Influence of conduction via a channel of an impurity δ-layer on themagneto-quantum effects in AlGaAs/GaAs/AlGaAs heterostructures

V.V. Vainberg a,n, A.S. Pylypchuk a, V.N. Poroshin a, O.G. Sarbey a,N.V. Baidus b, A.A. Biryukov b

a Institute of Physics, National Academy of Sciences of Ukraine, 46 pr. Nauki, Kiev 03028, Ukraineb Physical–technical Research Institute, Nizhni Novgorod State University, Nizhni Novgorod 603950, Russia

H I G H L I G H T S

� A quantum well with δ-doping in barrier is a system with parallel conduction channels.� The contribution of electrons in δ -doping well to the conductivity is relatively small.� The different electron mobilities in both channels determine high magnetoresistance.� Intersubband electron transitions between both wells may be neglected.� The existing theory satisfactorily describes magnetoresistivity including quantum effects.

a r t i c l e i n f o

Article history:Received 18 October 2013Received in revised form20 December 2013Accepted 31 January 2014Available online 8 February 2014

Keywords:HeterostructuresQuantum wellSelective dopingMagneto-quantum effects

a b s t r a c t

The lateral magnetoresistance of the multi-layer AlGaAs/GaAs/AlGaAs heterostructures with quantumwells and impurity δ-layers in the adjacent barriers has been studied at 4.2 K. It is shown that both theclassical magnetoresistivity tensor components and the magneto-quantum effects in such structuresconsiderably depend on the contribution to conduction from the channel with small mobility formed inthe impurity δ-layer. The obtained results are analyzed within the frame of the model of electrontransport via two parallel channels with different electron mobilities. Based on this model the electronconcentration in both the structural and δ-layer quantum wells and the dependence of the componentsof the magnetoresistivity tensor, including the magneto-quantum effects, on magnetic field strengthhave been calculated for samples with different doping level and manifest a good agreement betweenexperimental results and calculations.

& 2014 Elsevier B.V. All rights reserved.

1. Introduction

Investigations of magnetotransport in a quasi-two dimensionalquantum well, when the conduction is determined at least by twosize-quantized sub-bands with different electron mobilities, werereported in a number of theoretical and experimental papers (forexample, Refs. [1–5]). A common subject of such investigationswas a deep quantum well formed by heterojunctions of AlGaAs/GaAs or GaAs/InGaAs with moderate selective or extended dopingby shallow impurities in the barrier region. In such structuresalmost all electrons are located in the well region and fill states inseveral sub-bands. In order to control the sub-bands populationthe concentration of electrons in a well could be varied by an

external transverse electric field or illumination. By the way, insuch structures one may obtain very high electron mobilitieswhich are needed for a number of applications [6,7].

However, as is well known [8,9], if the concentration ofimpurities in the δ-layer is high enough the ionized impuritiesform a V-shaped potential well with the depth and number ofsize-quantized levels depending on the impurity concentration. Insuch quantum wells the electron mobility is much less than inquantumwells formed by heterojunctions. If the structural and theδ-layer potential wells are spatially located close to each otherthey form a system of tunnel-coupled wells with common sizequantized sub-bands and the electrons redistribute between bothwells. Further we will denote the level originated in the δ-well asthe δ-well level and that originated in the structural quantumwellas the quantum well level. The built-in field, arisen dueto separated charges of electrons and impurity ions, lowers theδ-well levels relative to the quantum well levels and favors the

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Physica E

http://dx.doi.org/10.1016/j.physe.2014.01.0351386-9477 & 2014 Elsevier B.V. All rights reserved.

n Corresponding author. Tel.: þ38 0445257951.E-mail address: vainberg@iop.kiev.ua (V.V. Vainberg).

Physica E 60 (2014) 31–36

electrons to remain in the δ-well. In this case the electrons occupydifferent energy levels in different quantum wells. This peculiaritydistinguishes such structures from a single quantumwell with twooccupied quantum levels.

The aim of this paper is investigation of the low temperaturelateral magnetotransport in such structures. For this purpose wehave chosen the multi-layer AlGaAs/GaAs/AlGaAs heterostructurescontaining many periods with structural quantum wells and δ-layers of impurity in adjacent barriers which attract now con-siderable interest because of their possibilities in applications [10].

2. Experimental details

Measurements were carried out on samples containing struc-tural quantum wells formed by heterojunctions of AlxGa1–xAs/GaAswith х¼15%. Consequently, the barrier height equals to 118 meV.This value is two times less than in widely used structures withx¼30%. That is, we investigate relatively shallow wells.

