- Email: [email protected]
A digital lock-in-technique for pulsed magnetic field experiments
IWllla ELSEVIER
Physica B 211 (1995) 355-359
A digital lock-in-technique for pulsed magnetic field experiments G. Machel*, M. von Ortenberg Fachbereich Physik, AG Magnetotransport, Humboldt-Universitiit zu Berlin, lnvalidenstr. 110, D-lOll5 Berlin, Germany
Abstract We have developed a digital lock-in (DLI) technique with improved signal-to-noise characteristics for magnetotransport experiments in pulsed magnetic fields. The basic idea of the DLI is to digitize both the high-frequency sample signal and the reference during the entire high field shot and to evaluate the data sets afterwards with respect to the p~inciples of the phase sensitive detection by appropriate software. Sophisticated noise elimination procedures, such as FET filtering, can easily be implemented. Each DLI channel is able to detect the in-phase and the quadrature signal simultar~eously. We demonstrate the superior properties of our DLI system by comparing actual measurement results with those itaken with the well-established analog lock-in technique.
1. Introduction
Recent progress in non-destructive pulsed magnetic field generation has pushed the maximum reachable fields into the 70T range [1], thus allowing solid state physics experiments under extreme conditions. Unfortunately the high field generation is inevitably accompanied by noise, especially in the case of magnetotransport experiments where the finite loops of the wiring to the sample and the sample itself lead to pick-up areas in which disturbing signals are induced. The noise mainly originates from the following sources: (i) induced voltage signals as a result of the time-variation of the magnetic field by sweeping and additional small magnetic field fluctuations, (ii) mechanical vibrations of the pick-up system in the acoustic frequency range, (iii) electromagnetic noise due to the high voltage switching process, and (iv) external noise due to insufficient shielding. Whereas different kinds of compensation techniques are able to * Corresponding author.
suppress only the low frequency noise, more sqphisticated methods have to be applied for the removal Of the high frequency components. One of the most effective noise-suppressing technique is lock-in or phase sensitive detection. However, its application to pulsed magnetic field experimen!s is not as easy as in dc fields. Since pulse durations are [ypically of the order of 5-50 ms, both high modulation irequencies and high signal bandwidths (i.e. low outpul time constants) are required for the detection of shart structures in the magnetoresponse. Commercially avaik ble lock-in amplifiers usually lack the combination of these two features so that a suitable analog lock-in syste na has been developed using a modular arrangement of diJ ferent electronic components [2]. This system is capable of working with modulation frequencies up to 1 MHz to ;ether with output time constants less than 10 las. Ho~vever, this system suffers from a limited dynamic range atthe input, the need for accurate phase adjustment before each measurement and poor flexibility in handli~ag the collected data. Furthermore, it would require !a large experimental set-up for two-channel measurements with
0921-4526/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 9 2 1 - 4 5 2 6 ( 9 4 ) 0 1 0 6 2 - 5
356
G. Machel, M. yon Ortenberg/Physica B 211 (1995) 355-359
vector-lock-in capabilities as is needed for the simultaneous measurement of magnet•resistance and the Hall voltage. These problems can be overcome by use of a digital lock-in (DLI) amplifier. The recent advances in the development of digitizing oscilloscopes and transient recorders with high sampling rates, high vertical bit-resolution and large memory made it possible to project a DLI application to pulsed magnetic field experiments. The fundamental concept of a DLI is to digitize both the modulated sample signal and the reference signal and to process the digitized data sets with respect to the principles of the phase sensitive detection. The DLI techniques published so far in the literature can be divided into two groups: (1) hardwarebased versions performing digital data processing via some sophisticated electronics [3, 4], and (2) softwarebased versions that transmit the digitized data directly to a computer for numerical analysis [5, 6-1. Despite the large amount of different DLI techniques described in the literature none of them is designed in a way to match the special conditions of a short-time experiment. Therefore, the version that we describe in this paper addresses exactly this problem. Since high speed digitizers are standard equipment in a pulsed magnetic field facility the main task was only to write the appropriate software. The principle of operation and the performance of our system is described in the following sections.
