1. Introduction

The trend in modern communication circuits is towards higher data rates and better bandwidth utilization. Due to this, very stringent requirements are placed on the spectral purity of oscillators, which increases the requirements on the local oscillator phase noise^{[1, 2]}.

Mostly, noise is sensitive to defects and non-ideality in semiconductor devices, which can impact device and circuit performances^[3]. Noise in devices can be up-converted in phase noise in oscillators and reduce the achievable spectral purity in communications systems.

SiGe technology is today one of the mainstream semiconductor technologies used in RF and microwave applications, which not only offers higher speed performance to bipolar transistors, but also lower noise and better linearity^{[4, 5]}.

The work in this paper is mainly focused on the 4 GHz Colpitts oscillator, which contains a Si/SiGe heterojunction bipolar transistor compatible with CMOS technology. First of all, we study the electrical characteristics of specified heterojunction bipolar transistors. In this regard, we report the electric behavior of a SiGe HBT with the presence of electrically active defects induced during extrinsic base implantation.

Next, we investigate substrate noise and its effect on the phase perturbation of oscillators. We study the contribution of device implantation defects to the output noise of an oscillator. These defects, which are responsible for the current fluctuations at the origin of noise in device, have an enormous effect on oscillator phase noise.

Finally we investigate oscillator phase noise; the study is based on the linear time varying model of phase noise developed by Hajimiri^[6]. This model relates the phase noise with the time-variant nature of the oscillator and provides a clear physical mechanism of phase noise generation. In this regard, an impulse sensitivity function (ISF) technique is proposed to describe the susceptibility of an oscillator to noise as a function of time.

2. Device structure

The device used in our study is a SiGe heterojunction bipolar transistor integrated in CMOS (complementary metal oxide semiconductor) technology. Integration of SiGe HBTs with CMOS devices combines the high density of CMOS for high levels of integration with the high performance of the HBT to produce low cost, low power, and high performance devices. BiCMOS technology presents the optimal choice for both high performances and integration level^{[7, 8]}.

The specific structure is a SiGe bipolar transistor that is integrated in a 0.35 $\mu$m BiCMOS technology, with a quasi-self-aligned polysilicon emitter (see Fig. 1).

Electrically active defects are suspected to be created during extrinsic base implantation. It is known that base extrinsic implantation damage is a source of excess interstitials, which enhance the boron diffusivity in the SiGe base layer of the transistor and degrade the static and high-frequency characteristics of the device considerably^[9].

The physical characteristics of the extrinsic base implantation defects are deduced by DLTS (deep level transient spectrometry) analysis. It established the presence of two types of implantation defect. The first defect is a hole trap with a cross capture section $\sigma$ $=$ 10²¹ cm², an effective density N_T $=$ 10¹⁶ cm^-3, and an activation energy E_T $=$ 0.1 eV. The second defect is also a hole trap with a cross capture section $\sigma$ $=$ 10¹⁸ cm² an effective density N_T $=$ 2.10¹⁶ cm^-3 and an activation energy E_T $=$ 0.25 eV^[10].

3. The HBT devise modeling

The electrical behavior of the device is obtained by numerical resolution of the basic semiconductor equations, which include drift-diffusion equations DDM. In this approximation, a simulator solves the continuity and transport equations along with the Poisson equation to describe the electrical characteristics of semiconductor devices^[11].

$\nabla^2 \phi =\frac{-q}{\varepsilon_{\rm S}/C}\left[p-n+N_{\rm D}^+-N_{\rm A}^-\right],$

(1)

$\frac{\partial n}{\partial t}=\frac{1}{q}\cdot {\rm div} \boldsymbol{J}_{\rm n}+G_{\rm n} -R_{\rm n},$

(2)

$\frac{\partial p}{\partial t}=-\frac{1}{q}\cdot {\rm div} \boldsymbol{J}_{\rm p} +G_{\rm p} -R_{\rm p},$

(3)

where $N_{\text{D}}^ +$ and $N_{\text{A}}^ -$ are the ionized impurity concentration, and $\varepsilon$ is the permittivity of the material. ${J_{\text{n}}}$, ${J_{\text{p}}}$, the electron and hole current densities are functions of carrier concentration.

