Experimental Results: Temperature Dependence of the Quality Factor, Gauge Factor and Thermomechanical Noise

Frequency [MHz]

5.4.4 Experimental Results: Temperature Dependence of the Quality Factor, Gauge Factor and Thermomechanical Noise

5.4.4 Experimental Results: Temperature Dependence of the Quality

10 100 400 40

60 80 100

Gauge Factor

Effective Phonon Temperature [K]

2000 4000 6000 8000 10000 12000

Quality Factor

58000 60000 62000 64000 66000

(c) (b) (a)

Resonant Frequency [Hz]

10 100 400

40 60 80 100

Gauge Factor

Effective Phonon Temperature [K]

2000 4000 6000 8000 10000 12000

Quality Factor

58000 60000 62000 64000 66000

(c) (b) (a)

Resonant Frequency [Hz]

(a) Resonance frequency; (b) Quality factor; and (c) Gauge factor. The data is for the thermomechanical resonance, collected at a bias of 3.3 µW. The temperature scale has been corrected to account for heating at this bias power as discussed in the text.

The solid gray lines provide a guide to the eye.

Fig. 5.4.B Force noise spectral density for the nanocantilever force sensor.

10 100 400

5 10 100 500

Force Sensitivity [aN/Hz

1/2

]

Effective Phonon Temperature [K]

65.68 65.70 65.72

110 120 130 140 150 160 170

SV [nV2 /Hz]

f [kHz]

10 100 400

5 10 100 500

Force Sensitivity [aN/Hz

1/2

]

Effective Phonon Temperature [K]

65.68 65.70 65.72

110 120 130 140 150 160 170

SV [nV2 /Hz]

f [kHz]

The temperature dependent magnitude of the thermomechanical force noise is shown here. Over the range shown, the data follow an approximately linear trend; a least- squares fit yields a T ^0.8 dependence (solid line). Inset: The measured voltage noise at a device temperature of 10.4K.

Sensing for NEMS

Apart from the high force sensitivity we have attained, perhaps the most compelling aspect of our present results is that they can be employed to elucidate the ultimate limits of semiconductor-based piezoresistive displacement sensing for NEMS. Specifically we can evaluate: (a) the lowest temperatures at which piezoresistive displacement transduction remains useful, (b) the optimal bias current for a given ambient temperature, and (c) the ultimate force sensitivity that is attainable with piezoresistive cantilevers at low temperatures where optimal performance is obtained. The analysis proceeds as follows. The transduction responsivity of a piezoresistive cantilever, R_T is proportional to the bias current applied through the device  but this bias, in turn, also determines the power dissipated within the piezoresistors, P_in =I R_b² _d. This power input, in turn, induces a temperature rise determined by the thermal conductance available from all mechanisms that serve to cool the piezoresistors. In steady state, a temperature

*

( , )0 _b

T T I is thereby established which depends upon the ambient temperature, T₀, and applied bias, I_b. We view T₀ as the independent variable since its value is typically dictated by the sensing application. Hence, for a given T₀ there exists an optimal bias current, I_b^(opt)( )T₀ , that yields the optimum force sensitivity, ^{1/ 2}_F ( )₀

S T opt at that temperature. However, below a characteristic temperature T₀ <T₀^(min), the steady-state device temperature, T T I^*( , )_{0 b} , is predominantly determined by Joule heating and not by the ambient temperature T₀. In the previous analysis a single pathway for dissipation of bias induced Joule heating was considered, namely heat transfer from holes to the

phonon bath and then to the environment via phonon transport. A second potential pathway for heat dissipation is available, specifically hole diffusion along the electrical conductors. These two mechanisms for thermal conduction act in parallel, as schematically depicted in Fig. 5.5.A. For the analysis here, which extends to lower temperatures than the experimental data presented earlier, we must consider both pathways. In doing so we will explicitly justify the omission of the hole diffusion pathway in the earlier analysis of our data at higher temperatures, T ≥6K.