2. The DLI
/".• / e~
• -"
.-'o.
/
"i...
sin (oat) + b cos (oat),
(1)
f
•.H
~'°I.......~ "
Time
Fig. 1. Time interval with synchronously digitized versions of signal and reference (sampling points are marked by filled circles). In order to determine the amplitude and phase of each data set a least-squares-fit procedure is performed (indicated by dotted curves). with oa = 2nfmoa (fmoa is the modulation frequency) and a, b as fit parameters. The least-squares-fit procedure can be solved analytically. The general result is b=
a =
F(t) = a
/ qi
e~
technique
Generally, the basic principle of lock-in detection is to shift the signal of interest into a frequency range with lower noise background (i.e. modulate it with a certain frequency), remove any undesired frequency components by band-pass filtering and demodulate the signal to its original frequency domain by mixing or multiplying it with a phase-locked reference signal. Finally, a low-pass filter at the output removes all frequency components in higher bands, thus leaving the clean signal of interest. The DLI analysis, however, although yielding the same results works in a quite different way. The crucial point of a DLI system is the numerical analysis of the digitized data sets with respect to amplitude and phase difference. For that reason, before continuing with the explanation of our complete system, this problem should be illustrated more clearly. In Fig. 1 the digitized versions of signal and reference are shown within a certain time interval. Since both signals are sinusoidal consisting of n samples each a least-squares-fit procedure to a sine wave can be applied. Hence, the fit function is of the form
"'o.-"
[y cos (tot)] [sin 2 (oat)] -- [y sin (tot)] [sin (oat)cos (oat)] [sin 2 (oat)] [cos 2 (oat)] - [sin (~ot)cos (oat)] (2) [y sin (oat)] - b [sin (oat)cos (cot)] [sin 2 (o90]
(3)
Here, the square brackets [---] denote a sum of the corresponding terms over all samples in the time interval, for example [y sin (cot)] = ~ Yisin (oati).
(4)
i=1
The Yi are the sampled values at the time ti. In case of an integer number of periods within the time interval Eqs. (2) and (3) can be simplified to 2 a = - [y sin (oat)], n
2 b = - [ y cos (oat)].
(5)
n
Consequently, the amplitude and the relative phase of the corresponding data set are determined by A = x / ~ + b 2,
4~ = arctan ~ .
(6)
Applying this technique to both data sets yields complete information about the amplitudes and the phase difference O = 4~sig- 4~rerbetween signal and reference within the time interval.
G. Machel, M. yon Ortenberg./ Physica B 211 (1995) 355-359
acquisition
"i ~ o n of measurement~ quency
reference. After that the entire data sets are scanned by applying our DLI technique to successive ismall time intervals of the length r, which determine the output bandwidth of the DLI. Every time interval is attached to exactly one output value of the DLI. The oulput values are strictly bound to the time pattern of tl~e sampled data. This means that successive time inter~als always overlap and contain at least two modulation l~eriods. The in-phase and quadrature components are calculated with
L
-1 b~d.pa~s filter
of initial phase
[,vu [ :=oll
It +°+"
357
I
1.o low-pass filter
hin ph(ti) = Z(ti)COS(~)(tl) -- Oo),
(7)
Aqu,d(ti) = A(ti)sin(O(tg) -- Oo),
(8)
where A(t~) and tg(tg) are the results of the DLI analysis within the corresponding time interval. Fihally, after a complete sweep through the data sets the ogtput waveforms can optionally be smoothed by a FFT low-pass filter. The result is then displayed and cani be further processed. All system parameters like the total numb+r of output data points, the output time constant z and the cut-off frequencies of the FFT filters are controlled c~nveniently via the software. Since the raw sampled data Iare usually saved the calculation can be repeated at any ~ime on the same data set in order to optimize the parameters for the qualitatively best output.
3. Performance and application l display result Fig. 2. Block diagram of the computer program performing the DLI analysis.