3.1 Model for the alloyed SiGe HBT

The incorporation of germanium in the base region significantly changes the properties of the base material. To model SiGe heterostructures, we need appropriate material physical parameters. Depending on germanium concentrations, ${\text{S}}{{\text{i}}_{\text{1}}}{\text{ - }}{x}{\text{Ge}}{x}$ has different material properties from those of silicon and germanium.

The compressive strain due to Ge incorporation reduces the bandgap of Si, leading to the narrow bandgap of SiGe alloy. Germanium in the base region induces a reduction of the energy gap, which depends on Ge mole fraction $x$ as illustrated by equation for ($x$ $\leqslant$ 0.245)^[11].

$E_{\rm g} =1.08+x(0.945-1.08)/0.245.$

(4)

For the band alignment, we assume that the entire bandgap difference appears in the valence band; the conduction band discontinuity $\Delta {E_{\text{C}}}$ is close to zero.

$\Delta {E_{\text{V}}} = \Delta {E_{\text{g}}}\;{\text{and }}\Delta {{E}_{\text{C}}}\;{\text{ = }}\;{\text{0}}.$

The electron mobility and its dependence on germanium content was taken as described by the Masettis model^[11]. This model assumes that the reduction in the electron and hole effective masses due to Ge incorporation induce higher carrier mobility in strained SiGe than that for Si.

The carrier lifetime of the SiGe alloy is significantly lower than that of silicon; according to Ref. [12] the device simulator is calibrated to:

$\tau_{\rm SiGe} =0.1\tau_{\rm Si}.$

(5)

3.2 Defect analysis in SiGe HBT

The recombination assisted by deep traps is considered by the simulator through the Shockley-Read-Hall (SRH) model^[13].

The generation recombination ratio is affected to: GR $=$ ${G_{{\text{SHR}}}}$.

$\begin{split} \label{eq6} G_{\rm SHR} = {}& np-n_{\rm i}^2 \tau _{\rm p} \left( {n+n_{\rm i} \mathit{\Gamma } \exp \dfrac{E_{\rm T} -E_{\rm F}}{kT}} \right) \\[2mm]& +\tau _{\rm n} \left( {p+n_{\rm i} \dfrac{1}{\mathit{\Gamma } }\exp \dfrac{E_{\rm F} -E_{\rm T}}{kT}} \right)^{-1}. \end{split}$

(6)

${\tau _{{\text{n}}, {\text{p}}}}$ are lengths of carrier life, they depend on the electric properties of the defect.

$\tau _{\rm n} =\frac{1}{c_{\rm n}}, \quad \tau _{\rm p} =\frac{1}{c_{\rm p}}.$

(7)

The rates of electron capture ($c_{\rm n})$ and hole capture ($c_{\rm p})$ determine the defect characteristics.

The capture rates for the two types of carriers are:

$c_{\rm n} =\sigma _{\rm n} n\left\langle {V_{\rm thn} } \right\rangle, \quad c_{\rm p} =\sigma _{\rm p} p\left\langle {V_{\rm thp} } \right\rangle.$

(8)

${\sigma _{{\text{n}}, {\mkern 1mu} {\text{p}}}}$ are the capture cross sections of the deep defect. They translate the surface in which the free carrier must approach the centre to be captured.

$\left\langle V_{\rm thn} \right\rangle$ and $\left\langle V_{\rm thp} \right\rangle$ are respectively electron and hole thermal velocities.

In Fig. 2, we report the effect of the implantation defects on Gummel characteristics and current gain. The result is obtained for two identical HBTs, one considered without defects and the other with an extrinsic base implantation defect positioned at the interface of the base/emitter junction. As seen, the results reveal that the base current is more important with the presence of the implantation defects, where an excess base current is observed at low injection V_be < 0.3 V. This increase in base current involves the generation of an additional recombination process. This process, related to the defect presence in device, represents the main contribution to the base current at low injection. The presence of implantation defects acts directly on the current gain by causing a notable degradation on this last of all, as seen in Fig. 2.

On the other hand, implantation defects do not have any influence on the dynamics performances of the study device.

4. Phase noise analysis of 4 GHz Colpitts oscillators

For the Colpitts oscillator study, a mixed-mode analysis is involved. Mixed-mode simulation provides the capability to simultaneously solve the device and circuit equations with SPICE and physical models. In mixed-mode simulation, the devices' capabilities can be characterized by their performances in a circuit as a function of transport models. A microscopic drift diffusion model was applied to the device, whereas the circuit is governed by Kirchhoff's laws. In this regard, the simulator combines the device and circuit equations into one single equation system.