Grown by MOVPE on (1 0 0)-plane of semi-insulating GaAssubstrates the structures contain 10 identical periods. Each periodconsists of a 80 A wide quantum well and a Si impurity δ-layerlocated at 100 A distance from a well edge. The distance betweenwells equals to 1000 A. The δ-impurity concentration rangesbetween 4� 1011 cm�2 and 2:5� 1012 cm�2 per period. Theeffective width of an impurity δ-layer is about 20 A. In allheterostructures the background impurity concentration is esti-mated to be about 5� 1015 cm�3.

The samples were formed as mesastructures in the shape ofHall bridges. The distances between the current and the sidepotential contacts are 3.5 and 1.05 mm, respectively, the width ofthe sample being 0.5 mm. The ohmic contacts were prepared bydeposition of Pd/Ge/Au layers in vacuum with a following heatingin the hydrogen atmosphere at 430 C. The current was directedalong ⟨1 1 0⟩.

During measurements of magnetoresistance the samples wereimmersed in liquid helium. The magnetic field strength variedfrom 0 to 4.5 T. Measurements were carried out in the dc regime(10 or 100 mA in dependence on the sample resistance). Thevoltages on the potential contacts were measured with an accu-racy of 10�4 during the magnetic field scan and digitally recordedwith the field strength step of 1 mT. Besides, the temperaturedependences of the conductivity and the Hall coefficient at a weakmagnetic field were measured in the range from 4 to 300 K.

3. Results and discussion

The measurements were carried out on five samples. The Hallelectron concentrations and mobilities were determined from valuesof the Hall coefficient RH measured at magnetic fields less than 0.5 Tand from the resistance in absence of magnetic field. These parametersare listed in Table 1. It is evident from this table that the samples showa large difference of the mobility values. It looks rather surprising.Indeed, in the AlxGa1–xAs/GaAs heterostructures at low temperatures

the electrons are scattered mainly by the ionized impurities. Usuallyone estimates their concentration as being equal to the Hall concen-tration of the electrons. In this case the measured mobility should notchange too much in the investigated samples because their Hallconcentration, nHall ¼ 1=eRHall, does not change by more than 60%,as follows from Table 1.

Figs. 1 and 2 depict the dependencies of the longitudinalmagneto-resistivity and the Hall resistivity vs magnetic fieldstrength for three samples. The results for samples #4 and #5 arequalitatively similar to those of samples #1 and #3, respectively.

Typical for all samples, except of the sample #2, is a negativemagneto-resistance in the range of weak magnetic fields (inset inFig. 1). With growing field magnetoresistance either tends to zero(samples #1 and 4) or changes sign. The positive magneto-resistance averaged over the oscillations may reach unusually highvalues in this case. For example, in the sample #2 the resistanceincreases almost up to five times, while in the structures withelectrons located only in one quantum well it increases at suchfields by several tenths of percent, only. Such a strong increase ofpositive magnetoresistance in the case of the sample #2 simplysuppress small negative magnetoresistance component already ata weak magnetic field.

With these data it is reasonable to suggest that in our samplesthe conduction is resulting from at least two size quantized sub-bands. To elucidate this we have performed the FFT spectrumanalysis of the resistance oscillations in Fig. 1 (eliminating slowchanges of the resistance). As an example two spectra are shown

Table 1Parameters of samples obtained from measurements.

Parameter Sample #

1 2 3 4 5

NHall, 1011 cm�2 5.7 8.9 6.5 5.54 11.5NSH, 1st peak (1011 cm�2) 5.6 7.55 6.4 5.3 7.8NSH, total (1011 cm�2) – 22.05 22.4 – 24.6μHall, cm2/V s 62,300 28,800 10,000 22,800 4600

Fig. 1. Dependences of longitudinal magneto-resistivity vs magnetic field strengthfor samples #1–3. T¼4.2 K. Inset: the initial part of the curve for the sample #1 inthe range 0–1 T. Squares – experiment. Solid lines – calculated fitting curves.

Fig. 2. Hall resistivity vs magnetic field strength dependences for samples #1–3.T¼4.2 K. Squares – experiment. Solid lines – calculated fitting curves.