The implementation of our software is represented schematically in Fig. 2. First, the data are digitized with the highest possible sampling rate of the transient recorder. Since a high magnetic field shot is an experiment on a short time scale the reference and the sample signal have to be sampled completely over the entire measurement before the DLI evaluation can take place. The next step is the determination of the modulation frequency fmod from the reference data set. This can be done either by FFT or by a simple period-counting routine. It is assumed that the frequency is a constant value throughout the measurement. After an optional FFT band-pass filter the above DLI process is carried out. Since every pulsed magnetic field shot consists also of a pretrigger measurement immediately before the beginning of the shot this part of the data sets can be used for the determination of the initial phase shift O o between signal and
The DLI system that we have developed is based on the completely computer-controlled F~ST/Comtec TRA/PC transient recorder. This digitize~ is able to sample at rates up to 2.5 MHz in four-channel mode with 12-bit resolution. The large memory of 256 k samples per channel allows sufficient sampling times. ]'he experimental set-up is shown in Fig. 3. Whereas ~hannel 1 is occupied by the magnetic field measurement in our experiments the remaining three channels areJfree for the DLI system. A Stanford Research Systems DS345 function generator is used as a signal source. It Can produce various waveforms from DC to 30 MHz with a frequency resolution of 1 I.tHz. The response signals oJthe experiment can optionally be conditioned by a tanford Research System SR560 low-noise preamplifier and a Krohn-Hite 3905B multi-channel filter. T lese devices have a limiting 3-db bandwidth of 1 and 3 1~[Hz, respectively. For the control of the transient reco :der and the DLI analysis an IBM-compatible computer equipped with a 66-MHz 486DX2 processor is use~t. The DLI processing time strongly depends on the 9ctual parameter settings; for a typical 10 ms measurement with 25000 data points per channel a time of 10-20s is needed.
358
G. Machel, M. yon Ortenberg/Physica B 211 (1995) 355-359
100
Transient Recorder
digitallock-in • analoglock-in •
Computer
90
Chl Ch2 Ch3 Ch4 "1:3
8 o ~o
80 Amplifier Filter
"5 70 z ~f
Experiment
-~ 6o
I Amplifier Filter
oll
50 40
10 j
2
5
10 2
2
5
10 3
Frequency (kI-Iz) Fig. 3. Schematical representation of the experimental set-up used for DLI measurements.
The limiting signal-to-noise ratio (SNR) of the DLI system was checked by feeding a clean 0.2 V sine wave into both the reference and the signal input at full scale. The output bandwidth fo, t was adjusted in such a way to keep the relation to the modulation frequency at a constant value fmod/fom = 5. Since the SNR of a DLI amplifier only depends on phase fluctuations between reference and signal [5], the RMS phase noise was measured within a time interval of 10 ms. In order to remove any random effects an average value of at least 10 measurements was calculated. The SNR is then given by S N R = 20log (~-~O),
Fig. 4. The signal-to-noise-ratio versus frequency for the DLI system (marked by filled circles) compared to the results for the analog lock-in-system described in Ref. [2] (marked by filled triangles). The lines are only a guide to the eye.
(c)
HgSe:Fe T=4.2K
O f-
(9)
(a)
t~
with 8 0 being the RMS phase noise. The frequency dependence of the SNR is shown in Fig. 4. F o r comparison the SNR of our analog lock-in set-up is included. At low frequencies a distinct enhancement of 20 dB can be achieved with the DLI system. With increasing frequency the SNR decreases and becomes comparable to the values of the analog lock-in at the high-frequency end. This can be explained by the reduced-quality digitizing process, since the maximum sampling frequency of the transient recorder is limited to 2.5 MHz. In o r d e r to demonstrate the efficiency of our DLI system we have applied it to magnetotransport experiments with our 50 k J, 2.5 kV capacitor bank. Fig. 5(a) shows a typical result of such measurements on MBEgrown samples of the semimagnetic semiconductor HgSe: Fe [7]. The high frequency quantum oscillations are clearly resolved although the pulse time of the magnetic field was only about 5 ms. This result was obtained
~
0
2
~
4
~
6
8 10 12 14 Magnetic Field (Tesla)
16
18
20
Fig. 5. Magnet0transport measurements on HgSe:Fe using (a) the DLI system in pulsed magnetic fields, (b) the analog lock-in system [2] in pulsed magnetic fields, and (c) conventional low frequency lock-in technique in quasi-stationary fields of a superconducting magnet.