A schematic circuit of an LC tuned oscillator can be represented as shown in Fig. 3.

Small signal oscillation conditions can be analyzed using the Barkhausen criteria in a feedback oscillator. In order to provide a stable oscillation at a given angular frequency the system must satisfy condition:

mod ($GB$) $=$ 1, arg($GB$) $=$ 0, where $G$ is the gain of amplifier (HBT) and $B$ is the gain of the LC tank.

The oscillation frequency is determined by the circuit properties, e.g., the LC tank resonant frequency.

$\omega _0 =\frac{1}{\sqrt {L C_{\rm eq}}}, \quad {\rm with}\; C_{\rm eq} =\frac{C_1 C_2}{C_1+C_2}+C.$

(9)

The oscillation frequency depends on the transistor junction capacitances ($C$) and the amplitude is defined by some limiting mechanism of device non-linearity.

Figure 4 shows the transient simulated output waveform of a study oscillator obtained by ISE TCAD simulation. In a real circuit the oscillation starts due to noise or an interferer coupling in the oscillator. In simulations, to start the oscillation it was necessary to provide a current impulse. The transistor gain and the operating point have been chosen in a simple way to obtain a very stable oscillation.

For an ideal oscillator, amplitude and phase are both constants. However, the active and passive devices used to implement a real oscillator introduce random noise into both the amplitude and phase of the output.

With the presence of different noise sources, a general oscillator output signal can be expressed as:

$V(t)=A(t) \times f(\omega_{0}t+ \phi(t)),$ where $A(t)$ is the amplitude and $\phi (t)$ is a small random excess phase.

The root cause of the phase noise in the oscillator was related to the noise sources of the active device. Several noise sources are present in bipolar transistors, the exact sources of this noise in devices are not yet completely understood, however it is basically agreed that it is a result of spontaneous fluctuations of current due to carrier recombination in traps caused by defects and contaminants in electronic devices. These traps randomly capture and emit carriers, thereby causing a fluctuation in the number of carriers available for current transport.

4.1 Influence of implantation defect on oscillator phase noise

In this part, we study the impact of electrically active defects created during technological processes on oscillator phase noise. The aim was to identify the parasitic impact of defects introduced by extrinsic base implantation. They are essentially hole trap defects localized at the base/emitter interface.

Figure 5 illustrates the output waveform of a study oscillator containing a heterojunction bipolar transistor with a SiGe base as an active device. Two devices are considered, one without defects and the other with an extrinsic base implantation defect introduced at the base/emitter interface. It is clearly seen that the output signal of the oscillator was sensitive to the implantation defect; the oscillator output waveform presents small output phase variations with the presence of an implantation defect.

The observed changes in the output phase are very low, in the order of 10-12 s (Fig. 5), which corresponds to a high frequency signal of about 1012 Hz. This phase variation corresponds to higher frequencies as compared to the carries lifetime of the transistor 10-6 s that corresponds to a low signal frequency.

In general, the noise conversion depends on the ratio of the cutoff frequency of the device. However, in the presence of implantation defects, the high frequency signal is not attenuated by the device because there is a product conversion of two signals.

This is can be confirmed theoretically because there is a signal proportional to the product of two signals LF and HF sin $2\pi c{f_0}$ sin $2\pi {f_1}$, where ${f_{\text{0}}}$ is the frequency of the LF and ${f_{\text{1}}}$ is the frequency of the HF signal. This product produces new signals at the sum ${f_{\text{1}}}$ +${f_{\text{0}}}$ and difference ${f_{\text{1}}}$ -${f_{\text{0}}}$ of the original frequencies. That creates new frequencies from two signals, and shifts the signal from the reference frequency to another frequency. This explains the observation of phase shifts to 10-12 s.

Figure 6 illustrates the variation of oscillator output waveform phase shift with trap energy activation E_T. In the considered device, the energy activation magnitude of hole traps created during the base implantation E_T is approximately 0.25 eV. In this part, we place the trap energy at 0.3 eV, 0.4 eV, 0.5 and 0.6 eV from the valence band, and we observe the output waveform of the study oscillator.