V.V. Vainberg et al. / Physica E 60 (2014) 31–3632

in Fig. 3. For the sample #1 it has only one peak corresponding toconcentration of 5:6� 1011 cm�2. For the sample #2 it consists oftwo well-expressed peaks with frequencies corresponding toconcentrations of 7:55� 1011 cm�2 and 1:45� 1012 cm�2. Thespectrum for sample #3 also shows two peaks corresponding to6:4� 1011 cm�2 and 1:6� 1012 cm�2.

The sums of concentrations corresponding to both peaksobtained by FFT analysis are listed in Table 1 together with theHall ones. It is seen that the concentration values determined bythese two methods differ from each other. Also the concentrationof electrons among samples differs up to four times. This differ-ence is one of the reasons for the divergence in the mobilities aswas mentioned above. Further, we suppose that the imperfectionof the hetero-structure interface leading to roughness scatteringmay play a remarkable role especially in samples #3 and 5.

The temperature dependence of the Hall coefficient and resis-tivity for samples #1–3 are shown in Figs. 4 and 5. The qualitativedifference in the temperature dependencies of the Hall coefficientof the samples (in particular, #1 and #2) correlates with differencein magnetoresistivity behavior.

The analysis of the field dependence of the magnetoresistivityallows us to determine some characteristics of the magneto-transport properties of the investigated samples. Generally thisdependence is determined by four different contributions: theweak localization, electron–electron interference, Shubnikov-deHaas effect and inclusion of two sub-bands in the conductivity.

We begin with the negative magneto-resistance. As is wellknown [11,12], in two-dimensional systems at low temperatures itis caused by the destruction of weak localization in the magnetic

field. To describe this phenomenon we used the consistent theoryof the correction to the conductivity δτj due to weak localizationsummarized and further developed in [13,14] (without taking intoaccount spin-related effects). The most important parameter of thetheory, the phase breaking time τφ, was evaluated according to thetheory formulated in [15].

The theory is not applicable if the condition kl ¼ 2πℏτn=m ⪢1is not fulfilled, where k is the wave number of electrons, l is thefree path length, τ is the transport relaxation time, n is theconcentration of electrons. For our samples it may be violatedfor electrons in the δ-layer well in the samples 3 and 5 only.However, the contribution of these electrons to the weak localiza-tion correction (and Shubnikov-de Haas oscillations) is negligible.

The second condition distinguishes between diffusive andballistic regimes of weak localization. In the diffusive regime,kTτ=ℏ51, one has to use the formula

1τφ

¼ 1þ 3F2

ð1þFÞð1þ2FÞ

" #kTℏkl

ln ½klð1þFÞ�

þπ

4ðkTÞ2ℏEF

1þ 3F2

ð1þFÞ2

!ln

EFτℏ

� �ð1Þ

In the opposite case of the ballistic regime (for the sample #1)the corresponding formula is

1τφ

¼ þπ

4ðkTÞ2ℏEF

ln2EFkT

� �þ 3F2

ð1þFÞ2ln

EFkT

ffiffiffiffiffiffibF

p� �

ð2Þ

were F and b are some functions of the product of wave numbersand Bohr radius of electrons defined in [15]. Note that forapproximation of experimental results we could use the calculatedphase breaking times without considerable corrections, while weused expressions (1) and (2) taking into account transport relaxa-tion times determined for each studied sample.

It turned out that for better agreement between the experi-mental and calculated data it is necessary to take into account theelectron–electron interference. The theory and correspondingformulae are given in [16] which imply assumption of thespherical shape for the Fermi surface that is well established forthe n-type III–V semiconductor compound. We used them in theform

Δsee ¼e2

πℏkTτℏ

1�38f

� �� 12π

lnEFkT

� ��

þkTτℏ

1�38t

� �2F1þF

� 32π

1�1Flnð1þFÞ

� �ln

EFkT

� ��ð3Þ

were f and t are functions of Tτ as defined in [16].To take proper account of two sub-bands conductivity we used

classical expressions for the magnetoconductivity tensor in the

Fig. 3. FFT spectra for samples #1 and #2. The first peak is due to a restrictedmeasurement range of magnetic field strength.

Fig. 4. Temperature dependences of the zero-field resistivity of the samples #1–3.

Fig. 5. Temperature dependences of the Hall coefficient in weak magnetic field forthe samples #1,.2,.3.

