Correlation between band structure and magneto-transport properties in far-infrared detector modulated nanostructures superlattice

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1 M. J. CONDENSED MATTER VOLUME, NUMBER JULY Correlation between band structure and magneto-transort roerties in far-infrared detector modulated nanostructures suerlattice R. Morghi a, A. Nafidi a*, M. Braigue a, H. Chaib a, J. Hemine a, A. Tirbiyine a, B. Er-Raha a, M. Garcia Enrique b and M. d'astuto c a Grou of Condensed Matter Physics, Deartment of Physics, Faculty of Sciences, P.O. Box 6 Dahla, University Ibn Zohr, Agadir, Morocco b Dto. de Física de la Materia Condensada, Universidad Autónoma de Madrid, 49 Madrid, Sain c IMPMC, CNRS UMR759 UPMC, rue de Lourmel, 75 Paris, France Abstract: We reort here carrier s magneto-transort roerties and the band structure results for II-IV semiconductors. HgTe is a zero ga semiconductor when it is sandwiched between CdTe layers to yield to a small ga suerlattice which is the ey of an infrared detector. Our samle, grown by MBE, had a eriod d ( layers) of 8 nm (HgTe) / 4.4 nm (CdTe). Calculations of the sectra of energy E( z ) and E( ), resectively, in the direction of growth and in the lane of the suerlattice were erformed in the enveloe function formalism. The angular deendence of the transverse magnetoresistance follows the two-dimensional (D) behavior with Shubniovde Haas oscillations. At low temerature, the samle exhibits tye conductivity with a hole mobility of 9 cm²/v.s. A reversal the sign of the wea-field Hall coefficient occurs at 5 K with an electron mobility of 4 cm /Vs. In intrinsic regime, the measured 8 mev agrees with calculated (Γ, K) = 4 mev which coincide with the Fermi level energy. The formalism used here redicts that this narrow ga samle is semi metallic, quasi-twodimensional and far-infrared detector. Keywords:Band structure, Enveloe function formalism, Magnetoresistance, Hall effect, suerlattice, Infrared detector, Narrow ga, quasi-two-dimensional system. *Corresonding author: nafidi@yahoo.fr, Phone: , Fax: + 58 I. Introduction The wor of Essai and Tsu in 9 [] caused a big interest in the study of suerlattices made from alternating layers of two semiconductors. The develoment of molecular beam eitaxy (MBE) was successfully alied to fabricate different quantum wells and suerlattices. Among them, III-V suerlattices (Ga -x Al x As-GaAs [-] - tye I), IV-IV (InAs/GaSb [] - tye II) and later II-VI suerlattice ( [4] - tye III). HgTe and CdTe crystallize in zinc blend lattice resectively. The lattice-matching within.% results in a small interdiffusion between HgTe and CdTe layers at low temerature near C by MBE. HgTe is a zero ga semiconductor (due to the inversion of relative ositions of Γ 6 and Γ 8 edges [5]) when it is sandwiched between the wide ga semiconductor CdTe (.6 ev at 4. K) layers yield to a small ga suerlattice which is the ey of an infrared detector. A number of aers have been ublished devoted to the band structure of this system [4] as well as its magnetootical and transort roerties [6]. In this aer we reort the band structure and magnetotransort results in semi metallic suerlattice grown by molecular beam eitaxy. II. Exerimental Techniques The suerlattice was grown by MBE on a [] CdTe substrate at C. The samle ( layers) had a eriod d = d +d where d (HgTe)=8 nm and d (CdTe) = 4.4 nm. It was cut from the eitaxial wafer, had tyical sizes of 5 mm. The ohmic contacts were obtained by chemical deosition of gold from a solution of tetrachloroauric acid in methanol after a roer masing to form the Hall crossbar. Carrier transort roerties were studied in the temerature range.5- K in magnetic field u to 8 Tesla. Conductivity, Hall effect [7], angular deendence of the transverse magnetoresistance and the Hall voltage with resect to the magnetic field were measured. The measurements at wea magnetic fields (u to. T) were erformed in standard cryostat equiment. The measurements of the magnetoresistance were done under a higher magnetic field (u to 8 Tesla). The samle was immersed in a liquid helium bath, in the center of a suerconducting coil. Rotating samles with resect to the magnetic field direction allowed us to study the angular deendence of the magnetoresistance and Hall voltage. The Moroccan Statistical Physical and Condensed Matter Society