with a modulation frequency of fmo~ = 500 kHz and a DLI output bandwidth of 100 kHz. The same sample was also measured: (1) with our analog lock-in set-up in pulsed magnetic fields (Fig. 5(b), identical settings as the DLI system), and (2) in the quasi-DC field of a 12-T
G. Machel, M. yon Ortenberg./Physica B 211 (1995) 355 359
359
positions. The DLI analysis, on the other hand~ can easily be programmed in a non-causal or symmetrical manner, therefore eliminating any hysteresis right from the beginning without loss in processing time.
vector
4. Conclusion
HgSe:Fe T=4.2K
e-,
0
5
10 15 20 Magnetic Field (Tesla)
25
Fig. 6. Simultaneous magnetotransport measurement of the inphase, quadrature and vector component on a HgSe: Fe sample showing a complex impedance behaviour.
superconducting magnet using a conventional lock-in amplifier (Fig. 5(c)). The difference in SNR between the three measurements is negligible. It follows from this that our lock-in techniques offer the same sensitivity and SNR, which has so far been exclusive to DC magnetic fields. However, our DLI system has some additional advantages that make the application to pulsed magnetic field experiments attractive. Each input channel of the DLI has complete vectorlock-in capabilities that can be helpful for the evaluation of complex sample signals. Fig. 6 shows a signal whose in-phase component is distorted by a strong phase shift in the high magnetic field range due to a complex sample impedance. Calculating both the in-phase and the quadrature component from the same data set, however, reveals valuable information for the interpretation of the measurement. As shown in Fig. 6, the calculation of the vector signal from the in-phase and quadrature components reproduces the true and undistorted magnetotransport signal. A problem that arises when using an analog lock-intechnique in combination with pulsed magnetic fields is the appearance of hysteresis between the up-sweep and down-sweep measurements. This is caused by the lowpass filter at the output of the lock-in amplifier that is used to suppress the high frequency noise. Such a filter works in a strictly causal or asymmetrical way since it takes into account only the preceding input signal to determine its output, thus producing problems in the exact determination of resonance or quantum oscillation
We have shown that high sensitive DLI techniques are also applicable to short-time pulsed magnetic field experiments. The central features of our DLI sysptem can be summarized as follows: (i) two independentl~ configurable input channels, each channel having v~ctor-lockin capabilities, (ii) 1.2 MHz maximum modulation frequency, 250kHz maximum output bandwidth, (iii) flexible data handling that allows the implementation of specific digital filter techniques, (iv) remqval of any hysteresis effects between up- and down-sweep measurements, (v) no need for any time-consuming piCk-up voltage compensation or phase adjustments. FUrthermore, it is possible in principle to measure also at higher harmonics of the modulation frequency by ~imply performing the above analysis with an integer multiple of fmod. Of course, this DLI technique is not limited to magnetotransport experiments; applicatiot~s to acsusceptibility or magneto-optic experiments axe conceivable. The experimental realization of such measurements is in progress.
Acknowledgements One of the authors (G.M.) would like to thank the Deutsche Forschungsgemeinschaft for financlal support.