V.V. Vainberg et al. / Physica E 60 (2014) 31–36 33

case of a two-level system. However, we introduce the correctionsfor the weak localization, Shubnikov-de Haas effect and electron–electron interaction in the following way [17,18]

sxx ¼e2

m∑j

njτ0j

1þðωτ1jÞ2þΔseej

" #ð4Þ

sxy ¼∑j

ωτ1jðsxx�seejÞ�e2

mnj

ω

δnj

nj

� �ð5Þ

where τ0j, and τ1j are the renormalized relaxation times accountingfor weak localization and Shubnikov-de Haas oscillations, respec-tively.

τ0j ¼τj

1þðδτj=τjÞ; τ1j ¼

τ0j1þðδnj=njÞ

; ð6Þ

τj – the classical transport relaxation time, δτj denotes the correc-tion due to weak localization, Δsее is the correction due toelectron–electron interference being independent of magneticfield, ω is the cyclotron frequency and

dnj

nj¼∑

j

2zsinhðzÞexp � π

ωτqj

� �cos

2πEFℏω

�π

� �ð7Þ

where z¼ 2π2kT=ℏω and qj denotes the quantum relaxation time.

Summation in Eqs. (4),(5) and (7) is done over two subbands.These formulae take into account that the electron–electron

interference introduces a correction to the longitudinal conductiv-ity only and does not introduce it into the transverse component.Further, we suppose that due to the spatial separation between theelectrons in the different sub-bands intersubband transitions areconsiderably less probable than intrasubband transitions andtherefore we neglect them.

Finally, the magnetoresistance components are related with sxx

and sxy as usual by

ρxx ¼sxx

s2xxþs2

xy; ρxy ¼

sxy

s2xxþs2

xyð8Þ

Our fitting program involves four fitting parameters, n1, n2, μ1,and μ2, to describe the non-oscillating part of the magnetoresis-tance and two additional parameters τqj for the dependence of theoscillation amplitudes on the magnetic field strength. The correc-tion due to electron–electron interference in each subband, Δseej,is calculated using Eq. (3) and fitting parameters n1, n2, μ1, and μ2.The Fermi energy regarding the subband bottom in each layer of astructure is calculated using values of n1 and n2, taking intoaccount well expressed spatial separation of electron density inboth subbands. The values of μ1 and μ2 are used to calculaterelaxation times. With these parameters we have to describe themagnetic field dependence of resistivity and Hall coefficient aswell as the FFT spectra of the samples.

The results of the numerical calculations are illustrated inFigs. 1 and 2. The fitting parameters (n1, n2, μ1, μ2, τq1) andcalculated parameters described above (τφ1, τφ2, Δsee1, Δsee2) arelisted in Table 2. It is seen that the measured and calculateddependences of magnetoresistance agree quite satisfactorily. Hereit is worth to emphasize once more that we got such agreementneglecting intersubband transitions. So, it means that thesetransitions do not play an essential role for the conductivity inthe studied samples. We connect this with the fact that electrondensities in our structures in both subbands are substantiallyseparated in real space in contrast to the case of a single-wellstructure with two filled subbands, i.e., the case when twosubbands are not spatially separated (see, for instance, [19]).

It should be noticed that in the region of strong magnetic fieldsthe calculated results for the Hall coefficient (Fig. 2) do not agreewell with experimental curves. Especially, the absence of

oscillations in the calculated curve for the sample #1 is striking.As suggested in [17] (see the discussion there), if ωτ1⪢1 the twoterms connected with the oscillation in formula (5) almost canceleach other (in the region, where oscillations in the Hall resistivityare visible frommeasurements for the sample #1, ωτ1 equals aboutto 15). A further cancellation of large terms takes place when ρxy isdetermined by inversion. The observed oscillations in this caseprobably are connected with localized states between the Landaulevels, as pointed out in [20]. It is worth to note that the presenceof the second channel of the conductivity removes the cancellationin the formula (8) and the calculated dependence shows smalloscillations again. This is the case for samples #2 and #3.

As is seen from Table 2, the electron mobility μ2 has asurprisingly low value – several hundred cm2/V s. The electronmobility in delta-wells was measured in a lot of works [9–21] andits value usually is about several thousand cm2/V s. We assumethat this discrepancy is caused by the following circumstance.In the case of delta-doped semiconductor the electron concentra-tion in a delta well equals to the concentration of positive charges.In our system the electron concentration in the δ-well is much lessthan that of positive charges. This circumstance causes weakerscreening of the charged impurities participating in scattering ofelectrons and, consequently, leads to considerably lower values ofmobility. The very low mobility is responsible for the smallcontribution of the electrons in the δ-layer well in the conductivityof the samples # 2, 3 and 5 (5–10%). Because of this one shouldemphasize particularly their role in the magnetoresistivity.