2 R. Morghi, A. Nafidi, M. Braigue, H. Chaib, J. Hemine, A. Tirbiyine, B. Er-Raha, M. Garcia Enrique and M. d'astuto III. Theory of electronic bands Calculations of the sectra of energy E( z ) and E( ), resectively, in the direction of growth and in lane of the suerlattice; were erformed in the enveloe function formalism [4,6] with a valence band offset between heavy holes bands edges of HgTe and CdTe of mev determined by the magneto-otical absortion exeriments [8]. The general disersion relation of the light article (electron and light hole) subbands of the suerlattice is given by the exression [4]: () ξ = r and r = ²( ²+ ²) - = E - Λ for H gt e m * H H ²( ²+ ²) - = E for C dt e m * H H m* HH =. m [5] is the effective heavy hole mass in the host materials. The energy E as a function of d, at 4. K, in the center of the first Brillouin zone and for d = 4. d, is shown in Fig.. (5) Where the subscrits and refer, resectively, to HgTe and CdTe. z is the suerlattice wave vector in the direction arallel to the growth axis. The twodimensional wave vector ( x, y ) describes the motion of articles erendicular to z. Here: ξ = r E - ε r = E + ε - Λ E is the energy of the light article in the suerlattice measured from the to of the Γ 8 valence band of bul CdTe, while ε i (i = or ) is the interaction band gas E(Γ 6 ) - E(Γ 8 ) in the bul HgTe and CdTe resectively. At given energy, the two band Kane model [9] gives the wave vector ( i ²+ ²) in each host material: () E(mev) T B E E E HH HH HH h d (Å) HgTe- CdTe d = 4. d T=4,K P² ² ( ² + ²) = (E-Λ) ( E - Λ + ε ) for HgTe, P² ² ( ² + ²) = E ( E -ε ) for CdTe () Figure : Energy osition and width of the conduction (E n ), heavy-hole (HH n ), and the first light-hole (h ) subbands calculated at 4. K in the center Γ of the first Brillouin zone as a function of layer thicness d for suerlattice with d = 4. d, where d and d are the thicnesses of the HgTe and CdTe layers, resectively. T is the oint of the transition semiconductorsemimetal. P is the Kane matrix element given by the relation: P² = ε m* Where m* =. m is the electron cyclotron mass in HgTe [5]. For a given energy E, a suerlattice state exists if the right-hand side of Eq. () lies in the range -π/d [-,] that imlies z π/d in the first Brillion zone. The heavy hole subbands of the suerlattice are given by the same Eq. () with: (4) The case of our samle (d = 44Ǻ) is indicated by the vertical solid line. Here the cross-over of E and HH subbands occurs. d controls the suerlattice band ga Eg = E -HH. For wea d the samle is semiconductor with a strong couling between the HgTe wells. At the oint T(d = 5.7 Ǻ, E= 8.7 mev) the ga goes to zero with the transition semiconductor- semimetal. When d increases, E and h states dros in the energy ga [, Λ] and become interface state with energy E =Λε / ε +ε =4 mev for infinite d. Then the suerlattice has the tendency to become a layer grou of isolated HgTe wells and thus assumes a semimetallic character. The ratio d /d governs the width of suerlattice subbands (i.e. the electron effective mass). A big d /d, as in our case,

3 Correlation between band structure and magneto-transort roerties in moves away the material from the two-dimensional behavior. In Fig. we can see that the band ga (Γ) increases, resents a maximum at mev and decreases when the valence band offset between heavy hole band edges of HgTe and CdTe increase. For each, (Γ) increases with T. Our chosen value of mev is indicated by a vertical dashed line. This offset agrees well with our exerimental results contrary to mev used by Bastard [4] and mev given by Johnson [8]. The later offset give a zero ga whereas, in intrinsic regime, our measured Eg 8 mev agree with calculated (Γ, K) = 4 mev. ( ) (mev) Å / 44 Å - - (m ev) T = K T = 77 K T = 4. K Figure : The band ga (Γ), in the center Γ of the first Brillouin zone, as function of temerature and valence band offset between heavy holes bands edges of HgTe and CdTe for the investigated suerlattice. Fig. shows the sectra of energy E( z ) and E( ), resectively, in the direction of growth and in lane of the suerlattice at 4. K. Along E( ), E and h increase with whereas HH n decreases. This yields