Here we evaluate the effective steady-state for the phonons and holes in the strain sensing region of the device, making it possible to evaluate the total force spectral density on resonance, referred to input (RTI). This is given by the formula

0 2 0

2 4

γ ,

= + ^J = π ^{B o}

∫

^{leg ph} + ^{B d}

∫

^{leg h}

F F F

o leg leg

k K T k R T

S S S dy dy

f Q T (5)

where S_F^γ is the thermomechanical noise and S_F^J = S_V^J /T² is the (Johnson) electrical noise (RTI). Here T =R_T ( /Q K_o) is the ganged system responsivity, the product of the transducer and on-resonance mechanical responsivities, S_V^J =4k T R_{B h d} is the (Johnson) voltage noise of the transducers, and T_h is the effective temperature of the holes. Note the separate roles played by the phonon and hole temperatures for the mechanical and electrical degrees-of-freedom, respectively.

The calculation proceeds as follows; the temperature profile for the holes is solved for using the diffusion equation: 2t_doped wl_egκ_h

(

d T dy² _h/ ²

)

= −Q_h/ _leg, where the hole thermal conductivity is given by the Widemann-Franz law

Fig. 5.5.A Model for low low temperature thermal transport in semiconducting piezocantilevers.

Joule heating of the transducer’s hole gas arising from the applied bias, P_in, is dissipated by both phonon emission, Q_{h ph−} , and hole diffusion down the electrical leads to the environment (i.e. substrate), Q_h. The induced heating of the resonator phonons is subsequently dissipated into the environment via phonon transport, Q_ph. The thermal resistances characterizing these processes are R_{h ph−} , R_h, and R_ph, respectively. In steady state, with the environment at temperature T₀, this branched heat flow raises the hole gas and resonator phonon temperatures to T_h^* and T_ph^* , respectively. The heat capacities of the hole gas and resonator phonons are C_h and C_ph, respectively.

environment

*

,

h h

T C

T

0 ph

,

ph

T C R

_h

R

h ph₋

R

ph

resonator phonons transducer

holes

P in

Q

^•h ph₋

Q

^•h

Q

^• ph

environment

*

,

h h

T C

T

0 ph

,

ph

T C

*

,

h h

T C

T

0 ph

,

ph

T C R

_h

R

h ph₋

R

ph

R

h

R

h ph₋

R

ph

resonator phonons transducer

holes

P in

Q

^•h ph₋

Q

^•h

Q

^• ph

P in

Q

^•h ph_{h ph−}

Q

^• ₋

Q Q

^••h_h

Q

^• ph_ph

Q

^•

*

environment

*

,

h h

T C

T

0 ph

,

ph

T C R

_h

R

h ph₋

R

ph

resonator phonons transducer

holes

P in

Q

^•h ph₋

Q

^•h

Q

^• ph

environment

*

,

h h

T C

T

0 ph

,

ph

T C

*

,

h h

T C

T

0 ph

,

ph

T C R

_h

R

h ph₋

R

ph

R

h

R

h ph₋

R

ph

resonator phonons transducer

holes

P in

Q

^•h ph₋

Q

^•h

Q

^• ph

P in

Q

^•h ph_{h ph−}

Q

^• ₋

Q Q

^••h_h

Q

^• ph_ph

Q

^•

*

( )

²

2 / / 3

κ_h = π k e_B σT_h . The temperature dependence of the thermal conductivity makes solution of this differential equation non-trivial; hence we evaluate it numerically.

In these equations Th represents the (local) effective steady state temperature of the holes, tdoped represents the thickness of the conducting layer of the piezoresistor, σ represents the conductivity of the piezoresistor, and e is the electronic charge. The temperature profile of the phonons is determined by Eq. (4) where the heat transferred, Q ,is now

Qph, the heat transferred via the phonon conduction pathway.