The proposed interpretation of the field dependence of themagnetoresistance is qualitatively confirmed by calculations of theenergy spectrum in our structures. These calculations were carriedout for one period of a structure by self-consistent solution of theone-dimensional Schroedinger and Poisson equations with cyclicboundary conditions. The procedure of calculations was describedin [22]. The impurity concentration is assumed to be equal to thesum of concentrations n1þn2. The calculated spectra for samples#1 and #3 are shown in Fig. 6. It is seen that in the samples withthe lowest electron concentration the Fermi level lies below thesecond size-quantized level and the electron density is located inthe region of the structural quantum well. For the other samplethe Fermi level is above the second level, and the electronconcentration in the delta-well is already higher than in thestructural well.

These results may be used for a qualitative interpretation of thetemperature dependencies of the Hall coefficient. As can be seenin Fig. 6, the Fermi level of the electrons in the sample #1 liesabout 20 meV above the quantum well level E1 and about 2 meVbelow the δ-well level E2. Therefore, with increasing the tempera-ture from 4.2 K the electrons have to transfer into the delta-well.As their mobility in this well is much less, the Hall concentrationhas to decrease. This decrease continues until the conditionkT⪢E1�E2 is fulfilled. The further behavior of the Hall

Table 2Parameters of samples obtained by fitting and calculations.

Parameter Sample #

1 2 3

n1, 1011 cm�2 5.37 7.55 6.5m1, cm2/V s 65,000 32,000 9700n2, 1011 cm�2 0.145 15.8 11m2, cm2/V s 1000 750 700τq1, s 2.7�10�13 2.5�10�13 1.05�10�13

τφ1, s 2.21�10�11 3�10�11 1.09�10�11

τφ2, s 1.4�10�12 5.92�10�12 –

Δsee1, 1⧸Ohm 5.8�10�6 �1.16�10�5 �2.37�10�5

Δsee2, 1⧸Ohm �1.16�10�6 �3.63�10�5 �3.62�10�5

V.V. Vainberg et al. / Physica E 60 (2014) 31–3634

concentration (and conductivity) is mainly determined by thetemperature dependence of mobility.

Contrary to this in the sample #3 the Fermi level is positionedabout 20–30 meV above the quantum levels E1 and E2. WhilekT⪡EF�E1; EF�E2, the position of the Fermi level is practicallyindependent of temperature and the Hall concentration does notchange with rising temperature. When kT becomes of the order ofE2�E1 the electrons begin transfer from the lower subband to theupper one. In the upper band of this sample the electron mobilityis much higher and the Hall concentration grows. This growth hasto continue until the mobilities in both subbands become compar-able. However, such temperatures are beyond the limits of ourmeasurements.

In conclusion one should notice that in the large interval ofpulsed magnetic fields (up to 50 T) a single-well GaA/GaInAS/GaAlAs structure with partly filled states in the delta layer ofimpurity in a barrier was investigated in detail in [3]. In this paperthe concentration of electrons in the sample was changed up to afactor of 4 by ionization of DX centers. At that the total concentra-tion of Si atoms was essentially higher than in our samples. Theexperimental results as well for the mobility of electrons in thedelta layer and the quantum well as for the quantum relaxationtime agree in both investigations very well. However, in ourinvestigated field region the influence of conductivity in the deltalayer is much more pronounced. Besides, the interference phe-nomena and influence of conductivity of the delta layer on the Hallresistance were not treated in [3].

4. Conclusion

Low temperature magnetotransport properties of the hetero-structures with quantumwells and δ-like doping layers in adjacentbarriers are investigated, when the conductivity is determined bytwo spatially separated subbands. The electron mobility in thedelta-wells of such systems is unusually small. It is due to adecrease in screening of the ionized impurity scattering potentialbecause of partial transfer of electrons from a delta-well into astructural one. The considerable difference between the electron

mobilities in both subbands causes a very large positive magne-toresistance. The existing theory, including the weak localization,electron–electron interference and oscillation phenomena, allowsto describe the magnetoconductivity in these structures with twochannels of conductivity quite satisfactorily. The obtained resultsare useful in the case of consideration of multi-layer heterostruc-tures which are perspective for modern applications, for examplein optoelectronics.

Acknowledgment

The work was supported by the State Scientific & Technicalprogram “Nanotechnology and Nanomaterials” for 2010–2014 years(Project #1.1.7.18/13 –H-18).

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