to an anti-crossing of HH and h at =.68 Ǻ -. The ga is (Γ,4. K) = mev. E (mev) 9 h HH -,5 -, -,5,,5, E HH HH z = Å / 44 Å = mev T = 4. K -,5 -, -,5,,5, - (Å - ) z (Å - ) E /d HH E F h Figure : Calculated bands along the wave vector z (a) and in lane ( x, y )(b), of the suerlattice at 4. K. E F is the energy of Fermi level. Please note that E F (4. K) =4 mev= E I and then the conduction is assumed by heavy and light holes. As seen in Fig. 4, along z and the light articles (electrons and light holes) subbands are not arabolic whereas the heavy hole subbands are arabolic. E (mev) 9 h z = HH z = -, -, -,,, E = z E =/d z -, -, -,,, - (Å- ) h z =/d HH = z HH z =/d i=; HH z = z = z =/d HH = HH = Figure 4: Calculated bands as a function of suerlattice at 4. K. Å / 44 Å = mev T = 4. K z HH = z (Å- ) and E = h = 9 of the For an anisotroic medium, such as the suerlattices, the effective mass is a tensor and its elements along and directions are given by the following exression []: E * m (6) By carrying out second derivative of the energy E, h and HH along z and in Fig. we calculated the effective mass bands in Fig. 5. Along, the effective mass of heavy holes m* HH = -. m and the effective mass of electrons m* E increases from. m to. m whereas the effective mass of the light holes h deceases from. m by half to a minimum of.74 m. After it increases and diverges at =. Å - assuming an electronic conduction. After it increases to -.4 m at the center Γ of the first Brillouin zone assuming a light hole conduction. m*/m e,,, -, -, E z = -,5 -, -,5,,5, Å / 44 Å = mev T = 4. K HH = z h z = -,5 -, -,5,,5, - (Å - ) z (Å - ) HH = Figure 5: Calculated relative effective mass bands along the wave vector z and in lane of the suerlattice at 4. K. Using the value of ε and ε at 4. K, 77 K and K [] and taing P temerature indeendent, this is h = E =,,, -, -,

4 4 R. Morghi, A. Nafidi, M. Braigue, H. Chaib, J. Hemine, A. Tirbiyine, B. Er-Raha, M. Garcia Enrique and M. d'astuto suorted by the fact that from eq.(4) P ε G (T)/m*(T) cte, we get the temerature deendence of the band ga, in the center Γ of the first Brillouin zone in Fig. 6. ()(mev) K z =, K = () = E - HH T(K) (m) Figure 6: Temerature deendence of the band ga and cut-off wavelength, at the center of the first Brillouin zone, in the investigated suerlattice. Note that increases from.6 mev at 4. K to 4 mev at K. We calculated the detection cut-off wave length by the relation: ( m) E ( mev ) (7) g In the investigated temerature range m m situates our samle as a far infrared detector. IV. Exerimental Results and discussion The transverse magnetoresistance ρ/ρ, in Fig. 7, follows the two-dimensional (D) deendence with manifestation of the beginning of the Shubniov-de Haas oscillations (with a wea [(Δ ρ/ρ ) max =.]). However, over the entire investigated magnetic field range, a non-vanishing magnetoresistance is observed when the field is arallel to the lane. This can be due to the inter-diffusion between HgTe well (great d /d and small d ) and/or to the widening of carriers subbands under the influence of the magnetic field along E( ) while the Hall voltage V H goes to zero at this configuration in Fig. 8. This suggests quasi twodimensional conductivity behavior (between threedimensional (D) and two-dimensional (D). / o T = 4, K H(Tesla) Figure 7: Transverse magnetoresistance of the samle, at various angles between the magnetic field and the normal to the suerlattice surface, at 4. K. V H (mv ) T = 4, K I = m A H(Tesla) Figure 8: Field deendence of Hall voltage, at various angles between the magnetic field and the normal to the suerlattice surface, at 4. K. At low temerature, the samle exhibits tye conductivity with a hole mobility μ 9 cm²/v.s. As seen in Fig. 9, a reversal of the sign of the wea-field Hall coefficient R H occurs at 5 K. It may be attributed to traing of carrier charges in intrinsic state E I. Such a reversal of the sign of the Hall voltage may be inferred by the existence of at least two tyes of carriers, which suggests a semimetallic character of the conduction mechanism. R H (cm²/c) 6 (R H ) max 5 (a) (R H ) sat, (-) (+),4,, (b) 4 / T (K - ) Figure 9: Temerature deendence of the conductivity (a) and wea-field Hall coefficient (b), in the investigated suerlattice.

