phys. stat. sol. (b) 236, No. 1, 55–60 (2003) / DOI 10.1002/pssb.200301504

© 2003 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 0370-1972/03/23603-0055 $ 17.50+.50/0

Band offset of quantum wells computed from charge measurements

Dipankar Biswas1, Satyajit Chakrabarti1, Sudipto Dasgupta1, Sudakshina Kundu*,2, and Resmi Datta2 1 Institute of Radio Physics and Electronics, Calcutta University, 92 Acharya Prafulla Chandra Road,

Calcutta-700009, India 2 Department of Electronic Science, Calcutta University, 92 Acharya Prafulla Chandra Road, Calcutta-

700009, India

Received 6 June 2002, revised 23 September 2002, accepted 26 September 2002 Published online 13 February 2003

PACS 73.40.Kp, 73.63.Hs

Band discontinuity is an important parameter for the design of heterojunction based electronic and opto- electronic devices. An alternative method for band offset measurement in quantum wells has been out-lined in this paper. The band offset is determined from a comparison of experimental results and theoreti-cal computations of the temperature dependent total charge content of a quantum well. In spite of the simplicity of the measurement procedures the result at its best has an error around ±2%.

1 Introduction

The heterojunction band offset is an important parameter for the design of heterostructure based electronic and optoelectronic devices. The total band discontinuity, distributed over the conduction and valence bands, depends on the semiconductors and the amount of mismatch strain at the interface. Several electrical and optical methods for the measurement of band offsets are well established [1–3]. In this paper a method for the measurement of band offset of a quantum well has been outlined. It is based on the comparison of the total charge as obtained from experimental (C–V) data and the theoreti-cally computed total charge content of the same quantum well at different temperatures.

2 Theory

In the capacitance–voltage (C–V) measurement technique for finding the carrier concentration N and the corresponding doping concentration N3d in bulk semiconductors, a reverse field is applied through a p–n junction or a Schottky barrier to the semiconductor. It varies the edge of the depletion capacitor formed by the Schottky barrier inside the semiconductor [4]. The carrier concentration N for an ionized donor concentration N3d at a depth x inside the bulk semiconductor corresponding to a reverse bias V,

* Corresponding author: e-mail: skundu@cucc.ernet.in

56 Dipankar Biswas et al.: Band offset of quantum wells computed from charge measurements

including the built-in potential of the Schottky barrier, is related to the measured capacitance by the equation [4]

( )

2

23d

1d

1 1

dC

V N xAε

= ,

where x = εA/C, A is the area of the diode and ε is the di-electric constant of the semiconductor. For the bulk semiconductors the C–V curves fall faster and the 1/C 2–V profiles become steeper with reverse bias, as the carrier concentration N decreases. In the presence of a quantum well inside the bulk semiconductor, the C–V and 1/C 2–V profiles change significantly as shown in Fig. 1 [2]. The effect of accumulation of carriers in the quantum well becomes prominent. The total charge content of the quantum well can be computed from the experimental 1/C 2–V curves.

Experimental samples of In0.24Ga0.76As/GaAs single quantum wells with the necessary buffer layers were grown on n+

⟨100⟩ GaAs substrates by Arora et al. [5] using metalorganic vapour phase epitaxy (MOVPE) at a pressure of 100 Torr and a temperature around 640 °C. The doping concentration was in the range of (2–4) × 1022 m–3. The quantum well was characterized by usual photoluminescence and X-ray measurements. Ohmic contacts of Au–Ge–Ni were alloyed to the back of the substrate and Au Schottky dots were evaporated on the top of the GaAs epilayer [5].

Fig. 1 Capacitance–voltage profiles in bulk semiconductor (dot-ted) and quantum well (continuous). a) Capacitance versus volt-age. b) Inverse of capacitance squared versus voltage.

