Physica E 6 (2000) 522–525
Probing the electronic density of states in semiconductor quantum wires using nonequilibrium acoustic phonons P. Hawker, I.A. Pentland, A.J. Kent ∗ , A.J. Naylor, M. Henini School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, UK
Abstract We have used nonequilibrium phonon-induced conductivity (phonoconductivity) measurements to probe the electronic states in semiconductor quantum wire devices. The devices were based on high mobility two-dimensional electron systems (2DESs) in GaAs=Al0:3 Ga0:7 As heterostructures and quantum wires formed using the well-known split-gate technique. Short (20 ns-long) pulses of nonequilibrium acoustic phonons were generated by heating a metal lm on the back surface of the substrate. These phonons propagated ballistically across the substrate and were incident on the quantum wire. The electron– phonon interaction was detected via the phonon-induced change in electrical conductance of the device. We observed giant oscillations of the phonoconductivity with increasing (negative) gate bias. Maxima occurred when the Fermi energy was coincident with the bottom of any one-dimensional electronic subband. In this paper we argue that the phonoconductance is due to phonon-induced backscattering of the electrons in the quantum wire and present evidence that the strength of the phonon signal is proportional to the density of electronic states in the quantum wire. ? 2000 Elsevier Science B.V. All rights reserved. PACS: 73.20.Dx; 72.10.Di Keywords: Quantum wires; Electrons; Phonons; Gallium arsenide
1. Introduction Much of the scienti c and technological interest in quantum wires stems from the particular nature of the one-dimensional (1D) electronic density of states, D(E), which, in an ideal wire, is singular at the 1D band edge. This may result in lower threshold currents and better spectral characteristics in quan∗ Corresponding author. Tel.: +44-115-9515143; fax: +44115-9515180. E-mail address: [email protected]
tum wire laser structures. Because the strength of the electron–phonon interaction is proportional to D(E), electron–phonon scattering measurements should be a more direct probe of D(E) in quantum wires than are conventional electronic transport measurements. The latter depend on the integral of D(E) and are also very sensitive to disorder related eects such as weak localization. In an earlier paper  we reported phonon-induced conductance measurements on a split-gate short quantum wire (point contact). A negative correction to the conductance was measured and this was
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P. Hawker et al. = Physica E 6 (2000) 522–525
attributed to backscattering of electrons. Giant oscillations in the phonoconductivity were observed, with the peaks occurring close to the steps in DC conductance, i.e. when the Fermi energy (EF ) was coincident with the bottom of a 1D subband. At that time, theoretical calculations of the phonoconductivity [2,3] predicted that the response due to direct backscattering of electrons in the short channel would be one or two orders of magnitude smaller than measured. It was therefore concluded that the conductance changes had to be due to heating of the electrons in the 2D contacts by the absorption of nonequilibrium phonons. If this explanation is correct then heating the electrons by a dierent means, e.g. warming up the entire sample, should have the same eect. However, more recent measurements of the equilibrium temperature dependence of the conductance  gave very dierent results to the nonequilibrium phonon measurement. Apparently, this rules out the heating argument despite the fact that it seemed to account for the magnitude of the phonoconductivity signal. New theoretical calculations of the electron–phonon interaction in a 2DES including the eects of acoustic anisotropy on the coupling matrix elements have recently been reported . These predicted a large enhancement of the interaction for particular phonon modes and directions of propagation when compared to earlier theories which do not include anisotropy. For, example, the coupling to transverse acoustic (TA) modes propagating close to [0 0 1] is at least an order of magnitude stronger than originally thought and this was shown to be consistent with the results of phonon emission measurements. It is reasonable to expect that acoustic anisotropy will have the same eect on the electron–phonon interaction in quantum wire devices and so could account for the magnitude of the phonoconductivity . If, on the basis of the above arguments, we can now attribute the negative phonoconductivity to direct backscattering of the 1D electrons by the incident phonons, then we should expect that the amplitude of the response should follow D(E) at EF . In this paper we present experimental measurements of the phonoconductivity in an applied magnetic eld that support this new interpretation.