5 Correlation between band structure and magneto-transort roerties in 5 The Hall constant at wea field in Fig. 9 is described by the formula: we have quasi-two-dimensional conductivity. and semimetallic -nb² μn (R ) = where b = (8) e +nb ² μ H w At low temerature in the saturation regime: ( R H ) sat e (9) Near the intrinsic regime, the Hall constant is given by: E) (mev) Å / 44 Å = mev E F (D) E F (D) E HH ( R ) H max ( b ) e 4b This results in the ratio: () 4( RH ) max ( b ) b () ( R ) b H sat Eq. (8) imlies for RH = : b 4 () n This imlies an electron mobility of μn= 4 cm²/v.s. Such a low value of the electron mobility can be connected, from the one side, with the suerlattice electron effective mass which is much higher than that of the bul material [] and, from the other side, with different tyes of imerfections of the investigated structures, including strong comensation. From Eq. () we have: P² ² F E (E E ) F F () g We extract the Fermi level energy: E E EF P² ² g g F (4) Where the Fermi wave vector is / F(D) = π² (D) = π / and F, resectively for three-dimensional and two-dimensional motion of carriers charges. Here we have chosen the minus sign because the samle is -tye at low temerature. Fig. shows that the band ga and the D Fermi level energy increase with temerature whereas D Fermi level energy and HH i band energy remain constant. In all cases the conductivity is assumed by light and heavy holes. So HH HH T (K) Figure : Temerature deendence of the energy for the threedimensional (D) and two-dimensional (D) Fermi levels in the investigated suerlattice. V. Conclusions The formalism used here redicts that the system is semimetallic when the HgTe to CdTe thicness ratio d /d is greater than 4. In our case, d /d = 4. and (Γ,4. K) = mev corresonding to thermal energy necessary to change the sign of R H (/T). In intrinsic regime, the measurements indicates 8 mev in good agreement with calculated (Γ, K) =4meV and E F (4.K)=E. In site of it, the samle exhibits the features tyical for the semimetallic conduction mechanism, which agree well with the overla between carrier subbands in E( ) with a quasi-twodimensional behavior and is a far-infrared detector. The theoretical and magnetotransort arameters are in good agreement for our samle with narrow ga near the oint T. Note that we had observed a -tye conduction mechanism in the medium-infrared detector, narrow ga and two-dimensional (5.6 nm / nm) suerlattice [-4]. Measurements erformed by us on others samles indicate an imrovement of quality of the material, manifested by higher mobility. VI. References [] L. Esai and R. Tsu, Suerlattice and negative differential conductivity in Semiconductors, IBM J. Res. Develoment, 6-65 (9). [] R. Dingle, A. C. Gossard, and W. Wiegmann, Direct Observation of Suerlattice Formation in a Semiconductor Heterostructure, Phys. Rev. Lett. 4, 7- (975). h

6 6 R. Morghi, A. Nafidi, M. Braigue, H. Chaib, J. Hemine, A. Tirbiyine, B. Er-Raha, M. Garcia Enrique and M. d'astuto [] H. Saai, L. L. Chang, G. A. Sai-Halasz, C. A. Chang, and L. Esai, Two-dimensional electronic structure in InAs-GaSb suerlattices, Solid State Communications 6, (978). [4] G. Bastard, Theoretical investigations of suerlattice band structure in the enveloefunction aroximation, Phys. Rev. B 5, (98). [5] J. Tuchendler, M. Grynberg, Y. Couder, H. Thomé, and R. Le Toullec, Submillimeter Cyclotron Resonance and Related Phenomena in HgTe, Phys. Rev. B 8, (97). [6] Ab. Nafidi, A. El Kaaouachi, H. Sahsah, and Ah. Nafidi, Band structure and magneto-transort in suerlattice, International Conference on Theoretical Physics (HT ), Paris, France -7 July, (). [7] Ab. Nafidi, A. El Kaaouachi, Ah. Nafidi, J.P. Faurie, A. Million, and J. Piaguet, Some Transort Proerties of Suerlattices, Physica Status Solidi (b) 9, (). [8] N. F. Johnson, P. M. Hui, and H. Ehrenreich, Valence-Band-Offset Controversy in Suerlattices. A Possible Resolution, Phys. Rev. Lett. 6, (988). [9] Evan O. Kane, Band structure of indium antimonide, Journal of Physics and Chemistry of Solids, 49-6 (957). [] C. Kittel, Introduction to solid state hysics, rd edition, John Wiley and Sons, Inc, New Yor, (97). [] M. H. Weiler, Semiconductors and Semimetals, edited by R. K. Willardson and A. C. Beer (Academic, New Yor), vol. 6, 9 (98). [] D. L. Smith, T. C. McGill, and J. N. Schulman, Advantages of the HgTe-CdTe suerlattice as an infrared detector material, Al. Phys. Lett. 4, (98). [] Ab. Nafidi, A. EL Abidi, A. El Kaaouachi, and Ah. Nafidi, Electronic Band Structure and New Magneto-transort Proerties in -tye Semiconductor Medium-infrared HgTe / CdTe Suerlattice, AIP Conference Proceedings, Volume 77, - (4). [4] A. Nafidi, A. El Abidi, and A. El Kaaouachi, Seebec and Shubniov-de Haas Effects in a Two-Dimensional -tye Suerlattice, AIP Conference Proceedings, Volume 8, 59- (6).