AlInAs semiconductor superlattices

Superlattices and Microstructures, Vol. 23, No. 6, 1998 Article No. sm960247

Millimetre-wave negative differential conductance in GaInAs/AlInAs semiconductor superlattices C. M INOT, N. S AHRI , H. L E P ERSON , J. F. PALMIER , J. C. H ARMAND , J. P. M EDUS , J. C. E SNAULT France Telecom, CNET/PAB, Laboratoire de Bagneux 196, avenue Henri Ravera, BP 107, 92225 Bagneux Cedex, France (Received 15 July 1996) We report microwave and temporal measurements on nonlinear transport in GaInAs/AlInAs superlattice n+ n− n+ diodes. The superlattices have wide minibands, wider than 80 meV. The microwave experiments are performed for 0–65 GHz. The diodes exhibit reflection gain up to 65 GHz over a bias-dependent Hakki resonance. The temporal experiments use electro-optic sampling to push the measurement range beyond 200 GHz. Harmonics of the fundamental Hakki resonance are observed up to order three at 150 GHz. A simple admittance model for negative differential conductance devices accounts for the results in both experimental approaches. They are in agreement with miniband transport and demonstrate superlattice negative differential conductance far in the millimetre-wave domain. c 1998 Academic Press Limited

Key words: negative differential conductance, GaInAs/AlInAs superlattices, nonlinear transport, millimetre waves, electro-optic sampling.

1. Introduction The nonlinear transport properties of semiconductor superlattices [1] have recently been investigated both experimentally [2–10] and theoretically [11–16]. In particular, electron negative differential velocity (NDV) has been proved in superlattice unipolar diodes whose static characteristics exhibit negative differential conductance (NDC) [6, 7, 9]. The effect is adequately described by miniband transport models relying upon the Boltzmann equation or balance equations [13, 16]. Nevertheless, the microscopic mechanism of NDV is not understood in detail because precise knowledge of the involved electronic interactions is not available. Important missing parameters are the ratio of elastic and inelastic collisions, and the amount of electronic thermalization due to interelectronic interactions. Some assessments obtained from different theoretical models show that the dynamics of transport should be very sensitive to the magnitude of the various collision rates at moderately high frequency [14, 15]. Dynamic experiments could thus give a deeper insight into the interaction mechanisms responsible for NDV, as well as into the frequency limitations of NDC. Microwave experiments have been performed on a variety of superlattice diodes of different miniband widths and doping levels [2, 4, 6, 7, 9]. Hakki resonances [17] of the conductance are observed at the fundamental frequency, up to the millimetre waves in the widest minibands [6, 9]. However, the resonance behaviour is often 0749–6036/98/061323 + 10

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Superlattices and Microstructures, Vol. 23, No. 6, 1998 Table 1: Structural parameters of the samples. The miniband width is obtained from an envelope function calculation, taking nonparabolicity into account according to the Kane model.

Sample A B C

Well (Å) (aimed) 45 45 60

Barrier (Å) (aimed) 20 20 20

Period (Å) (X-rays) 63 64 75.5

Thickness (µm) 1 1 0.71

Doping (cm−3 ) ∼5 × 1015 ∼1 × 1016 1.2 × 1016

Miniband (meV) 85 85 86

obscured by parasitic elements or insufficient bandwidth. In this paper, we present and discuss millimetrewave measurements on GaInAs/AlInAs superlattice diodes. The distinct voltage dependence of the Hakki resonance enables us to extract the electron transit time through the superlattice as a function of applied voltage, and the electron velocity law. We also report on experiments in the temporal domain based on an electro-optic sampling technique, which extends the measurement range beyond 200 GHz with a reduced accuracy. A Fourier transform to the frequency domain shows harmonics of the fundamental Hakki resonance up to 150 GHz.