References I'1] L. van Bockstal, G. Heremans and F. Herlach, Meas. Sci. Technol. 2 (1991) 1159. I-2] M. von Ortenberg, W. Staguhn, F. B~bel, Si Takeyama, T. Sakakibara and N. Miura, J. Phys. E 22 (1~)89) 359. 13] A.I. Vistnes, D.I. Wormald, S. lsachsen and D.!Schmalbein, Rev. Sci. Instrum. 55(4) (1984) 527. [4] E. Iacopini, B. Smith, G. Stefanini and S. CarusOtto, J. Phys. E 16 (1983) 844. [5] P.K. Dixon and L. Wu, Rev. Sci. lnstrum. 60!(1989) 3329. 1"6] S.Y. Zhang and A.V. Soukas, Meas. Sci. Technol. 2 (1991) 13. [7] D. Schikora, T. Widmer, M. Barczewski, G. Machel, M. yon Ortenberg, W. Jantsch and K. Lischka, in: Prt)c. 22nd Int. Conf. on the Physics of Semiconductors, VancOuver (1994), to be published.
Physica B 211 (1995) 355-359
A digital lock-in-technique for pulsed magnetic field experiments G. Machel*, M. von Ortenberg Fachbereich Physik, AG Magnetotransport, Humboldt-Universitiit zu Berlin, lnvalidenstr. 110, D-lOll5 Berlin, Germany
Abstract We have developed a digital lock-in (DLI) technique with improved signal-to-noise characteristics for magnetotransport experiments in pulsed magnetic fields. The basic idea of the DLI is to digitize both the high-frequency sample signal and the reference during the entire high field shot and to evaluate the data sets afterwards with respect to the p~inciples of the phase sensitive detection by appropriate software. Sophisticated noise elimination procedures, such as FET filtering, can easily be implemented. Each DLI channel is able to detect the in-phase and the quadrature signal simultar~eously. We demonstrate the superior properties of our DLI system by comparing actual measurement results with those itaken with the well-established analog lock-in technique.
1. Introduction
Recent progress in non-destructive pulsed magnetic field generation has pushed the maximum reachable fields into the 70T range [1], thus allowing solid state physics experiments under extreme conditions. Unfortunately the high field generation is inevitably accompanied by noise, especially in the case of magnetotransport experiments where the finite loops of the wiring to the sample and the sample itself lead to pick-up areas in which disturbing signals are induced. The noise mainly originates from the following sources: (i) induced voltage signals as a result of the time-variation of the magnetic field by sweeping and additional small magnetic field fluctuations, (ii) mechanical vibrations of the pick-up system in the acoustic frequency range, (iii) electromagnetic noise due to the high voltage switching process, and (iv) external noise due to insufficient shielding. Whereas different kinds of compensation techniques are able to * Corresponding author.
suppress only the low frequency noise, more sqphisticated methods have to be applied for the removal Of the high frequency components. One of the most effective noise-suppressing technique is lock-in or phase sensitive detection. However, its application to pulsed magnetic field experimen!s is not as easy as in dc fields. Since pulse durations are [ypically of the order of 5-50 ms, both high modulation irequencies and high signal bandwidths (i.e. low outpul time constants) are required for the detection of shart structures in the magnetoresponse. Commercially avaik ble lock-in amplifiers usually lack the combination of these two features so that a suitable analog lock-in syste na has been developed using a modular arrangement of diJ ferent electronic components [2]. This system is capable of working with modulation frequencies up to 1 MHz to ;ether with output time constants less than 10 las. Ho~vever, this system suffers from a limited dynamic range atthe input, the need for accurate phase adjustment before each measurement and poor flexibility in handli~ag the collected data. Furthermore, it would require !a large experimental set-up for two-channel measurements with
0921-4526/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 9 2 1 - 4 5 2 6 ( 9 4 ) 0 1 0 6 2 - 5
356
G. Machel, M. yon Ortenberg/Physica B 211 (1995) 355-359
vector-lock-in capabilities as is needed for the simultaneous measurement of magnet•resistance and the Hall voltage. These problems can be overcome by use of a digital lock-in (DLI) amplifier. The recent advances in the development of digitizing oscilloscopes and transient recorders with high sampling rates, high vertical bit-resolution and large memory made it possible to project a DLI application to pulsed magnetic field experiments. The fundamental concept of a DLI is to digitize both the modulated sample signal and the reference signal and to process the digitized data sets with respect to the principles of the phase sensitive detection. The DLI techniques published so far in the literature can be divided into two groups: (1) hardwarebased versions performing digital data processing via some sophisticated electronics [3, 4], and (2) softwarebased versions that transmit the digitized data directly to a computer for numerical analysis [5, 6-1. Despite the large amount of different DLI techniques described in the literature none of them is designed in a way to match the special conditions of a short-time experiment. Therefore, the version that we describe in this paper addresses exactly this problem. Since high speed digitizers are standard equipment in a pulsed magnetic field facility the main task was only to write the appropriate software. The principle of operation and the performance of our system is described in the following sections.