Fig. 2 Experimental 1/C 2 –V curves at two widely different temperatures whose slopes show carrier concentration in the barrier region and the well.

phys. stat. sol. (b) 236, No. 1 (2003) 57

Experimental 1/C 2–V curves of a strained In0.24Ga0.76As/GaAs quantum well of 80 Å width at two widely different temperatures are shown in Fig. 2. At low reverse bias the 1/C 2–V curve is a straight line with a slope corresponding to the doping concentration in the bulk. With increase in bias the depletion edge penetrates into the semiconductor. As it reaches the depletion region near the barrier adjacent to the well, the curve becomes steeper due to depletion of charge. With further increase in the reverse bias the depletion edge enters the quantum well where the density of accumulated carriers is high. The change of capacitance with voltage decreases, and the slope of the curve becomes flat, particularly at low tempera-tures. The bands bend due to increased fields. The carrier distribution changes inside the well in response to the field and also start spilling continuously [6]. The change of capacitance with voltage increases again when the density of carriers in the quantum well becomes low due to spilling of the carriers from the quantum well which are present in absence of the applied field. Finally the well is emptied. Once the depletion edge crosses the quantum well the curve gets steeper than the bulk as it reaches the depletion region at the other end. This is apparently more pronounced because of the enhanced depletion and stronger band bending at high field. Depletion or accumulation can be confirmed from the 1/C 2–V curve, when the change of slope is either way from the nominal slope of the bulk. Figure 2 shows that the confinement of carriers in the quantum well increases at low temperatures. The total charge content of the quantum well at any temperature is found from the region between the turning points on the 1/C 2–V curves and is directly calculated for the accumulation region of the experi-mental curve as

1

,n

iQ C V= ∆∑ (2)

where ∆V are the small voltage increments and Ci the corresponding capacitances. The band diagram for the n-doped quantum well for theoretical computation is shown in Fig. 3. For the 80 Å wide well used, only the lowest sub-band is populated with two dimensional electron gas as the next higher sub-band is above the well. Theoretically the position of the conduction band with respect to the Fermi level can be obtained from the three dimensional carrier concentration N in the barrier layer using the three dimensional distribution function. The carrier concentration in the barrier layer is

( ) ( )

( )CB

1 23 2

CB2B

FB

d*4 21 expE

E E EN m

E E kTπ

∞ −=

+ − ∫� , (3)

where mB* is the effective mass of electrons in the barrier, ECB and EFB are the conduction band edge and the Fermi level in the barrier, respectively, and N = N3d. Under equilibrium, charge from either side of the barrier flow into the quantum well and the bands in the barrier bend by δVB for proper alignment of the Fermi level across the structure as shown in Fig. 3.

( )FW CW C FB CB BE E E E E V− = ∆ − − − δ , (4)

where ∆EC is the conduction band offset, ECW is the conduction band and EFW the Fermi position in the well and δVB is the bending of the barrier layer. The areal density of the two dimensional electrons in the quantum well is

( )( )

C

0

W2d 2

FW CW

* d

1 exp

E

E

m EN

E E E kTπ

∆

= + − −

∫�

, (5)

where E0 is the first sub-band in the quantum well. The bending δVB in the barrier layer is obtained by solving Poisson’s equation and is given by

2B 3d B 2 ,V qN L εδ = (6)

where LB = N2d/2N3d is the depletion width in the barrier layer. This accounts for electrons coming from both sides of the well.

58 Dipankar Biswas et al.: Band offset of quantum wells computed from charge measurements

The above equations are all interdependent. They are solved using self-consistent iterative method. The total charge is calculated as Q = N2dAq where A is the area of the interface and q is the electronic charge. The dielectric constant ε was assumed to be constant across the interface and the doping N3d in the barrier layer was found directly from the 1/C 2–V curves to be 3.6 × 1022 m–3. The area of the Schott- ky interface was 0.45 × 10–6 m–2 [5]. The effective mass of the electrons in In0.24Ga0.76As was obtained from a linear extrapolation given by Osbourne [7]:

( )0* 0.067 0.44 ,m m x= − (7)

where x is the mole fraction of InAs present in the alloy. The band gaps of the semiconductors forming the interface are temperature dependent. The effects of temperature [8] and mole fraction x have been included in the computation as

( )g 1.52 1.1 2.5 1 3000 .E x x T= − + − (8)

3 Results and Discussion

Figure 4 shows the variation of the carrier concentration in the well at different temperatures. The theo-retical values obtained by iterative method for three different but closely spaced band offset values are compared with the experimental carrier concentration calculated directly from the experimental 1/C 2–V curve at different temperatures [5]. These results show very clearly how the confinement of carriers in the quantum well increases with the decrease of temperature. This is also evident in the primary 1/C 2–V curves where the flat accumulation region increases with decrease in temperature and a larger voltage is necessary to deplete the enhanced accumulated charge. Figure 4 reveals how the carrier confinement increases with increase in the band offset. The conduction band offset that gives the best fit is 62 ± 2% of the total offset, which is close to that obtained by Subramanian et al. [9]. The accuracy of the experimental data depends grossly on the accuracy of the C–V measurement which is less than 2% with almost all modern equipments [3]. The best accuracy is obtained for low temperatures where the assumption of the lowest sub-band cor-responding to an infinite well (for T < 190 K) is highly justified. The values of kT at such temperatures are much lower than the conduction band offset and the wells may be taken to be approximately infinite with almost complete confinement of the carriers. At high temperatures, it may be necessary to incorpo-

Fig. 3 Energy band profile in a symmetric quantum well.

phys. stat. sol. (b) 236, No. 1 (2003) 59

rate the change in the value of the effective mass m* with temperature. However, it may be noted that even in cases where fairly accurate results are necessary this small variation is ignored and the effective mass is usually taken to be constant with temperature. In fact, although the temperature of the sample is widely varied in deep level transient spectroscopic (DLTS) measurements, no temperature correction for m* is incorporated [2, 3]. The temperature dependence of the dielectric constant has not been included in the calculations as it is expected to introduce only an insignificant change in the calculations. The variation of the dielectric constant per degree Kelvin is about 1.2 × 10–4 [10]. This change introduces an error much below that expected otherwise. The error obtained by different workers while measuring band offsets by C–V techniques are as fol-lows. Kroemer [11] gives a value of band offset where the error is of the order of 10% whereas the error is around 6% in the results of Debber et al. [2] and around 7% in the results of Subramanian et al. [9]. All the analyses based on C–V measurements assume that the density of interface states is small, which is obtained through proper growth of the heterostructure. It is aptly suggested by Subramanian et al. [9] that in order to get accurate values of the doping concentration in the barrier, flat carrier profiles on either side away from the interface must be present. When an InGaAs alloy is grown there is a chance of In segregation which is more pronounced in thicker strained layers, namely quantum dots and quantum wires, and is reflected in the Photolumines-cence spectra [12, 13]. No such segregation or deviation from sharp interfaces were revealed in the pho-toluminescence spectra of our sample. A detailed discussion of the different types of discrepancies, artefacts, and errors arising from experi-ments and theories of C–V measurements for determining band offsets is to be presented as a separate communication. We may conclude that a simple method for the determination of band offsets of quantum wells has been developed which requires few low temperature C–V measurements and a simple computer program instead of involved experimental facilities needed for other experimental methods [2, 3]. The doping concen-tration N3d can be determined directly from the C–V profiles. This is an added advantage of the method. The method with its simplicity can be used for routine checkup of band discontinuity of quantum wells.

Acknowledgement The authors are grateful to Prof. B. M. Arora of Tata Institute of Fundamental Research, India, for providing the experimental data and for valuable discussions.

Fig. 4 Total charge content in the quantum well at different temperatures obtained experimentally (circles) and theoretically for different band off-sets.

60 Dipankar Biswas et al.: Band offset of quantum wells computed from charge measurements

References

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