2. Experimental details The samples were based on an GaAs=Al0:3 Ga0:7 As heterojunction, grown by MBE on a 380 m-thick semi-insulating GaAs substrate. The 2DES carrier density was 4:4 × 1015 m−2 and the mobility 100 m2 V−1 s−1 . A TiAu split gate structure was fabricated on top of the heterostructure by electron-beam lithography. The quantum wire channel was de ned by 0.4 m-wide gap in the gate structure. Two channel lengths were used: 0.2 and 2 m. On the back face of the substrate, directly opposite the channel, a 100 m × 10 m CuNi heater was made. The sample was mounted in a liquid helium cryostat at T = 1:3 K and the drain–source contacts connected via a low-capacitance (14 pF) coaxial line to a high-input impedance, wide-bandwidth preampli er at room temperature. The preampli er output was fed to a high-speed digitiser and signal averager. Nonequilibrium phonon pulses were generated by applying 20 ns-long pulses of up to 5 V amplitude to the heater. The phonon pulse had an approximately Planckian energy spectrum characterised by the temperature of the heater, Th (=28 K for the maximum 5 V excitation pulse). These phonons propagated ballistically in the GaAs substrate and were incident on the quantum wire which was biased with a current of 100 nA. The transient phonoconductivity was detected via the change in voltage drop across the device. It was necessary to average over about 106 pulses to achieve an adequate signal-noise ratio. To minimise the eect of electrical pickup of the heater excitation pulse, the phonoconductivity was measured in a 50 ns-wide gate window starting 90 ns after the pulse, 90 ns being the minimum time of ight of phonons across the substrate. Fig. 1 shows a typical example of the amplitude of the phonoconductivity response as a function of the (negative) gate bias for a 0.2 m-long wire. The amplitude has been corrected for the RC time constant of the device, coaxial line and pre-ampli er input, and expressed as a change in conductance as a fraction of the conductance quantum G0 = 2e2 =h. Also shown in Fig. 1 is the DC conductance of the device as a function of the gate voltage.
P. Hawker et al. = Physica E 6 (2000) 522–525
Fig. 1. Phonoconductivity of a short quantum wire (point-contact) as a function of split-gate bias. Also shown is a measurement of the DC conductance of the wire.
Fig. 2. Dierence between the DC conductivity measurements at 1.3 and 4.2 K to show the eects of heating.
3. Results and discussion In Fig. 1 it is seen that the phonoconductivity peaks near a step in the DC conductance (when the steps are resolvable), i.e. when EF is close to the edge of a 1D subband. This is where the density of states at EF is a maximum. Compare this with Fig. 2 which shows the dierence between the DC conductance at 1.3 K and 4.2 K, i.e. the eect of warming up the sample, and it is clear that the eect of the nonequilibrium phonons is very dierent. Firstly, the conductance oscillations due to heating pass through zero at steps in the DC conductance. Secondly, the phonoconductance is always negative whereas the heating causes both positive and negative conductance changes.
Fig. 3. Phonoconductivity of 2 m-long wire.