2. The DLI
/".• / e~
• -"
.-'o.
/
"i...
sin (oat) + b cos (oat),
(1)
f
•.H
~'°I.......~ "
Time
Fig. 1. Time interval with synchronously digitized versions of signal and reference (sampling points are marked by filled circles). In order to determine the amplitude and phase of each data set a least-squares-fit procedure is performed (indicated by dotted curves). with oa = 2nfmoa (fmoa is the modulation frequency) and a, b as fit parameters. The least-squares-fit procedure can be solved analytically. The general result is b=
a =
F(t) = a
/ qi
e~
technique
Generally, the basic principle of lock-in detection is to shift the signal of interest into a frequency range with lower noise background (i.e. modulate it with a certain frequency), remove any undesired frequency components by band-pass filtering and demodulate the signal to its original frequency domain by mixing or multiplying it with a phase-locked reference signal. Finally, a low-pass filter at the output removes all frequency components in higher bands, thus leaving the clean signal of interest. The DLI analysis, however, although yielding the same results works in a quite different way. The crucial point of a DLI system is the numerical analysis of the digitized data sets with respect to amplitude and phase difference. For that reason, before continuing with the explanation of our complete system, this problem should be illustrated more clearly. In Fig. 1 the digitized versions of signal and reference are shown within a certain time interval. Since both signals are sinusoidal consisting of n samples each a least-squares-fit procedure to a sine wave can be applied. Hence, the fit function is of the form
"'o.-"
[y cos (tot)] [sin 2 (oat)] -- [y sin (tot)] [sin (oat)cos (oat)] [sin 2 (oat)] [cos 2 (oat)] - [sin (~ot)cos (oat)] (2) [y sin (oat)] - b [sin (oat)cos (cot)] [sin 2 (o90]
(3)
Here, the square brackets [---] denote a sum of the corresponding terms over all samples in the time interval, for example [y sin (cot)] = ~ Yisin (oati).
(4)
i=1
The Yi are the sampled values at the time ti. In case of an integer number of periods within the time interval Eqs. (2) and (3) can be simplified to 2 a = - [y sin (oat)], n
2 b = - [ y cos (oat)].
(5)
n
Consequently, the amplitude and the relative phase of the corresponding data set are determined by A = x / ~ + b 2,
4~ = arctan ~ .
(6)
Applying this technique to both data sets yields complete information about the amplitudes and the phase difference O = 4~sig- 4~rerbetween signal and reference within the time interval.
G. Machel, M. yon Ortenberg./ Physica B 211 (1995) 355-359
acquisition
"i ~ o n of measurement~ quency
reference. After that the entire data sets are scanned by applying our DLI technique to successive ismall time intervals of the length r, which determine the output bandwidth of the DLI. Every time interval is attached to exactly one output value of the DLI. The oulput values are strictly bound to the time pattern of tl~e sampled data. This means that successive time inter~als always overlap and contain at least two modulation l~eriods. The in-phase and quadrature components are calculated with
L
-1 b~d.pa~s filter
of initial phase
[,vu [ :=oll
It +°+"
357
I
1.o low-pass filter
hin ph(ti) = Z(ti)COS(~)(tl) -- Oo),
(7)
Aqu,d(ti) = A(ti)sin(O(tg) -- Oo),
(8)
where A(t~) and tg(tg) are the results of the DLI analysis within the corresponding time interval. Fihally, after a complete sweep through the data sets the ogtput waveforms can optionally be smoothed by a FFT low-pass filter. The result is then displayed and cani be further processed. All system parameters like the total numb+r of output data points, the output time constant z and the cut-off frequencies of the FFT filters are controlled c~nveniently via the software. Since the raw sampled data Iare usually saved the calculation can be repeated at any ~ime on the same data set in order to optimize the parameters for the qualitatively best output.