We can see that the closer the trap is to the valence band, the more important the output waveform phase shift is. The phase noise in the oscillator is very sensitive to trap and defects in the device; it is mostly associated with trapping physical processes between an energy band and a discrete energy level (trap) in the bandgap. In fact, the closer the trap is to the valence band, the easier the capture of the holes becomes; resulting in large current fluctuations. The phase noise in the oscillator is strongly related to current flow in the devices due to random transitions of charge carriers in semiconductor.

4.2 Linear time varying (LTV) model for excess phase

Various phase noise models have been developed to explain the processes involved in converting device noise into phase noise. The linear time variant (LTV) phase noise model proposed by Hajimiri and Lee involves the conversion of excess injected current into excess phase, which is done via linear time variation.

The linear time varying (LTV) model takes into account the important time variant nature of phase noise in oscillators. The key point of this model is that the response of the oscillator to any impulse injection is time dependent^{[14, 15]}.

This model is based on the impulse sensitivity function (ISF), which represents the instantaneous sensitivity of the oscillator phase after applying a parasitic impulse on an oscillator circuit^[16]. According to the ISF theory, the phase perturbation of an oscillator depends on where in the oscillation period the impulse is injected. In this fact, the same perturbation occurring at different times will result in different phase shifts due to the time-variant nature of oscillators^[17].

Based on this time-variant characteristic and the linear assumption, the unit impulse response for the excess phase of an oscillator can be expressed as:

$h_\phi (t, \tau )=\frac{\mathit{\Gamma } (\omega _0 \tau )}{q_{\rm max } }u(t-\tau),$

(10)

where $u(t$-$\tau)$ is the unit step function and ${q_{{\text{max}}}}$ is the maximum charge displacement across the capacitor on the node, which makes the function $\mathit{\mathit{\Gamma } }(x)$ independent of signal amplitude.

For susceptibility analysis of the oscillator, we consider harmonic disturbance (current or voltage) defined by its magnitude $A$ and its angular frequency ${\omega _{\text{p}}}$.

$p(t)=A\cos ({\omega _{\rm p} t})u({t-t_0}).$

(11)

The output excess phase $\phi$$(t)$ can be calculated in response to any injected disturbance using a superposition integral.

$\begin{split} \label{eq12} \phi (t){}& =\int\nolimits_{-\infty}^t {\mathit{\Gamma }^\phi \left( {\omega _0 \tau} \right) p(\tau) {\rm d}\tau} \\[2mm]& =\int\nolimits_{t_0 }^t {\mathit{\Gamma } ^\phi \left( {\omega _0 \tau} \right)A\cos \left( {\omega _{\rm p} \tau } \right){\rm d}\tau}. \\ \end{split}$

(12)

Due to its periodicity, $\mathit{\Gamma } (\omega_{0}t)$ function can be extended in a Fourier series as follows:

$\mathit{\Gamma } ^\phi \left( {\omega _{\rm c} \tau} \right)=\frac{c_0 }{2}+\sum\limits_{n=1}^\infty {c_{\rm n} \cos \left( {n\omega _0 \tau +\theta _{\rm n}} \right)} .$

(13)

Then, the phase shift can be written as follows:

$\begin{split} \phi (t)={}& A \frac{c_0 \sin \left( {\omega _{\rm p} t} \right)}{2\omega _{\rm p}}+ \\& A\cdot \sum\limits_1^\infty {\frac{c_{\rm n} \sin \left[{\left( {n\omega _0 \pm \omega _{\rm p}} \right)t+\theta _{\rm n}} \right]} {2\left( {n\omega _0 \pm \omega _{\rm p}} \right)}} +\phi _0(t_0). \\ \end{split}$

(14)

If ${\omega _{\text{p}}} \ll {\omega _{\text{0}}}$, then $\phi (t)$ is approximated by the first term.

$\phi (t)=A \frac{c_0 \sin \left( {\omega _{\rm p} t} \right)}{2\omega _{\rm p}}.$

(15)

Since the ISF is determined by the waveform, the first co-efficient c₀, can be significantly reduced if certain symmetry properties exist in the waveform.

In ISE-TCAD simulation, the sensitivity of the oscillator to noise was obtained as the response to the distribution source. A disturbance pulse is injected into the Colpitts oscillator nodes, and their effect on excess phase is observed for several oscillation periods after the injections. The disturbance sources can be defined as a voltage source in series or current sources in parallel with the tank interested nodes.