Once the temperature profiles for holes and phonons have been evaluated, the heat transferred from holes to phonons can be calculated using the equation:

( ( ))

− =

∫ ∫

^T_ph^h −

h ph T h ph

Q dV dT G T y , where the volume integral is over the conducting region of the device and Gh-ph is the hole-phonon thermal conductance per unit volume given by

( ) ( ) ( ) 4

− = Γ − = −

h ph h h ph h ph

G T C T T g T , where Ch is the electronic heat capacity per unit volume and Γ_{h ph−} is the hole-phonon scattering rate.¹⁵ We model the hole heat capacity as that of Sommerfeld free hole gas, C T_h( ) = π²pk T_B² /ε_F =γ_hT ,where p is the hole density, ε_F = ²(3 p)π^{2 2/ 3}/ 2m_h is the Fermi energy, and m_h the (light) hole mass in the valence band. Assuming the holes and phonons are Fermi and Bose distributed, respectively, the deformation potential hole-phonon scattering rate, Γ_{h ph−} , is given by

( )

^{3 2} 3 3

4 4

s F

3 3 v v

ς α

π ρ

− −

Γ_{h ph} = k D^B = _{h ph}

T T , where D=8.3eV is the deformation potential^16,17 and

( )

³

ς =1.202. The relative values for the heat conducted via the two thermal conduction

convergence is attained, yielding Q_{h ph−} =Q_h (subject to the constraint Pin=Q_h+ Q_ph).

The effective, electrically-transduced force noise referred to the input (RTI) calculated with this model is shown at four different ambient temperatures in Fig. 5.5.B (a) for the experimental device geometry. For very low bias currents the total force noise increases due to the smaller transducer responsivity and, consequently, the dominance of S_F^J referred to input. At high bias currents, the total noise increases due to the increased device temperature. It is clear that by minimizing Eq. (5) we can determine the optimum bias power, P_in, yielding the highest sensitivity for a given ambient temperature and device geometry. The effective temperature of both the phonons and holes is shown in Fig. 5.5.B (b). It is evident that below a few degrees Kelvin, Joule heating is very significant in determining the effective device temperature. The temperature dependence of the optimum force sensitivity obtained by this procedure is shown in Fig. 5.5.B (c) for the experimental geometry. The corresponding effective temperatures for the holes and phonons are shown in Fig. 5.5.B (b). As seen, a limiting sensitivity of S_F =13aN/ Hz can be achieved for an ambient temperature T₀<1K.

This analysis allows us to assess the sensitivity improvements possible through optimization of geometry – for example, by reducing the width of the cantilever legs and decreasing the total device thickness. To illustrate the improvements that are easily

Fig. 5.5.B Optimization of transduced force sensitivity.

(a,d) Evaluation of the bias power dependence of the effective force noise, which includes both mechanical- and electrical- domain contributions (referred to input).

Calculated performance is displayed for two device geometries (see text): the microdevice on which experiments were conducted in this paper (plots a,b,c) and a smaller, but realistic, nanodevice (plots d,e,f). At each ambient temperature, T₀, the minima define an optimal bias point, P_in^(opt) =I_B^(opt)R²_T. At low bias power the responsivity of the transducer decreases, thereby leading to an increase in the effective force noise (referred to input). At high bias power the total force noise increases due to Joule heating. (b, e) At the optimal bias power the transducer hole gas and the resonator phonons attain elevated steady-state temperatures, T_h^* and T_ph, which depend upon T₀. (c, f) The optimum force sensitivity attained at P_in^(opt)( )T₀ is shown as a function of T₀. Note that an improvement in force sensitivity of greater than an order of magnitude is observed for the nanoscale device.

3 5 10 20

T_h T_ph Effective Temperature [K]

(b)

(c) Tph

*

Th

0.1 1 10 20

10 13 20 30

Ambient Temperature [K]

Optimum Force Sensitivity [ aN/√Hz ]

1n 10n 100n 1µ 10µ

12 20 30 40

Bias Power [W]

T₀=8K T₀=4K T₀=1K T₀=0.1K

Effective Force Sensitivity [ aN/√Hz ]

x x optimal points x

of operation

= x (a)

3 5 10 20

T_h T_ph Effective Temperature [K]

(b)

(c) Tph

*

Th

Tph

*

Th

0.1 1 10 20

10 13 20 30

Ambient Temperature [K]