3. Performance and application l display result Fig. 2. Block diagram of the computer program performing the DLI analysis.
The implementation of our software is represented schematically in Fig. 2. First, the data are digitized with the highest possible sampling rate of the transient recorder. Since a high magnetic field shot is an experiment on a short time scale the reference and the sample signal have to be sampled completely over the entire measurement before the DLI evaluation can take place. The next step is the determination of the modulation frequency fmod from the reference data set. This can be done either by FFT or by a simple period-counting routine. It is assumed that the frequency is a constant value throughout the measurement. After an optional FFT band-pass filter the above DLI process is carried out. Since every pulsed magnetic field shot consists also of a pretrigger measurement immediately before the beginning of the shot this part of the data sets can be used for the determination of the initial phase shift O o between signal and
The DLI system that we have developed is based on the completely computer-controlled F~ST/Comtec TRA/PC transient recorder. This digitize~ is able to sample at rates up to 2.5 MHz in four-channel mode with 12-bit resolution. The large memory of 256 k samples per channel allows sufficient sampling times. ]'he experimental set-up is shown in Fig. 3. Whereas ~hannel 1 is occupied by the magnetic field measurement in our experiments the remaining three channels areJfree for the DLI system. A Stanford Research Systems DS345 function generator is used as a signal source. It Can produce various waveforms from DC to 30 MHz with a frequency resolution of 1 I.tHz. The response signals oJthe experiment can optionally be conditioned by a tanford Research System SR560 low-noise preamplifier and a Krohn-Hite 3905B multi-channel filter. T lese devices have a limiting 3-db bandwidth of 1 and 3 1~[Hz, respectively. For the control of the transient reco :der and the DLI analysis an IBM-compatible computer equipped with a 66-MHz 486DX2 processor is use~t. The DLI processing time strongly depends on the 9ctual parameter settings; for a typical 10 ms measurement with 25000 data points per channel a time of 10-20s is needed.
358
G. Machel, M. yon Ortenberg/Physica B 211 (1995) 355-359
100
Transient Recorder
digitallock-in • analoglock-in •
Computer
90
Chl Ch2 Ch3 Ch4 "1:3
8 o ~o
80 Amplifier Filter
"5 70 z ~f
Experiment
-~ 6o
I Amplifier Filter
oll
50 40
10 j
2
5
10 2
2
5
10 3
Frequency (kI-Iz) Fig. 3. Schematical representation of the experimental set-up used for DLI measurements.
The limiting signal-to-noise ratio (SNR) of the DLI system was checked by feeding a clean 0.2 V sine wave into both the reference and the signal input at full scale. The output bandwidth fo, t was adjusted in such a way to keep the relation to the modulation frequency at a constant value fmod/fom = 5. Since the SNR of a DLI amplifier only depends on phase fluctuations between reference and signal [5], the RMS phase noise was measured within a time interval of 10 ms. In order to remove any random effects an average value of at least 10 measurements was calculated. The SNR is then given by S N R = 20log (~-~O),
Fig. 4. The signal-to-noise-ratio versus frequency for the DLI system (marked by filled circles) compared to the results for the analog lock-in-system described in Ref. [2] (marked by filled triangles). The lines are only a guide to the eye.