2. Samples and experimental set-up The samples are grown by molecular beam epitaxy on Fe doped semi-insulating InP substrates. All the layers are lattice-matched to InP. The superlattices are sandwiched between n+ Si-doped (1019 cm−3 ) contact layers, which consist of a GaInAs layer and a gradual AlInAs/GaInAs alloy layer to smooth away the conduction band discontinuities. Two superlattices (samples A and B) are 1 µm thick (154 periods) and have identical well and barrier thicknesses and compositions. They only differ by the n-type Si doping level Nd : residual ∼5 × 1015 cm−3 and ∼1016 cm−3 , respectively. Sample C is designed with thicker wells, smaller overall thickness (0.71 µm, 94 periods) and slightly higher doping: Nd = 1.2 × 1016 cm−3 . The barrier height is also smaller since the barrier material is a 50% quaternary alloy of lattice matched GaInAs and AlInAs. Table 1 sums up the characterizations provided by X-ray diffraction and secondary ion mass spectroscopy. The samples are processed in a passivated mesa technology [9]. The upper contact metal (AuGeNi) is deposited at first. The active region of the device is delimited by dry-etching the main mesa. Wet etching of the lower contact layer isolates the devices electrically. Then a Si3 N4 passivation layer is deposited before the final AuGeNi and TiAu interconnection. All steps use conventional optical lithography, but require very careful alignment to obtain very small area devices. In the following, the sample area is either 500 µm2 (sample A) or 200 µm2 (samples B and C). The layout of the metallic access lines provides two distinct patterns: 50  coplanar waveguides for onwafer measurements using wide band Microtech microwave probes and a Wiltron 360B network analyser, and striplines for experiments in the temporal domain on cleaved chips. The latter are mounted inside a microwave housing on inverted microstrip lines deposited on a fused-silica substrate. The distance between the striplines and the ground plane is fixed by calibrated silica blocks. The sample is in close proximity to a photoconductive switch made of either intrinsic GaAs or low-temperature (LT) grown GaAs (see the inset in Fig. 5). Both devices are glued with conductive epoxy to the metallic lines and can be biased independently. The photoconductor is excited by a train of 80 fs light pulses emitted by a mode-locked Ti:sapphire laser, and generates ultrashort electrical pulses which propagate up to the supperlattice diode as fast polarization transients. The transmitted electrical signal is measured by electro-optic sampling in a LiTaO3 crystal laid on the ground plane below the stripline and just behind the sample [18]. The time resolution of the measurement system has been characterized in a simpler configuration: the superlattice diode has been removed and the electro-optic crystal laid close to the photoconductive switch. The LT-GaAs switch produces short symmetric pulses whose measured duration is 2 ps. The intrinsic GaAs switch generates asymmetric pulses with a very

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Table 2: Critical voltage Uc and current Ic for the onset of NDC in static characteristics. The peak velocity V p is derived from the critical current using Ic = eNd AV p , or calculated from the Boltzmann transport equation.

Sample A B C

Uc (V) 1.3 0.6

Ic (mA) ∼30 20.1 16.6

s−1 )

V p (cm 7.5 × 106 6.3 × 106 5.2 × 106

V p (cm s−1 ) (calculated) 8 × 106 8 × 106 5.4 × 106

long fall time. Their measured rise time is 1.8 ps, and they can be considered as steps on the timescale of the present experiments. The overall bandwidth of the system results from both the finite duration of the electrical transients and the finite resolution of the electro-optic sampling technique. In particular, the electro-optic crystal is a dielectric resonator at 200 GHz, which causes parasitic electromagnetic reflections. However, the latter are efficiently removed from the measured temporal responses by adequate filtering. When this is done, the 3 dB bandwidth is estimated at 230 GHz [10].

3. Static characterizations The static current voltage characteristics of samples B and C show current saturation or weak NDC above a critical voltage Uc . In sample A, the doping level is too low to turn NDV into NDC, but NDV is clearly revealed by the concavity of the characteristics. The series resistance Rs , mainly due to contact resistances, hinders a straightforward proportionality between the voltage and the field. Measurements on short-circuited devices give Rs ∼ 4 in the 200 µm2 samples, so that the voltage drop over the superlattice is not very different from the applied voltage. Rs is somewhat larger (8–12 ) in the 500 µm2 samples. Table 2 gives the raw values of Uc and Ic , as well as estimates of the peak velocity V p according to the relation: Vp =

Ic eNd A

(1)

where A is the sample area. It is not possible to evaluate the critical field directly from the measured characteristics. Table 2 also reports theoretical values of V p computed from the Boltzmann transport equation [9, 16], in good agreement with (1). The small dimensions of the samples result in quite low capacitance values. The geometrical capacitance of sample B is 22 fF. The RC product is thus about 1.1 ps for a 50  load, which gives a 3 dB bandwidth of 145 GHz (58 GHz and 103 GHz for samples A and C, respectively).