Figure 7 shows the dynamic behavior of the phase shift obtained by mixed-mode simulations when a fixed current impulse (2000 $\mu$A during 20 ps) is applied in parallel with C₁ tank capacitance. The pulse is applied at different specific moments of a wave form period (at the zero-crossing, the bottom and at the peak), and the resulting phase shifts are observed.

We can observe that an injected current impulse induces instantaneous change in the voltage waveform amplitude. The voltage fluctuation is related to the overall capacitance C_tot in the LC circuit and can be expressed as: $\Delta V$ $=$ $\Delta q/C$.

The variation of the amplitude was more important when the pulse is injected at the waveform oscillation peak amplitude and minimal at zero-crossing times.

On other hand, we observe that the amount of phase shift depends also on the delay of pulse injection in the oscillation cycle. As seen, a current pulse that is injected at the zero crossing moment causes a maximum phase shift. On the other hand, the same pulse occurring at the moment of the output waveform peak has no effect on the phase of the oscillator; only the amplitude is modified. The oscillator is therefore a periodically time varying system; it is more susceptible to the phase variation at a zero crossing moment and less sensitive at a maximum of the output waveform.

The time variant relationship between distribution noise and phase shift can be characterized using the impulse sensitivity function (ISF). By applying impulses at various phases of the period, and recording the resulting offset we can access the ISF function. In this fact, the position of the impulse is shifted with respect to the oscillation waveform period and the simulation is re-run to evaluate the ISF at a different time point of one period.

Figure 8 presents the sensitivity of every point of the periodic waveform to an input fixed current impulse (2000 $\mu$A during 20 ps) applied in parallel with the tank capacitance C₁. As can be seen, the ISF shape is sinusoidal with oscillation period of 268 ps, the operating conditions of the transistors in the oscillator are periodic; as a consequence the ISF function is periodic with the same period as the oscillator.

In addition it can be viewed from Fig. 8 that the oscillator phase sensitivity to a disturbance impulse current is a time dependant. The oscillator is most sensitive to noise phase at the waveform transitions when the phase variations is maximum and completely immune to disturbances at the peaks.

The ISF function of the study oscillator has its maximum value at the zero crossings of the oscillation waveform, and its minimum value at the maximum, which represent a 90 degree phase shift from the output voltage. So in an LC oscillator the ISF function $\mathit{\Gamma } (x)$ is approximately proportional to the derivative of the oscillator waveform.

In this part, we treated the case where the injected pulse is a voltage. The charge displacement on the interested node caused by the injection depends on the voltage source amplitude but it will not change the linearity of the phase noise response. In this fact, a voltage pulse of 0.5 mV magnitude and 10 ps duration is simultaneously applied in parallel with the $C_1$ tank capacitance.

Figure 9 presents the ISF curves for the study Colpitts oscillator; calculated when a voltage impulse is applied at different instants in the oscillation period. For voltage impulse we obtain a different dynamic behavior of ISF function and observe a high sensitivity to noise near the extremes of the output waveform.

In the case of voltage impulse perturbation, ISF analysis of the LC oscillator suggests that the phase noise is less sensitive near the waveform transitions. Therefore, the resulting phase noise can be reduced if the disturbance pulse is injected into the tank at the right moment, i.e., when the ISF is at its minimum.

5. Conclusion

This paper presents mixed-mode simulations of a radiofrequency LC oscillator implemented in silicon germanium technology. As first, we have investigated the effect of electrically active defects induced during extrinsic base implantation on the electrical characteristics of the SiGe HBT. If static performances of the HBT are strongly penalized by the presence of these defects, conversely, the dynamic performances remain unchanged.

Second we analyzed the influence of electrically active implantation defects on the phase noise of an actual 4 GHz LC oscillator. The energy position of the trap in the semiconductor gap has a real influence on the phase shift of the output oscillator waveform. The simulation predicts an increase in the phase shift of output waveform with defect energy.

Finally, based on the LTV model, which makes use of the impulse sensitivity function (ISF), we have investigated the susceptibility of a radiofrequency oscillator to noisy disturbance sources. Through ISF analysis, we have verified the time variance of the oscillator and described the sensitivity of the oscillator over time. The oscillator is rather sensitive at certain instances of the delay and is less sensitive at other instances.