Optimum Force Sensitivity [ aN/√Hz ]

1n 10n 100n 1µ 10µ

12 20 30 40

Bias Power [W]

T₀=8K T₀=4K T₀=1K T₀=0.1K

Effective Force Sensitivity [ aN/√Hz ]

x x optimal points x

of operation

x=optimal points of operation

= x (a)

2 5 10 20

Effective Temperature [K]

T_h T_ph

1p 10p 100p 1n 10n 50n

0.6 1 2 4

Bias Power [W]

Effective Force Sensitivity [ aN/√Hz ] _T

0=8K T₀=4K T₀=1K T₀=0.1K

x optimal points of operation

= (d)

(e)

(f) Tph

*

Th

0.1 1 10

0.6 1 2 1.5

Optimum Force Sensitivity [ aN/√Hz ]

Ambient Temperature [K]

x x x

2 5 10 20

Effective Temperature [K]

T_h T_ph

1p 10p 100p 1n 10n 50n

0.6 1 2 4

Bias Power [W]

Effective Force Sensitivity [ aN/√Hz ] _T

0=8K T₀=4K T₀=1K T₀=0.1K

x optimal points of operation

x=optimal points of operation

= (d)

(e)

(f) Tph

*

Th

Tph

*

Th

0.1 1 10

0.6 1 2 1.5

Optimum Force Sensitivity [ aN/√Hz ]

Ambient Temperature [K]

x x x

3 5 10 20

T_h T_ph Effective Temperature [K]

(b)

(c) Tph

*

Th

0.1 1 10 20

10 13 20 30

Ambient Temperature [K]

Optimum Force Sensitivity [ aN/√Hz ]

1n 10n 100n 1µ 10µ

12 20 30 40

Bias Power [W]

T₀=8K T₀=4K T₀=1K T₀=0.1K

Effective Force Sensitivity [ aN/√Hz ]

x x optimal points x

of operation

= x (a)

3 5 10 20

T_h T_ph Effective Temperature [K]

(b)

(c) Tph

*

Th

Tph

*

Th

0.1 1 10 20

10 13 20 30

Ambient Temperature [K]

Optimum Force Sensitivity [ aN/√Hz ]

1n 10n 100n 1µ 10µ

12 20 30 40

Bias Power [W]

T₀=8K T₀=4K T₀=1K T₀=0.1K

Effective Force Sensitivity [ aN/√Hz ]

x x optimal points x

of operation

x=optimal points of operation

= x (a)

2 5 10 20

Effective Temperature [K]

T_h T_ph

1p 10p 100p 1n 10n 50n

0.6 1 2 4

Bias Power [W]

Effective Force Sensitivity [ aN/√Hz ] _T

0=8K T₀=4K T₀=1K T₀=0.1K

x optimal points of operation

= (d)

(e)

(f) Tph

*

Th

0.1 1 10

0.6 1 2 1.5

Optimum Force Sensitivity [ aN/√Hz ]

Ambient Temperature [K]

x x x

2 5 10 20

Effective Temperature [K]

T_h T_ph

1p 10p 100p 1n 10n 50n

0.6 1 2 4

Bias Power [W]

Effective Force Sensitivity [ aN/√Hz ] _T

0=8K T₀=4K T₀=1K T₀=0.1K

x optimal points of operation

x=optimal points of operation

= (d)

(e)

(f) Tph

*

Th

Tph

*

Th

0.1 1 10

0.6 1 2 1.5

Optimum Force Sensitivity [ aN/√Hz ]

Ambient Temperature [K]

x x x

within the scope of our present, top-down nanofabrication capabilities, we consider nanocantilever devices with w_leg =100nm, w=300nm, t=30nm, tdoped=7nm, 1_leg = µm that are otherwise identical to the device of Fig. 1. For such a device one obtains an optimal sensitivity S_F =0.6aN/ Hz for T₀<1K. A quality factor of 10 000 was assumed for this device. This seems reasonable based on a survey of work that has gone on in the field.¹⁸ The optimum sensitivity versus ambient temperature is shown in Fig. 5.5.B (f).