4. Microwave measurements The microwave measurements are performed in a one-port configuration on 200 µm2 samples whose lower contact is connected to the ground strip of a coplanar waveguide. The network analyser directly gives the S11 parameter, i.e. the ratio of the complex amplitudes of reflected and incident waves. Figure 1 shows the module of S11 as a function of frequency for sample B from 0 to 65 GHz, for successive values of the applied voltage. When the conductance is negative, |S11 | is larger than unity and there is a reflection gain. A gain resonance structure appears above 0.8 V. It shifts regularly to lower frequencies as the applied voltage increases, becoming sharper up to 1.8 V and broadening again beyond. Reflection gain is observed up to 65 GHz at 1 V. The cut-off frequency can be extrapolated linearly to ∼75 GHz. A similar behaviour can be inferred from the results on sample C (Fig. 2), although the resonance is out of range. In this sample, the

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7

2.0 Reflection gain |S11|

6

1.5

–1 V

1.0

–0.8 V –0.7 V

0.5

5 Reflection gain |S11|

Sample B

0.0

4

–0.5 V

–1.8 V

0 10 20 30 40 50 60 Frequency (GHz)

–1.4 V –2.4 V

3

–1 V 2

–2.8 V

1

0

0

10

20

30

40

50

60

Frequency (GHz) Fig. 1. Reflection gain for sample B, as a function of frequency and applied voltage. Solid lines: experiment. Dotted lines: fit using the admittance model for NDC devices.

highest values of gain are observed at the maximum measurement frequency, and the cut-off frequency cannot be precisely extrapolated. The resonance behaviour can be described by a simple drift-diffusion model [19] which neglects diffusion and field inhomogeneity (Fig. 3A), and gives the sample admittance as: Y (ω) = Yc

e−s

s2 +s−1

where Yc =

C T

and

s=

T + iωT τ

(2)

where C is the sample capacitance, T is the electron transit time through the superlattice, ω is the pulsation and τ is the dielectric relaxation time τ = ε/eµ− Nd , where µ− is the differential mobility and ε the dielectric constant. The relation between admittance and S11 parameter is given by: S11 =

Y0 − Y Y0 + Y

with

1 = 50. Y0

(3)

A least-squares fit of |S11 (ω)| for each applied voltage yields the voltage dependence of the transit time and the dielectric relaxation time. The electron velocity law is then obtained in the region of NDC if the electric field is written as F = U/L and the velocity as V = L/T , where U is the potential drop over the total superlattice thickness L. This point-by-point determination of the V (F) relation is shown in Fig. 4 for

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Sample C

2.4 2.2

Reflection gain |S11|

2.0

0.8 V

1.8

0.6 V

1.6

1.1 V

1.4

1.2 V

1.2 1.4 V

1.0 0.8

0V

0.6 0.4 V

0.4 0.2 0.0 0

10

20

30 40 Frequency (GHz)

50

60

Fig. 2. Reflection gain for sample C, as a function of frequency and applied voltage.

sample B, together with a fit to an Esaki–Tsu velocity law: V (F) = 2V p

b 1 + bη

b=

with

F Fc

and

η = 2.

(4)

The fit parameters are the critical field Fc = 5.7 kV cm−1 and the mobility µ = 2200 cm2 V−1 s−1 , which yield V p = 6.4 × 106 cm s−1 in excellent agreement with the peak velocity assessed from the critical current or calculated from the Botzmann equation (Table 2). In the fitting procedure, C is a free parameter, the value of which is close to the geometrical capacitance near the critical voltage (C = 27 fF at 1 V), but significantly larger above 1 V (between 44 and 56 fF). This can be explained by the field inhomogeneity which progressively develops above the critical field but is not taken into account by the model.

