/T of weak-field Hall concentration and (b) Variation with T of Hall mobility in the investi‐ gated HgTe/CdTe superlattice

**Figure 8.** a) Variation of transverse magneto-resistance with magnetic field at 4.2 K, (b) Measured thermoelectric power as a function of temperature in the investigated HgTe/CdTe superlattice.

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

$$\alpha = \frac{\mathbb{E}\{\tau k\_{\beta}\}^2 T \text{(s} + 1\text{)}\}}{3eE\_F} \tag{14}$$

In conclusion, the HgTe/CdTe superlattice is a stable alternative for application in infrared

Correlation Between Band Structure and Magneto- Transport Properties in n-type HgTe/CdTe Two-Dimensional...

http://dx.doi.org/10.5772/52101

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Measurements performed by us on others' samples indicate an improvement of quality of

Laboratory of Condensed Matter Physics and Nanomaterials for Renewable Energy Univer‐

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optoelectronic devices than the alloys Hg1-xCdxTe.

the material manifested by higher mobility.

sity Ibn-Zohr 80000 Agadir, Morocco

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**Author details**

Abdelhakim Nafidi

**References**

where the collision time τ Es-(1/2). This permits us to estimate the Fermi energy at E<sup>F</sup> = 91 meV (in "Figure 4(a)"), with s= 2.03 corresponding to electrons diffusion by ionized impurities, in agreement with the calculated EF=88 meV in formula (13). In intrinsic regime for T>150 K, the measure of the slope of the curve RH T3/2 indicates a gap Eg = 98 meV which agree well with calculated Eg (Γ, 300 K) = E1-HH1= 105 meV. Here αT-3/2 indicates electrons scattering by phonons.

This HgTe/CdTe superlattice is a stable alternative for application in far infrared optoelec‐ tronic devices than the random alloys Hg0.99Cd0.01Te because the small composition x=0.01, with Eg (Γ, 300 K) =100 meV given by the empiric formula for Hg1-xCdxTe [16]

$$E\_{\rm g}(\mathbf{x}, \mathbf{T}) = -0.302 + 1.93 \mathbf{x} \cdot \text{0.810x} + 0.832 \mathbf{x}^3 + 5.035 \times 10^4 \text{(1-2x)} \,\mathrm{T} \tag{15}$$

is difficult to obtain with precision while growing the ternary alloys and the transverse ef‐ fective 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:



We reported here remarkable correlations between calculated bands structures and magne‐ to- transport properties in HgTe/CdTe nanostructures superlattices. Our calculations of the specters of energy E(d2), E(kz) and E(kp), 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 d1/d2 = 2.69, when d2 < 100 nm. In our case, d2=3.2 nm and E<sup>g</sup> (Γ, 4.2 K) = 48 meV. In spite of it, the sample exhibits the features typical for the semiconductor n-type con‐ duction 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 semi‐ metallic 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 Hg1-xCdxTe.

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