Fig. 3 shows the phonoconductivity of the 2 m-long wire. Again, strong peaks are present, but in this case the DC conductance steps were not seen and warming the sample slightly had no discernable eect on the conductance. The absence of DC conductance steps is probably due to disorder in the longer wire. However, it seems that the phonoconductivity measurement is still able to reveal the nature of the 1D density of states. It is interesting to note the splitting of the peaks in Fig. 3, this is possibly due to
uctuations in width along the channel. In , the change in conductance, G, of a 1D channel of length Ly due to phonon scattering was calculated: Ly m∗ C X 1 G √ =− G0 (2)2 ˜2 MN EF − EN Z Z dqx dqz q N (q) | Z(qz )|2 |XMN (qx )|2 p : × EF − EM + ˜sq Here q ia the phonon wave vector, s the phonon speed, C is the electron–phonon coupling constant, = +1 for deformation potential coupling and −1 for piezoelectric coupling and F(q) is the Bose distribution function for the phonons. The form-factors Z(qz ) and XMN (qx ) arise from the electron con nement in the heteojunction growth direction and con nement by the gate potential, respectively. EN and EM are, respectively, the energies of the N th and M th 1D subbands, for intrasubband scattering N = M . All the other symbols have their usual meanings. This expression is able to explain the main features of the observed signals: (i) that G is negative, (ii) that G is larger for the longer wire, and (iii) that G is strongly peaked at EF ≈ EN . The latter is due to the term 1=(EF − EN )1=2
P. Hawker et al. = Physica E 6 (2000) 522–525
measuring the relative amplitudes of the N = 0 peak we can determine the ratio m∗ (B)=m∗ (0), see inset to Fig. 4, the solid line is given by m∗ (B)=m∗ (0) = !2 =!02 , with !0 = 2:4 × 1012 s−1 . This value of !0 implies a characteristic width of lw = (˜=m∗ !0 )1=2 = 26 nm which seems entirely reasonable. 4. Conclusions
Fig. 4. Phonoconductivity measurements at dierent values of applied magnetic eld. The inset shows m∗ (B)=m∗ (0) deduced from the magnetic eld dependence of the amplitude of the N = 0 peak.
which re ects the singularities in the ideal 1D density of states. Further evidence that the phonoconductivity measurement is sensitive to the 1D density of states is provided by the results obtained in a magnetic eld, B = BZ . Fig. 4 shows the phonoconductivity oscillations as a function of gate voltage at BZ = 0; 0:34 and 0.67 T (note that these curves have not been corrected for the RC time constant of the system). Application of the magnetic eld has the eect of increasing the amplitude of the oscillations well as broadening and increasing the separation of the peaks. The strong positive phonoconductivity peak at Vgate ≈ −1:9 V is believed to be due to phonon-activated conduction when the channel is just pinched-o. It is well known that the application of a magnetic eld perpendicular to the wire axis leads to the formation of hybridized electric-magnetic subbands. Assuming the gate con nement potential is parabolic in pro le and the characteristic harmonic oscillator frequency is !0 , the characteristic frequency of the hybridized state ! = (!02 + !c2 )1=2 ¿ !0 , where !c is the cyclotron frequency. The energy of the N th subband is given by EN (k) = EN (B) + ˜2 k 2 =2m∗ (B), where the magnetic eective mass m∗ (B) = m∗ (0)!2 =!02 accounts for a attening of the subband dispersion and an increased density of states at the subband edge. In the above formula for G=G0 , we see that the phonoconductivity is proportional to m∗ , so by
We have described phonoconductivity measurements on quantum wire samples in zero and applied magnetic eld. The conductance changes produced by incident nonequilibrium phonon pulses are dierent to those caused by warming up the sample. The phonoconductivity is attributed to direct phonon-induced backscattering of electrons and the main features of the experimental measurement are in agreement with theoretical predictions. We have shown that the phonoconductivity measurement is able to reveal the form of the density of 1D electronic states in quantum wires, even in samples where the DC conductance
uctuations are smeared out by disorder. Acknowledgements The authors would like to acknowledge the Engineering and Physical Sciences Research Council of the UK for its nancial support for this work under grant no. GR=K55561. We also thank Dr M Blencowe, Professor A Shik and Dr D Lehmann for very helpful discussions. References  A.J. Naylor, A.J. Kent, I.A. Pentland, P. Hawker, M. Henini, Phys. Low-Dim. Strict. 1=2 (1998) 167.  M. Blencowe, A. Shik, Phys. Rev. B 54 (1996) 13 899.  M. Blencowe, in: P. Serena, N. Garcia (Eds.), Nanowires, Kluwer, Dordrecht, 1997.  A.J. Naylor, Ph.D. Thesis, University of Nottingham, 1998.  D. Lehmann, Cz. Jasiukiewicz, A.J. Kent, Physica B 249 –251 (1998) 718.  D. Lehmann, Cz. Jasiukiewicz, A.J. Kent, Physica E, these Proceedings.