5. Temporal experiments Relation (2) expresses the ratio of current-to-voltage amplitudes at pulsation ω. The Fourier transform of (2) is thus a model of the current impulse response of the superlattice samples. Except for a multiplicative factor, it is given by:  X ∞ Zi t Zi e (5) e−αt I (t) = 2 Re i=1

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70

20

Module of reduced admittance (YT/C)

60

Real part (YT/C)

α = T/τ r = RC/T r = 0.125 r=0 α = –1.9

50 40

10 0 –10 –20 –30

0

1

2 3 Frequency (vT)

30 20 10 0

A

1

0

2 Frequency (vT)

40

3

α = –1.9

20

Current

0 –20 –40 –60

n=4

–80

n=2

–100

n=1

–120 0.0 B

0.5

1.0

1.5

2.0

2.5

3.0

Time (T)

Fig. 3. A, Admittance function according to the admittance model for NDC devices, for two values of the series resistance: R = 0 (dotted line) and R = 0.125T /C (solid line). The inset shows the conductance. B, Current impulse response given by the Fourier transform of the admittance model for NDC devices. n is the number of harmonics taken into account in the calculation. α = T /τ is the ratio of the transit time over the dielectric relaxation time.

where α = T /τ and the Z i are the poles of the admittance function Y (s). Their real part is negative. The derivation of (5) requires the addition of infinitesimal parallel capacitance and series inductance to Y in order to ensure causality. In the region of NDC (α < 0), continuous oscillation occurs when α < Re(Z 1 ), which amounts to the usual condition on the Nd L product for Gunn oscillation. I (t) is the sum of harmonics of the fundamental oscillation. Their frequency Im(Z i )/2π is approximately given by the multiples of the inverse of the transit time, and their damping factor Re(Z i ) − α increases with order. Successive inclusion of additional harmonics in I (t) is represented in Fig. 3B. The larger the number of harmonics, the faster the instantaneous response is, but the highest frequencies rapidly vanish so that only the fundamental oscillation remains. Such a behaviour is clearly observed in the temporal measurements by electro-optic sampling (Fig. 5) in sample A. The diode is biased at 2.5 V, i.e. far into the NDV region if one refers to the critical voltage of sample B, whose structural parameters are identical except for the doping. The fundamental frequency is 50 GHz. The sample is excited by polarization steps of amplitude 1U between 5 and 100 mV. When 1U

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7 Sample B 6

Velocity (106 cm s–1)

5

4

3 µ = 2200 cm2 V–1 s–1 Fc = 5700 V cm–1

2

η=2 1

0

0.0

0.5

2.0 1.0 1.5 Field (104 V cm–1)

2.5

3.0

Fig. 4. Electron velocity law in sample B. The open circles are obtained from the fit of the microwave spectra (Fig. 1) as superlattice thicknesses divided by transit time. The curve is a fit to an Esaki–Tsu-type velocity law.

is small, the temporal responses clearly contain high-order harmonics at the outset since the first oscillation is pulse-shaped, whereas the following oscillations are more sinusoidal. When 1U increases, the contribution of harmonics becomes weaker. This nonlinear behaviour cannot be explained by the above admittance linear model. Returning to the spectral domain through an inverse Fourier (IFT) of the responses (Fig. 6) provides a quantitative assessment of the harmonic content and allows for a comparison with Equation (5), after normalization by the amplitude of the Fourier components of the incident excitation step. The load resistance R must be taken into account. When it is not, the amplitude of the successive harmonics increases (Fig. 3A, dotted line). In fact, the load resistance and the capacitive admittance of the superlattice diode (Im(Y )) already cut off the resonances of orders 2 and 3 (Fig. 3A, solid line). The value r = 0.125 with r = RC/T is appropriate for sample A and R = 50, and yields satisfactory agreement with the measured amplitudes. Higher-order resonances are not accessible to the measurement system.

6. Discussion The microwave measurements show that superlattice diodes can be designed to exhibit NDC in the millimetre-wave range. In a theoretical model based on balance equations [15], a cut-off frequency f c as low as 70 GHz is predicted for NDC in a superlattice of similar miniband width but larger doping than in

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4.0 Sample A

hv

3.5 S

Normalized current response

3.0

P

SL

EO

G

2.5 2.0

∆U= 100 mV 1.5 ∆U= 60 mV 1.0 ∆U= 30 mV 0.5 ∆U= 15 mV 0 ∆U= 5 mV

–0.5 –1.0 0

20

40

60

80

100

120

Time (ps) Fig. 5. Current impulse responses of sample A for varying amplitudes for the excitation step, measured by electro-optic sampling on a picosecond timescale. The bias voltage is 2.5 V. The inset shows the measurement set-up schematically: (G) and (S) are the ground plane and the substrate of the interrupted inverted-microstrip transmission line, (P) is the photoconductive generator, (SL) is the superlattice sample, and (EO) is the electro-optic crystal.