(c)
HgSe:Fe T=4.2K
O f-
(9)
(a)
t~
with 8 0 being the RMS phase noise. The frequency dependence of the SNR is shown in Fig. 4. F o r comparison the SNR of our analog lock-in set-up is included. At low frequencies a distinct enhancement of 20 dB can be achieved with the DLI system. With increasing frequency the SNR decreases and becomes comparable to the values of the analog lock-in at the high-frequency end. This can be explained by the reduced-quality digitizing process, since the maximum sampling frequency of the transient recorder is limited to 2.5 MHz. In o r d e r to demonstrate the efficiency of our DLI system we have applied it to magnetotransport experiments with our 50 k J, 2.5 kV capacitor bank. Fig. 5(a) shows a typical result of such measurements on MBEgrown samples of the semimagnetic semiconductor HgSe: Fe [7]. The high frequency quantum oscillations are clearly resolved although the pulse time of the magnetic field was only about 5 ms. This result was obtained
~
0
2
~
4
~
6
8 10 12 14 Magnetic Field (Tesla)
16
18
20
Fig. 5. Magnet0transport measurements on HgSe:Fe using (a) the DLI system in pulsed magnetic fields, (b) the analog lock-in system [2] in pulsed magnetic fields, and (c) conventional low frequency lock-in technique in quasi-stationary fields of a superconducting magnet.
with a modulation frequency of fmo~ = 500 kHz and a DLI output bandwidth of 100 kHz. The same sample was also measured: (1) with our analog lock-in set-up in pulsed magnetic fields (Fig. 5(b), identical settings as the DLI system), and (2) in the quasi-DC field of a 12-T
G. Machel, M. yon Ortenberg./Physica B 211 (1995) 355 359
359
positions. The DLI analysis, on the other hand~ can easily be programmed in a non-causal or symmetrical manner, therefore eliminating any hysteresis right from the beginning without loss in processing time.
vector
4. Conclusion
HgSe:Fe T=4.2K
e-,
0
5
10 15 20 Magnetic Field (Tesla)
25
Fig. 6. Simultaneous magnetotransport measurement of the inphase, quadrature and vector component on a HgSe: Fe sample showing a complex impedance behaviour.
superconducting magnet using a conventional lock-in amplifier (Fig. 5(c)). The difference in SNR between the three measurements is negligible. It follows from this that our lock-in techniques offer the same sensitivity and SNR, which has so far been exclusive to DC magnetic fields. However, our DLI system has some additional advantages that make the application to pulsed magnetic field experiments attractive. Each input channel of the DLI has complete vectorlock-in capabilities that can be helpful for the evaluation of complex sample signals. Fig. 6 shows a signal whose in-phase component is distorted by a strong phase shift in the high magnetic field range due to a complex sample impedance. Calculating both the in-phase and the quadrature component from the same data set, however, reveals valuable information for the interpretation of the measurement. As shown in Fig. 6, the calculation of the vector signal from the in-phase and quadrature components reproduces the true and undistorted magnetotransport signal. A problem that arises when using an analog lock-intechnique in combination with pulsed magnetic fields is the appearance of hysteresis between the up-sweep and down-sweep measurements. This is caused by the lowpass filter at the output of the lock-in amplifier that is used to suppress the high frequency noise. Such a filter works in a strictly causal or asymmetrical way since it takes into account only the preceding input signal to determine its output, thus producing problems in the exact determination of resonance or quantum oscillation
We have shown that high sensitive DLI techniques are also applicable to short-time pulsed magnetic field experiments. The central features of our DLI sysptem can be summarized as follows: (i) two independentl~ configurable input channels, each channel having v~ctor-lockin capabilities, (ii) 1.2 MHz maximum modulation frequency, 250kHz maximum output bandwidth, (iii) flexible data handling that allows the implementation of specific digital filter techniques, (iv) remqval of any hysteresis effects between up- and down-sweep measurements, (v) no need for any time-consuming piCk-up voltage compensation or phase adjustments. FUrthermore, it is possible in principle to measure also at higher harmonics of the modulation frequency by ~imply performing the above analysis with an integer multiple of fmod. Of course, this DLI technique is not limited to magnetotransport experiments; applicatiot~s to acsusceptibility or magneto-optic experiments axe conceivable. The experimental realization of such measurements is in progress.
Acknowledgements One of the authors (G.M.) would like to thank the Deutsche Forschungsgemeinschaft for financlal support.
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