the present samples, due to interelectronic interactions and electronic thermalization. This is not consistent with the results reported here, especially in sample C whose cut-off frequency is certainly far above 70 GHz. However, since the doping level is significantly lower than in the modelled superlattice, electronic thermalization is expected to be weaker and this may explain the apparent contradiction. It should be noted that the electronic temperature remains quite moderate above the critical field in semiclassical models based on the Boltzmann equation, and the cut-off frequency continuously increases with the field. In Ignatov’s model for example [11]: s 1 b4 − 1 ed F −→ (6) 2π f c = τ 2 τε τε b + 1 b→∞ h¯ where τ and τε are the momentum and energy relaxation times, and d is the superlattice period, the cut-off frequency grows as the Bloch frequency at high field (but this result may not be strictly valid for doped superlattice diodes in the absence of a rigorous treatment of interelectronic interactions in semiconductor superlattices in the Boltzmann equation based models). In the conditions of the experiment, the values of τ (86 fs) and τε (366 fs) can be determined using Ignatov’s expressions for Fc = 5.7 k V cm−1 and V p =

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1.1 3

Sample A

Normalized current response

1.0

Normalized current amplitude

0.9 0.8

2 1 0

0.7 –1 10

0.6

20

30

40

Time (ps)

0.5

1U = 5 mV 1U = 15 mV

0.4

1U = 30 mV

0.3

1U = 60 mV

0.2

1U = 100 mV

0.1 0 0

50

100

150

200

250

300

350

400

Frequency (GHz) Fig. 6. Fourier transforms of the current impulse responses of sample A, a detailed view of which is shown in inset.

6.4 × 106 cm s−1 . Equation (6) then gives f c ∼ 1.2 THz for b = 2, a value compatible with the results in the temporal domain. Indeed, the generation capabilities of the superlattice diode are experimentally evidenced by the temporal measurements, up to 150 GHz at least when the doping level is ≤ 5 × 1015 cm−3 . The amplitudes of the second and third harmonics of the fundamental oscillation frequency are consistent with the admittance model (3) for NDC devices in series with a resistive load, which bring in a low pass filtering. As a consequence, (3) can be considered as a satisfactory model for the superlattice diode itself up to 150 GHz, exhibiting NDC on the second and third harmonics (see the dotted line in the inset of Fig. 3A). The cut-off frequency for NDC is thus larger than 150 GHz in sample A. In fact, the real part of the IFT of the temporal responses directly shows NDC on the third harmonics, but an exhaustive analysis of the phase is out of the scope of this paper. Finally, further investigations are also required to explain the nonlinear dependence of the amplitudes of the harmonics as a function of excitation amplitude. It can be attributed a priori either to some nonlinearity of the charge distribution in the n+ n− n+ structure, or to the intrinsic nonlinearity of the superlattice, since, in the miniband transport approaches, it is a nonlinear (Bloch) harmonic oscillator [11] when the electric field is time dependent.

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7. Conclusion In conclusion we have shown that NDC in lightly doped semiconductor superlattices can cover a broad portion of the millimetre-wave domain. Miniband transport accounts for superlattice nonlinear conduction quite generally, but special attention should be paid to electronic thermalization in order to gain deeper insight into carrier dynamics in doped samples. Practically, it is possible to design superlattice diode amplifiers and oscillators in the millimetre-wave, for example in a reflection configuration using the reflection gain [20]. The efficiency of such devices improves when the doping increases, but the question of the cut-off frequency of NDC then arises. To study more strongly doped samples with the same experimental approaches would require smaller device areas to reduce the capacitance, and less thick superlattice active layers to avoid selfoscillation (Nd L condition), i.e. larger oscillation frequencies, a challenge for both the device technologies and the temporal measurement techniques. Acknowledgements—We are very indebted to G´erard Caumont and Sylvian Lamy whose skillfulness has been essential to the realization of the delicate optical and mechanical parts. We also acknowledge Nouredine Bouadma, Daniel Arquey and Fran¸coise Heliot for their technological assistance, Guy Leroux for X-ray characterizations and Alain Sibille for fruitful discussions.

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